Worked example: Using the reaction quotient to predict a pressure change

WEBVTT
Kind: captions
Language: en

00:00:00.516 --> 00:00:02.092
- [Voiceover] A one litre reaction vessel

00:00:02.092 --> 00:00:05.563
contains 1.2 moles of carbon monoxide,

00:00:05.563 --> 00:00:07.732
1.5 moles of hydrogen gas,

00:00:07.732 --> 00:00:10.688
and 2.0 moles of methanol gas.

00:00:10.688 --> 00:00:13.138
How will the total pressure change

00:00:13.138 --> 00:00:14.866
as the system approaches equilibrium

00:00:14.866 --> 00:00:17.038
at constant temperature?

00:00:17.038 --> 00:00:20.177
So, our carbon monoxide is
reacting with our hydrogen

00:00:20.177 --> 00:00:21.597
in a one to two ratio

00:00:21.597 --> 00:00:23.965
to give us methanol.

00:00:23.965 --> 00:00:26.367
And this reaction is reversible.

00:00:26.367 --> 00:00:29.669
We also know the equilibrium
constant for this reaction

00:00:29.669 --> 00:00:32.336
is 14.5 at some temperature.

00:00:33.452 --> 00:00:37.471
And we know that the
temperature is staying constant.

00:00:37.471 --> 00:00:42.419
So, we are going to break this
problem up into two parts.

00:00:42.419 --> 00:00:44.628
In part one, we're
gonna try to figure out,

00:00:44.628 --> 00:00:46.231
using the reaction quotient,

00:00:46.231 --> 00:00:50.532
whether our system is
at equilibrium or not.

00:00:50.532 --> 00:00:52.829
So, for this reaction,

00:00:52.829 --> 00:00:55.408
our reaction quotient 'Q'

00:00:55.408 --> 00:00:57.410
is the product concentration.

00:00:57.410 --> 00:01:01.327
"C-H-3-O-H" for methanol.

00:01:02.217 --> 00:01:05.693
Divided by the concentration
of our hydrogen gas;

00:01:05.693 --> 00:01:06.956
to the second power,

00:01:06.956 --> 00:01:10.372
because of that
stoichiometrical efficient.

00:01:10.372 --> 00:01:11.984
And then also in the denominator,

00:01:11.984 --> 00:01:15.484
we have our carbon monoxide concentration.

00:01:16.930 --> 00:01:18.144
We can calculate 'Q'

00:01:18.144 --> 00:01:21.000
by plugging in the
concentrations of these,

00:01:21.000 --> 00:01:23.800
at this particular moment in time.

00:01:23.800 --> 00:01:25.884
And we can calculate the concentrations

00:01:25.884 --> 00:01:29.490
using the volume of the vessel,

00:01:29.490 --> 00:01:30.896
which is one liter

00:01:30.896 --> 00:01:33.285
and the mole quantities.

00:01:33.285 --> 00:01:35.986
We know that concentration

00:01:35.986 --> 00:01:38.678
is just moles divided by volume.

00:01:38.678 --> 00:01:41.249
And since we're dividing
everything by one,

00:01:41.249 --> 00:01:42.782
the initial concentrations

00:01:42.782 --> 00:01:46.139
will be the same as the number of moles.

00:01:46.139 --> 00:01:47.368
So, if we write that out,

00:01:47.368 --> 00:01:49.731
for carbon monoxide,

00:01:49.731 --> 00:01:53.322
the initial concentration is 1.2 molar.

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For hydrogen, it's 1.5 molar.

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And for methanol,

00:01:58.723 --> 00:02:01.640
it is 2.0 molar.

00:02:02.720 --> 00:02:05.237
So now we can plug these
concentrations into

00:02:05.237 --> 00:02:08.530
our expression for 'Q'.

00:02:08.530 --> 00:02:09.875
And then we get,

00:02:09.875 --> 00:02:12.270
in our numerator, 2.0.

00:02:12.270 --> 00:02:16.631
And our denominator, is 1.5

00:02:16.631 --> 00:02:19.156
squared times 1.2.

00:02:19.156 --> 00:02:21.299
So if we plug this all
into our calculators,

00:02:21.299 --> 00:02:23.563
what I got is at our 'Q',

00:02:23.563 --> 00:02:25.438
for this particular moment in time,

00:02:25.438 --> 00:02:29.334
with these concentrations is 0.74.

00:02:29.334 --> 00:02:30.518
So this tells us,

00:02:30.518 --> 00:02:32.825
first of all, we know that 'Q'

00:02:32.825 --> 00:02:35.492
is not equal to 'KC'.

00:02:37.209 --> 00:02:40.179
So that means we are not at equilibrium.

00:02:40.179 --> 00:02:43.346
"Not at equilibrium".

00:02:46.414 --> 00:02:48.139
Which means that our pressures

00:02:48.139 --> 00:02:50.029
are indeed going to change

00:02:50.029 --> 00:02:52.928
because the system is going
to try to reach equilibrium.

00:02:52.928 --> 00:02:54.279
The second thing we can do,

00:02:54.279 --> 00:02:55.908
using the reaction quotient to,

00:02:55.908 --> 00:03:00.036
is figure out how the
concentrations will change.

00:03:00.036 --> 00:03:03.161
Now that we know our
reaction quotient 'Q',

00:03:03.161 --> 00:03:04.651
we know that

00:03:04.651 --> 00:03:06.295
our reaction quotient 'QC'

00:03:06.295 --> 00:03:08.295
is less than 'K'.

00:03:09.434 --> 00:03:13.007
We can visualize this on a number line.

00:03:13.007 --> 00:03:16.283
So if we look at all
possible values of 'Q',

00:03:16.283 --> 00:03:18.000
we know that when 'Q' is zero,

00:03:18.000 --> 00:03:20.583
we have all reactants.

00:03:22.805 --> 00:03:26.638
When 'Q' is infinitely large, we have

00:03:27.489 --> 00:03:31.489
all reactants, we have all products.

00:03:34.618 --> 00:03:38.785
And then we have all of the
possible values in between.

00:03:42.275 --> 00:03:43.966
What we're really worried about here,

00:03:43.966 --> 00:03:47.109
is just looking at the
relative value of 'Q' and 'K'.

00:03:47.109 --> 00:03:51.158
And seeing how reaction
concentrations are going to shift.

00:03:51.158 --> 00:03:53.642
So 'Q', we can put on our number line,

00:03:53.642 --> 00:03:55.725
is somewhere around here.

00:03:56.599 --> 00:04:00.399
And 'K' is 14.5, so
we'll say it's somewhere

00:04:00.399 --> 00:04:03.626
around here.

00:04:03.626 --> 00:04:06.876
So this is our 'Q' and this is our 'K'.

00:04:08.271 --> 00:04:12.916
We can see that 'Q' is less
than 'K' on our number line.

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So what's gonna happen is,

00:04:14.654 --> 00:04:16.835
in order to reach equilibrium,

00:04:16.835 --> 00:04:19.888
our concentrations are
going to shift to the right

00:04:19.888 --> 00:04:21.648
to get 'Q' closer to 'K'.

00:04:21.648 --> 00:04:24.906
Which means, what's going to happen is,

00:04:24.906 --> 00:04:25.906
the reaction

00:04:27.448 --> 00:04:29.615
is going to shift to favor

00:04:30.692 --> 00:04:32.442
making more products.

00:04:36.891 --> 00:04:40.223
So if we look back at
the balance reaction,

00:04:40.223 --> 00:04:42.139
what's going to happen here is,

00:04:42.139 --> 00:04:45.639
it's going to shift to favor the products.

00:04:46.490 --> 00:04:49.461
So I'm making that top arrow
a little bit more bold.

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And to tie this into

00:04:50.825 --> 00:04:52.464
what the problem wants to know,

00:04:52.464 --> 00:04:54.761
we can figure out how the shift

00:04:54.761 --> 00:04:58.444
to make more products will
affect the total pressure.

00:04:58.444 --> 00:04:59.569
So the total pressure

00:04:59.569 --> 00:05:02.647
for a system that has a
bunch of gas molecules in it,

00:05:02.647 --> 00:05:03.647
we know that

00:05:04.742 --> 00:05:07.075
total pressure is related to

00:05:08.911 --> 00:05:12.276
the moles of gas in the system.

00:05:12.276 --> 00:05:15.872
So since we're shifting
to favor the reactants.

00:05:15.872 --> 00:05:18.339
And on the reactants side, we have,

00:05:18.339 --> 00:05:20.252
we're making one mole of gas.

00:05:20.252 --> 00:05:23.928
And we're starting with
three moles of reactant gas.

00:05:23.928 --> 00:05:27.457
We're favoring the side that
has fewer gas molecules.

00:05:27.457 --> 00:05:31.519
So that means as we shift
to favor the products,

00:05:31.519 --> 00:05:34.356
we're going to reduce number
of gas molecules in the system,

00:05:34.356 --> 00:05:36.936
and that's gonna reduce our 'P' total.

00:05:36.936 --> 00:05:38.213
So the answer is,

00:05:38.213 --> 00:05:41.880
that P total is going to decrease

00:05:43.691 --> 00:05:47.365
as our reaction approaches equilibrium

00:05:47.365 --> 00:05:48.962
and that is because

00:05:48.962 --> 00:05:52.545
our reaction quotient
'Q' is less than 'K'.


Sign convention for passive components

WEBVTT
Kind: captions
Language: en

00:00:00.868 --> 00:00:01.769
- [Voiceover] Today we're
gonna talk about the

00:00:01.769 --> 00:00:03.814
sign convention for passive components.

00:00:03.814 --> 00:00:07.119
It's a big mouthful, but
it's a fairly simple idea.

00:00:07.119 --> 00:00:09.299
Well first of all, let's
look at this word passive.

00:00:09.299 --> 00:00:11.026
Passive is the way we describe

00:00:11.026 --> 00:00:13.572
components that do not create power

00:00:13.572 --> 00:00:17.548
or components that do not amplify signals.

00:00:17.548 --> 00:00:20.271
So particularly resistors,
as we have one here.

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Does not amplify a signal.

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Capacitors and inductors do not either.

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So all three of those
are passive components.

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And sometimes we say
something like a battery

00:00:31.251 --> 00:00:33.527
or a voltage source is a passive component

00:00:33.527 --> 00:00:36.168
because it can't amplify a signal.

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It does provide power, but it
does not make signals bigger,

00:00:39.337 --> 00:00:43.759
it can't make a voltage
larger than it already is.

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So we're gonna start with the
way to label this resistor

00:00:47.261 --> 00:00:49.273
with its current and voltage.

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First we'll resort to Ohm's law,

00:00:51.548 --> 00:00:52.626
our friend,

00:00:52.626 --> 00:00:56.687
voltage equals current,
which is i times r,

00:00:56.687 --> 00:00:59.370
and when we label this,
when we label this device,

00:00:59.370 --> 00:01:02.540
here's a resistor, so
we'll give that a label,

00:01:02.540 --> 00:01:04.183
and we're gonna put a voltage across this,

00:01:04.183 --> 00:01:06.581
something outside of this is
gonna put a voltage across it

00:01:06.581 --> 00:01:10.603
and this end will be positive
with respect to this end.

00:01:10.603 --> 00:01:13.021
And the question is, the
question we want to answer is

00:01:13.021 --> 00:01:15.053
which direction does the current go?

00:01:15.053 --> 00:01:16.435
When the positive voltage is here

00:01:16.435 --> 00:01:17.592
and the negative voltage is here,

00:01:17.592 --> 00:01:19.970
which way's the current
flow through this resistor?

00:01:19.970 --> 00:01:23.261
Does it go from top to
bottom, or bottom to top?

00:01:23.261 --> 00:01:27.650
The answer is the positive
current goes this way,

00:01:27.650 --> 00:01:31.367
it goes down, so that's i in this.

00:01:31.367 --> 00:01:35.086
And the current is going into

00:01:35.086 --> 00:01:39.248
the end with the positive
sign on the voltage.

00:01:39.248 --> 00:01:41.462
Now I'm gonna draw another one of these,

00:01:41.462 --> 00:01:45.384
same sort of thing, and
we'll do it a different way.

00:01:45.384 --> 00:01:47.650
We'll do this one sideways.

00:01:51.053 --> 00:01:54.202
And I'll label the voltage, here's r,

00:01:54.202 --> 00:01:57.737
and I'll put a voltage on it,

00:01:57.737 --> 00:02:00.096
and the voltage let's label it this way,

00:02:00.096 --> 00:02:01.496
we label the plus end this way

00:02:01.496 --> 00:02:03.528
and label the minus end this way.

00:02:03.528 --> 00:02:05.302
The question is which
way does the current go?

00:02:05.302 --> 00:02:07.375
Well the current goes in the positive end

00:02:07.375 --> 00:02:09.223
so it goes in that way.

00:02:09.223 --> 00:02:11.043
And that's the sign convention

00:02:11.043 --> 00:02:14.756
for passive components.

00:02:14.756 --> 00:02:18.886
The current goes into

00:02:18.886 --> 00:02:21.704
the voltage on the positive
terminal or the voltage.

00:02:21.704 --> 00:02:24.446
If I drew the arrow over here like this,

00:02:24.446 --> 00:02:25.828
that would still mean the same thing,

00:02:25.828 --> 00:02:28.001
it's still going in the positive

00:02:28.001 --> 00:02:30.420
and coming out the negative side.

00:02:30.420 --> 00:02:32.933
So when we draw this for
capacitors and inductors,

00:02:32.933 --> 00:02:34.997
let's draw a capacitor over here,

00:02:40.450 --> 00:02:43.830
and there's c, and if I
labeled this with a voltage,

00:02:43.830 --> 00:02:46.809
plus and minus v,

00:02:46.809 --> 00:02:49.085
then the passive sign convention tells me

00:02:49.085 --> 00:02:53.433
that I label the positive
current going in, i.

00:02:53.433 --> 00:02:55.302
The same for an inductor.

00:02:55.302 --> 00:02:58.308
We'll draw a curly inductor,

00:02:58.308 --> 00:03:02.624
like that, we'll put a
voltage on it this time.

00:03:02.624 --> 00:03:07.254
I'll put the voltage on
this side, plus, minus v,

00:03:11.094 --> 00:03:13.777
this is our inductor, like that,

00:03:13.777 --> 00:03:15.568
and the current,

00:03:22.026 --> 00:03:23.752
the current goes in the top like that.

00:03:23.752 --> 00:03:25.987
So it goes in the same side as the plus.

00:03:25.987 --> 00:03:28.943
And the current comes
out on the minus side.

00:03:28.943 --> 00:03:32.653
You can draw it either way you want.

00:03:32.653 --> 00:03:33.859
Let's practice some more,

00:03:33.859 --> 00:03:36.749
this time we'll do it
with a resistor, again.

00:03:36.749 --> 00:03:40.175
Here's another resistor, and
this time I'm gonna label

00:03:40.175 --> 00:03:41.333
the current first.

00:03:41.333 --> 00:03:43.284
I'm gonna say the current goes

00:03:43.284 --> 00:03:47.631
this way through the resistor.

00:03:47.631 --> 00:03:50.130
And now, using the passive component

00:03:50.130 --> 00:03:53.727
sign convention, how do
we label the voltage?

00:03:53.727 --> 00:03:55.738
I wanna label the voltage.

00:03:55.738 --> 00:03:56.957
Where's the plus sign?

00:03:56.957 --> 00:04:00.249
Plus sign is on the side
that the current goes in.

00:04:00.249 --> 00:04:02.614
The current's going in to this terminal,

00:04:02.614 --> 00:04:05.700
so that is the plus side

00:04:05.700 --> 00:04:09.886
of the resistor.

00:04:09.886 --> 00:04:12.405
And that's basically the
idea of a sign convention

00:04:12.405 --> 00:04:13.441
for passive components.

00:04:13.441 --> 00:04:14.823
When we're setting up a circuit,

00:04:14.823 --> 00:04:16.225
if we have a complicated circuit,

00:04:16.225 --> 00:04:18.337
we're setting it up, we have to give names

00:04:18.337 --> 00:04:20.552
to all the voltages and all the currents.

00:04:20.552 --> 00:04:23.323
And this is the way you
do it to make sure that

00:04:23.323 --> 00:04:26.018
Ohm's law comes out with the correct sign.

00:04:26.018 --> 00:04:28.541
That's why we have this sign convention.


Current direction

WEBVTT
Kind: captions
Language: en

00:00:02.286 --> 00:00:03.851
- [Voiceover] In the last
video, we talked about

00:00:03.851 --> 00:00:05.125
the meaning of current,

00:00:05.125 --> 00:00:08.608
and current is defined to
be the movement of charge,

00:00:08.608 --> 00:00:11.101
amount of charge per second.

00:00:11.101 --> 00:00:12.461
We looked at a copper wire

00:00:12.461 --> 00:00:15.961
where electrons are carrying the current,

00:00:15.961 --> 00:00:18.661
and we looked also at a salt solution

00:00:18.661 --> 00:00:20.861
where both positive and negative ions

00:00:20.861 --> 00:00:23.916
are carrying the current
across this imaginary boundary

00:00:23.916 --> 00:00:28.083
where we keep track of how
many charges are moving.

00:00:29.481 --> 00:00:31.241
So now, next we need to talk about

00:00:31.241 --> 00:00:33.763
how do we define a positive current?

00:00:33.763 --> 00:00:35.421
What does a positive current mean?

00:00:35.421 --> 00:00:37.558
So, I'm gonna move this up a little bit.

00:00:37.558 --> 00:00:40.904
And we're gonna get to something
that causes some confusion.

00:00:40.904 --> 00:00:43.321
But, we'll try to clear it up here.

00:00:43.321 --> 00:00:47.141
The positive sign, this is
a convention that we use.

00:00:47.141 --> 00:00:48.761
The positive sign for current

00:00:48.761 --> 00:00:52.865
is the direction that
positive charges move.

00:00:52.865 --> 00:00:55.762
So, that's the direction...

00:00:55.762 --> 00:00:59.901
this is the direction a
positive sodium ion moves.

00:00:59.901 --> 00:01:04.621
This is the direction a
negative chlorine ion moves,

00:01:04.621 --> 00:01:07.861
When I assign a positive
direction to the current,

00:01:07.861 --> 00:01:09.963
the overall positive direction

00:01:09.963 --> 00:01:13.201
is gonna be in this direction.

00:01:13.201 --> 00:01:16.034
That's plus current. Plus current.

00:01:22.439 --> 00:01:25.473
Alright, and the same thing
actually holds for a wire.

00:01:25.473 --> 00:01:29.799
If I draw, let's go back
and draw my wire again.

00:01:29.799 --> 00:01:31.882
If I draw my copper wire,

00:01:34.383 --> 00:01:38.550
and inside there we know that
all these little electrons,

00:01:40.723 --> 00:01:43.343
all the electrons are
moving in that direction.

00:01:43.343 --> 00:01:46.805
So q- is moving in that direction.

00:01:46.805 --> 00:01:50.703
And when I assign a direction
for the arrow for a current,

00:01:50.703 --> 00:01:53.620
the current arrow points this way.

00:01:54.703 --> 00:01:55.786
Plus current.

00:02:00.573 --> 00:02:03.602
So for instance, if I had
10 electrons per second

00:02:03.602 --> 00:02:06.149
that were moving in this direction here,

00:02:06.149 --> 00:02:11.083
I would say the current is
plus 10 charges per second

00:02:11.083 --> 00:02:12.583
in this direction.

00:02:13.883 --> 00:02:14.823
And ask yourself:

00:02:14.823 --> 00:02:18.383
If it's always electrons
carrying a current,

00:02:18.383 --> 00:02:22.283
why on Earth do we point
the arrow in the other way?

00:02:22.283 --> 00:02:24.943
And that actually goes
back, way back in history,

00:02:24.943 --> 00:02:27.383
to out friend Ben Franklin.

00:02:27.383 --> 00:02:29.723
Ben Franklin is the person who said,

00:02:29.723 --> 00:02:32.140
who basically made a decision

00:02:33.102 --> 00:02:35.982
that said the charge on
an electron is minus.

00:02:35.982 --> 00:02:38.002
Now, when he made that decision,

00:02:38.002 --> 00:02:40.702
nobody knew there was such
a thing as an electron.

00:02:40.702 --> 00:02:42.035
So Ben Franklin.

00:02:50.702 --> 00:02:53.202
And that was in the year 1747.

00:02:56.485 --> 00:03:00.062
Ben Franklin's a person
who assigned plus and minus

00:03:00.062 --> 00:03:03.310
as being the two types
of charge that existed.

00:03:03.310 --> 00:03:05.042
And when he did that he had no idea.

00:03:05.042 --> 00:03:07.107
Nobody had any idea that
there was such a thing

00:03:07.107 --> 00:03:08.697
as an electron.

00:03:08.697 --> 00:03:12.864
Electron was discovered by a
physicist named J. J. Thomson,

00:03:18.802 --> 00:03:20.219
In the year 1897.

00:03:24.996 --> 00:03:28.059
That was when the electron was discovered.

00:03:28.059 --> 00:03:30.528
And believe it or not, if
we do the subtraction here,

00:03:30.528 --> 00:03:32.945
look at this, it's 150 years.

00:03:34.398 --> 00:03:37.326
For 150 years of electric research,

00:03:37.326 --> 00:03:40.706
we were able to figure out
a lot of really good things,

00:03:40.706 --> 00:03:42.346
assigning current

00:03:42.346 --> 00:03:45.366
going in the opposite direction
of electrons in metal.

00:03:45.366 --> 00:03:48.471
And it's all okay. It's all okay.

00:03:48.471 --> 00:03:51.246
One of the questions that's
often asked by new students,

00:03:51.246 --> 00:03:52.986
is why don't we switch it around

00:03:52.986 --> 00:03:54.526
so the current points this way?

00:03:54.526 --> 00:03:57.734
Why don't we assign an electron,

00:03:57.734 --> 00:04:00.901
why don't we make current go this way?

00:04:04.959 --> 00:04:07.326
So, why didn't we just switch it around?

00:04:07.326 --> 00:04:09.446
Well, we had a 150 years of experience,

00:04:09.446 --> 00:04:12.146
and now we've actually had another 120,

00:04:12.146 --> 00:04:15.364
since the discovery of the electron.

00:04:15.364 --> 00:04:16.938
And we've managed to get by with this.

00:04:16.938 --> 00:04:18.669
We have not chosen to do it.

00:04:18.669 --> 00:04:20.846
One of the reasons we
don't do it is because

00:04:20.846 --> 00:04:24.926
it would basically, it
would be a big mess.

00:04:24.926 --> 00:04:27.606
Imagine if for instance,
say in the United States

00:04:27.606 --> 00:04:30.926
we decided, Oh, driving on
the right side of the street

00:04:30.926 --> 00:04:32.566
is the wrong side of the street.

00:04:32.566 --> 00:04:34.266
We want to all drive on the left.

00:04:34.266 --> 00:04:36.837
Imagine the chaos that would cause.

00:04:36.837 --> 00:04:38.374
Now, before we made the change,

00:04:38.374 --> 00:04:40.379
everybody managed to
get where we were going,

00:04:40.379 --> 00:04:42.789
and after the change, everybody
would get there going,

00:04:42.789 --> 00:04:47.026
but the change over would
be just so, so costly.

00:04:47.026 --> 00:04:48.986
And the same thing with electricity.

00:04:48.986 --> 00:04:50.716
We can do perfectly well

00:04:50.716 --> 00:04:52.861
talking about current going the direction

00:04:52.861 --> 00:04:54.201
of a positive charge,

00:04:54.201 --> 00:04:57.319
just like we did here
with the positive sodium

00:04:57.319 --> 00:04:59.319
going in this direction.

00:05:00.246 --> 00:05:03.786
This is actually how electricity
is conducted in your body.

00:05:03.786 --> 00:05:08.686
So, it's not uncommon to have
positive charge moving around.

00:05:08.686 --> 00:05:10.981
That's the definition of
current in a nutshell.

00:05:10.981 --> 00:05:13.041
We put a boundary across,

00:05:13.041 --> 00:05:16.846
and you watch the charges going through,

00:05:16.846 --> 00:05:20.506
in either direction, which
ever direction they do,

00:05:20.506 --> 00:05:24.006
and you count up and you get q per second.

00:05:25.363 --> 00:05:26.613
That's current.


Current

WEBVTT
Kind: captions
Language: en

00:00:00.816 --> 00:00:01.751
- [Voiceover] All right,
now we're gonna talk

00:00:01.751 --> 00:00:04.169
about the idea of an electric current.

00:00:04.169 --> 00:00:08.598
The story about currents
starts with the idea of charge.

00:00:08.598 --> 00:00:11.808
We've learned that we
have two kinds of charges,

00:00:11.808 --> 00:00:13.636
positive and negative charge.

00:00:13.636 --> 00:00:17.517
We'll just make up two
little charges like that.

00:00:17.517 --> 00:00:20.646
And we know if they're the opposite sign,

00:00:20.646 --> 00:00:24.019
that there'll be a force
of attraction between them.

00:00:24.019 --> 00:00:27.005
And if they have two like signs,

00:00:27.005 --> 00:00:31.333
here's two charges that are both positive,

00:00:31.333 --> 00:00:34.128
and these charges are
gonna repel each other.

00:00:34.128 --> 00:00:37.257
So this is the basic electrostatics idea,

00:00:37.257 --> 00:00:39.939
and the same thing for two minus charges.

00:00:39.939 --> 00:00:40.934
They also repel.

00:00:40.934 --> 00:00:45.101
So like charges repel, and
unlike charges attract.

00:00:47.741 --> 00:00:48.594
That's one idea.

00:00:48.594 --> 00:00:49.955
We have the idea of charge.

00:00:49.955 --> 00:00:52.942
And now we need a place
to get some charge.

00:00:52.942 --> 00:00:54.791
One of the places we
like to get charge from

00:00:54.791 --> 00:00:56.791
is copper, copper wires.

00:00:58.024 --> 00:00:59.619
A copper atom looks like this.

00:00:59.619 --> 00:01:01.702
Copper atom has a nucleus

00:01:03.296 --> 00:01:05.286
with some protons in it,

00:01:05.286 --> 00:01:09.010
and it also has electrons
flying around the outside,

00:01:09.010 --> 00:01:11.348
electrons in orbits around the outside.

00:01:11.348 --> 00:01:13.994
So we'll draw the electrons like this.

00:01:13.994 --> 00:01:17.244
There'll be orbits around this nucleus.

00:01:19.682 --> 00:01:21.349
Pretty good circles.

00:01:22.508 --> 00:01:25.737
And there'll be electrons in these.

00:01:25.737 --> 00:01:27.525
Little minus signs.

00:01:27.525 --> 00:01:30.608
There's electrons stacked up in this.

00:01:32.706 --> 00:01:36.039
And even farther out, there's electrons.

00:01:39.167 --> 00:01:43.332
So there's kind of a
interesting looking copper atom.

00:01:43.332 --> 00:01:46.563
Copper, the symbol for copper is Cu,

00:01:46.563 --> 00:01:48.896
and its atomic number is 29.

00:01:51.357 --> 00:01:53.917
That means there's 29 protons inside here,

00:01:53.917 --> 00:01:56.580
and there's 29 electrons outside.

00:01:56.580 --> 00:01:59.139
It turns out, just as a
coincidence for copper,

00:01:59.139 --> 00:02:01.639
that the last orbital out here

00:02:03.182 --> 00:02:05.844
has just one electron in it,

00:02:05.844 --> 00:02:08.140
that guy right there.

00:02:08.140 --> 00:02:09.640
And that's the one

00:02:10.477 --> 00:02:13.281
that is the easiest to
pull away from copper

00:02:13.281 --> 00:02:17.303
and have it go participate in conduction,

00:02:17.303 --> 00:02:18.970
in electric current.

00:02:21.354 --> 00:02:22.881
If I have a chunk of copper,

00:02:22.881 --> 00:02:26.464
every copper atom will
have the opportunity

00:02:28.326 --> 00:02:32.493
to contribute one, this one
lonely electron out here.

00:02:33.507 --> 00:02:37.063
If we look at another element,
like for instance silver,

00:02:37.063 --> 00:02:41.148
silver has this same kind
of electron configuration,

00:02:41.148 --> 00:02:42.691
where there's just one out here.

00:02:42.691 --> 00:02:44.194
And that's why silver and copper

00:02:44.194 --> 00:02:46.777
are such good, good conductors.

00:02:48.624 --> 00:02:49.680
Now we're gonna build,

00:02:49.680 --> 00:02:51.712
let's build a copper wire.

00:02:51.712 --> 00:02:54.129
Here's sort of a copper wire.

00:02:55.897 --> 00:02:57.462
It's just made of solid copper.

00:02:57.462 --> 00:02:59.962
It's all full of copper atoms.

00:03:01.430 --> 00:03:04.763
And I'm gonna put a voltage across this.

00:03:08.685 --> 00:03:09.780
There's our little battery.

00:03:09.780 --> 00:03:12.645
This is the minus sign,
this is the plus side.

00:03:12.645 --> 00:03:15.645
And we'll hook up a battery to this.

00:03:19.634 --> 00:03:20.528
What's going on in here?

00:03:20.528 --> 00:03:24.528
Inside this copper is a
whole bunch of electrons

00:03:27.438 --> 00:03:28.271
that are

00:03:31.723 --> 00:03:32.861
associated with atoms.

00:03:32.861 --> 00:03:35.644
It's a neutral piece of metal.

00:03:35.644 --> 00:03:38.898
There's the same number of
protons as there is electrons.

00:03:38.898 --> 00:03:40.866
But these electrons
are a little bit loose.

00:03:40.866 --> 00:03:43.385
So if I put a plus over there,

00:03:43.385 --> 00:03:44.441
that's this situation right here,

00:03:44.441 --> 00:03:46.881
where a plus is attracting a minus.

00:03:46.881 --> 00:03:49.236
So an electron is gonna sort of wander

00:03:49.236 --> 00:03:51.451
over this way and go like that.

00:03:51.451 --> 00:03:55.129
And that's gonna leave a net
positive charge in this region.

00:03:55.129 --> 00:03:56.266
So these electrons are all gonna

00:03:56.266 --> 00:03:58.849
start moving in this direction.

00:04:00.188 --> 00:04:02.402
And down at the end,

00:04:02.402 --> 00:04:06.669
here, an electron is gonna
come out of this battery,

00:04:06.669 --> 00:04:08.050
travel in here,

00:04:08.050 --> 00:04:10.976
and it's gonna go in there
and make up the difference.

00:04:10.976 --> 00:04:13.171
So if I had a net positive charge here

00:04:13.171 --> 00:04:16.584
from the electrons leaving
and going to the left,

00:04:16.584 --> 00:04:18.656
this battery would fill those in.

00:04:18.656 --> 00:04:21.236
And I'm gonna get a
net movement of charge,

00:04:21.236 --> 00:04:26.072
of negative charge, around
in this direction, like this.

00:04:26.072 --> 00:04:28.104
The question is, how do I measure that?

00:04:28.104 --> 00:04:29.892
How do I measure or give a number

00:04:29.892 --> 00:04:32.391
to that amount of stuff that's going on?

00:04:32.391 --> 00:04:34.910
So we wanna quantify that,
we wanna assign a number

00:04:34.910 --> 00:04:37.003
to the amount of current happening here.

00:04:37.003 --> 00:04:39.136
What we do is, in our heads,

00:04:39.136 --> 00:04:40.904
we put a boundary across here.

00:04:40.904 --> 00:04:43.139
So just make that up in your head.

00:04:43.139 --> 00:04:45.355
And it cuts all the
way through the copper.

00:04:45.355 --> 00:04:49.272
And what we know, we're
gonna stand right here.

00:04:50.204 --> 00:04:52.496
We're gonna keep our eye

00:04:52.496 --> 00:04:55.910
right on this boundary down in here.

00:04:55.910 --> 00:04:57.596
As we watch, what we're gonna do is,

00:04:57.596 --> 00:05:01.030
we're gonna count the number of electrons

00:05:01.030 --> 00:05:02.530
that move by here,

00:05:04.160 --> 00:05:05.418
and we're gonna have a stop watch

00:05:05.418 --> 00:05:07.630
and we're gonna time that.

00:05:07.630 --> 00:05:08.463
So we're gonna get, basically,

00:05:08.463 --> 00:05:09.713
this is charge,

00:05:10.560 --> 00:05:12.103
it's negative charge,

00:05:12.103 --> 00:05:13.140
and it's moving to the side.

00:05:13.140 --> 00:05:15.151
What we're gonna do is, at
one little spot right here,

00:05:15.151 --> 00:05:18.868
we're just gonna count the
number that go by in one second.

00:05:18.868 --> 00:05:21.951
So we're gonna get charge per second.

00:05:23.765 --> 00:05:27.252
It's gonna be a negative charge moving by.

00:05:27.252 --> 00:05:29.585
That's what we call current.

00:05:32.088 --> 00:05:34.770
It's the same as water
flowing by in a river.

00:05:34.770 --> 00:05:36.520
That's the same idea.

00:05:38.792 --> 00:05:42.471
Now I'm gonna set up a different situation

00:05:42.471 --> 00:05:43.914
that also produces a current.

00:05:43.914 --> 00:05:45.459
And this time, we're
gonna do it with water,

00:05:45.459 --> 00:05:46.709
water and salt.

00:05:49.133 --> 00:05:53.300
Let's build a tube of salt,
of salt water, like this.

00:05:55.086 --> 00:05:56.589
We're gonna pretend this is some tube

00:05:56.589 --> 00:05:57.964
that's all full of water.

00:05:57.964 --> 00:05:59.851
I'm also gonna put a battery here.

00:05:59.851 --> 00:06:02.018
Let's put another battery.

00:06:07.694 --> 00:06:10.721
And we'll stick the wire into there.

00:06:10.721 --> 00:06:13.444
We'll stick the wire into there.

00:06:13.444 --> 00:06:15.110
This is the plus side of the battery,

00:06:15.110 --> 00:06:18.320
and this is the minus side of the battery.

00:06:18.320 --> 00:06:19.403
Water is H20,

00:06:23.196 --> 00:06:24.822
and this does not conduct.

00:06:24.822 --> 00:06:26.984
There's no free electrons available here.

00:06:26.984 --> 00:06:27.879
But what I'm gonna do is,

00:06:27.879 --> 00:06:29.300
I'm gonna put some table salt in it.

00:06:29.300 --> 00:06:31.962
This is ordinary salt
that you put on your food.

00:06:31.962 --> 00:06:36.057
It's made of sodium, that's
the symbol for sodium,

00:06:36.057 --> 00:06:38.474
and chloride, Cl is chloride.

00:06:39.552 --> 00:06:41.581
Sodium chloride is table salt.

00:06:41.581 --> 00:06:44.852
If we sprinkle some table salt into water,

00:06:44.852 --> 00:06:46.682
what happens is, these dissolve

00:06:46.682 --> 00:06:49.566
and we get a net plus charge here

00:06:49.566 --> 00:06:52.816
and a net minus charge on the chlorine.

00:06:53.792 --> 00:06:58.540
So out here is floating
around Na's with plus signs

00:06:58.540 --> 00:07:02.707
and Cl's nearby, really
close nearly, with minuses.

00:07:13.241 --> 00:07:14.824
Let's keep it even.

00:07:18.565 --> 00:07:21.458
Now, when I dip my battery
wires into this water,

00:07:21.458 --> 00:07:22.558
what's gonna happen is,

00:07:22.558 --> 00:07:23.858
this plus charge,

00:07:23.858 --> 00:07:27.069
this plus charge over
here from the battery

00:07:27.069 --> 00:07:30.035
is gonna attract the minus Cl's.

00:07:30.035 --> 00:07:34.038
So the Cl's gonna move
that way a little bit.

00:07:34.038 --> 00:07:35.502
And over here, the same
thing is happening.

00:07:35.502 --> 00:07:37.715
There's a minus sign here.

00:07:37.715 --> 00:07:40.465
There's a minus from the battery.

00:07:42.835 --> 00:07:45.720
That's going to attract this,

00:07:45.720 --> 00:07:49.582
and it's also gonna repel Cl minuses.

00:07:49.582 --> 00:07:52.203
So what we get is a net motion

00:07:52.203 --> 00:07:55.703
of positive charge, plus q going this way,

00:07:59.253 --> 00:08:02.086
and we get minus q going this way.

00:08:06.769 --> 00:08:08.476
How do we measure that current?

00:08:08.476 --> 00:08:09.593
How do we measure that current?

00:08:09.593 --> 00:08:12.173
Well, we do it the same way as
we did up there with copper.

00:08:12.173 --> 00:08:14.776
We put a boundary through
here in our heads.

00:08:14.776 --> 00:08:15.859
We stand here

00:08:17.498 --> 00:08:21.072
and we watch the charges moving by.

00:08:21.072 --> 00:08:24.739
What we're gonna get
is some sodiums, Na's,

00:08:25.815 --> 00:08:27.148
moving this way,

00:08:28.070 --> 00:08:30.387
and chlorines moving this way.

00:08:30.387 --> 00:08:31.951
Just like we showed here.

00:08:31.951 --> 00:08:32.784
Na

00:08:35.364 --> 00:08:36.766
moving this way.

00:08:36.766 --> 00:08:38.961
So there's gonna be plus charges
moving through the boundary

00:08:38.961 --> 00:08:41.115
and minus charges moving
through the boundary

00:08:41.115 --> 00:08:42.597
in the opposite direction.

00:08:42.597 --> 00:08:45.159
If I take the total sum of that.

00:08:45.159 --> 00:08:47.778
For example, if I see one Na go this way

00:08:47.778 --> 00:08:50.440
and one Cl go this way,

00:08:50.440 --> 00:08:52.370
that's equivalent to two charges

00:08:52.370 --> 00:08:54.828
moving through the boundary.

00:08:54.828 --> 00:08:55.906
Hope that makes sense.

00:08:55.906 --> 00:08:58.039
It's equivalent to two charges,

00:08:58.039 --> 00:09:00.111
one going this way and one going this way.

00:09:00.111 --> 00:09:02.326
Because they have opposite signs,

00:09:02.326 --> 00:09:05.536
they add together and make two charges.

00:09:05.536 --> 00:09:07.286
In this case, current

00:09:12.078 --> 00:09:15.245
is equal, again, to charge per second.


Voltage

WEBVTT
Kind: captions
Language: en

00:00:00.408 --> 00:00:01.305
- [Voiceover] Voltage is one of

00:00:01.305 --> 00:00:04.808
the most important quantities
and ideas in electricity,

00:00:04.808 --> 00:00:07.301
and in this video we're gonna develop

00:00:07.301 --> 00:00:11.181
an intuitive feeling
for what voltage means.

00:00:11.181 --> 00:00:13.266
But it has to do with potential energy

00:00:13.266 --> 00:00:16.155
of electrical charges
and that's what we're

00:00:16.155 --> 00:00:19.203
going to cover here, we're
not gonna do a derivation,

00:00:19.203 --> 00:00:22.145
but we are going to do
an intuitive description

00:00:22.145 --> 00:00:23.628
of what voltage means.

00:00:23.628 --> 00:00:26.579
We're gonna start with
an analogy to gravity.

00:00:26.579 --> 00:00:30.329
Gravity and voltage are
really similar ideas.

00:00:30.329 --> 00:00:32.380
I'm going to draw a mountain here.

00:00:32.380 --> 00:00:35.880
Here's some mountain-side with snow on it.

00:00:38.369 --> 00:00:39.827
And I'm going to put a mass here,

00:00:39.827 --> 00:00:43.021
here's a mass of some mass M.

00:00:43.021 --> 00:00:46.229
And it was lifted up to the
top of the hill somehow,

00:00:46.229 --> 00:00:50.116
by a ski lift, by a mountain
climber, something like that.

00:00:50.116 --> 00:00:53.566
And if I put it on top of
the mountain and I let it go,

00:00:53.566 --> 00:00:56.387
the potential energy that it has is going

00:00:56.387 --> 00:00:59.352
to be dissipated as kinetic energy

00:00:59.352 --> 00:01:03.519
and that mass is going to
roll down the hill to here.

00:01:05.431 --> 00:01:07.201
And as it does, it could do some work.

00:01:07.201 --> 00:01:10.620
It could hit some trees,
let's draw a tree.

00:01:10.620 --> 00:01:13.862
And it could run into a tree
and knock that tree around.

00:01:13.862 --> 00:01:16.846
It could hit a bear, it
could bounce off rocks,

00:01:16.846 --> 00:01:21.579
all kinds of things, so that's
a mass rolling down a hill.

00:01:21.579 --> 00:01:24.495
Now if I draw, this is a
way to think about voltage,

00:01:24.495 --> 00:01:28.578
think about voltage as
being another mountaintop,

00:01:29.846 --> 00:01:31.834
and this time we'll put a battery in here.

00:01:31.834 --> 00:01:35.404
This is our battery, this is
what a battery does for us,

00:01:35.404 --> 00:01:38.071
it actually builds our mountain.

00:01:39.596 --> 00:01:42.846
The battery actually delivers electrons

00:01:45.446 --> 00:01:48.019
to the top of the hill, so
here's an electron coming

00:01:48.019 --> 00:01:52.003
out of the battery terminal,
the negative battery terminal.

00:01:52.003 --> 00:01:56.170
And if I release this, it is
going to roll down the hill.

00:02:00.501 --> 00:02:02.511
And eventually it's going to return

00:02:02.511 --> 00:02:03.901
to the bottom side of the battery.

00:02:03.901 --> 00:02:06.109
But the same thing, this
is the image you have

00:02:06.109 --> 00:02:08.448
in your head when we hook up a circuit.

00:02:08.448 --> 00:02:12.031
Along the way I could put in
different circuit components

00:02:12.031 --> 00:02:16.199
like resistors or capacitors
or anything like that,

00:02:16.199 --> 00:02:18.029
and I could make this electron do work

00:02:18.029 --> 00:02:20.249
and bump into things as it goes down.

00:02:20.249 --> 00:02:22.748
So, the amount of voltage here

00:02:22.748 --> 00:02:24.861
is proportional to the
height of this mountain.

00:02:24.861 --> 00:02:27.205
A high voltage is a high mountain

00:02:27.205 --> 00:02:29.388
and a low voltage is a low mountain.

00:02:29.388 --> 00:02:33.194
The electrons are pushed
out the top by the battery

00:02:33.194 --> 00:02:35.983
and roll down to the bottom
doing work along the way,

00:02:35.983 --> 00:02:39.566
and this is where we
do our circuit design.

00:02:43.419 --> 00:02:44.608
That's what we're doing over here,

00:02:44.608 --> 00:02:48.067
we buy batteries and we
do our circuit design

00:02:48.067 --> 00:02:50.019
and study over here.

00:02:50.019 --> 00:02:53.095
So this is a pretty good
analogy for thinking

00:02:53.095 --> 00:02:57.012
about voltage as you begin
to build your circuits.

00:02:57.012 --> 00:02:58.163
See you next time.


Simplifying resistor networks

WEBVTT
Kind: captions
Language: en

00:00:00.831 --> 00:00:01.740
- [Voiceover] We've learned about series

00:00:01.740 --> 00:00:04.666
and parallel resistors,
we've learned how to

00:00:04.666 --> 00:00:06.990
simplify series and parallel resistors

00:00:06.990 --> 00:00:10.991
into an equivalent resistor,
and just to review,

00:00:10.991 --> 00:00:15.158
for the series resistor, R
series, an equivalent R series

00:00:16.366 --> 00:00:18.816
is equal to the sum of
resistors in series.

00:00:18.816 --> 00:00:19.733
R1 plus R2.

00:00:23.136 --> 00:00:26.898
And we learned that if we
have resistors in parallel,

00:00:26.898 --> 00:00:30.021
meaning they share the same
nodes, if they're in parallel,

00:00:30.021 --> 00:00:33.179
we can get a parallel equivalent resistors

00:00:33.179 --> 00:00:37.080
R parallel, and if
there was two resistors,

00:00:37.080 --> 00:00:40.656
if there's two resistors in
parallel, the formula was

00:00:40.656 --> 00:00:41.989
R1 times R2 over

00:00:45.578 --> 00:00:47.245
R1 plus R2, plus R2.

00:00:50.954 --> 00:00:54.321
And if there's three
resistors or more in parallel

00:00:54.321 --> 00:00:55.714
it's a little more complicated.

00:00:55.714 --> 00:00:57.095
This formula look like this.

00:00:57.095 --> 00:01:01.262
One over R parallel, this is
for three or more resistors.

00:01:02.180 --> 00:01:03.097
One over R1

00:01:05.072 --> 00:01:06.489
plus one over R2,

00:01:09.368 --> 00:01:11.618
plus one over R3, etcetera,

00:01:14.890 --> 00:01:16.906
for as many resistors as you have.

00:01:16.906 --> 00:01:19.431
These two parallel
formulas do the same thing.

00:01:19.431 --> 00:01:21.172
This one's a little more convenient to use

00:01:21.172 --> 00:01:23.922
when you just have two resistors.

00:01:24.969 --> 00:01:27.209
So now, this problem in this video,

00:01:27.209 --> 00:01:30.240
we're going to look at a
complicated resistor circuit.

00:01:30.240 --> 00:01:32.237
There's a lot of resistors here.

00:01:32.237 --> 00:01:34.593
The question we're going
to ask is, if we plug in

00:01:34.593 --> 00:01:37.519
this voltage source here
into our resistor network

00:01:37.519 --> 00:01:40.436
there's the two little plug points,

00:01:41.652 --> 00:01:43.916
the question is what kind
of current is going to be

00:01:43.916 --> 00:01:45.774
pulled out of this voltage source?

00:01:45.774 --> 00:01:47.597
That's what I want to know.

00:01:47.597 --> 00:01:49.791
And if I stare at this, I go,

00:01:49.791 --> 00:01:52.798
well how am I going to figure that out?

00:01:52.798 --> 00:01:55.213
The important part of what
we're doing here is to

00:01:55.213 --> 00:01:58.649
stop for a minute and just see
what this circuit looks like,

00:01:58.649 --> 00:02:00.983
just be with it for a second.

00:02:00.983 --> 00:02:03.363
Let your eyes wander over it and find

00:02:03.363 --> 00:02:05.557
resistor patterns that you recognize.

00:02:05.557 --> 00:02:08.057
And you just wait and do that.

00:02:09.098 --> 00:02:11.919
All right, so we're looking
for series in parallel patterns

00:02:11.919 --> 00:02:16.018
and there's one, and this
is not in series, no.

00:02:16.018 --> 00:02:19.122
Okay, oh, there's a possibility,
here's a possibility

00:02:19.122 --> 00:02:21.584
of parallel resistors.

00:02:21.584 --> 00:02:24.161
This, is there a series one over here.

00:02:24.161 --> 00:02:27.481
I don't see a series head
to tail connection here

00:02:27.481 --> 00:02:32.381
and there's a series connection
here that we can use.

00:02:32.381 --> 00:02:34.819
Okay, so there's a couple of possibilities

00:02:34.819 --> 00:02:36.468
here for simplification.

00:02:36.468 --> 00:02:40.183
And one of the questions
is, okay, where do we start?

00:02:40.183 --> 00:02:43.724
We want to start this problem
at the end farthest away

00:02:43.724 --> 00:02:45.373
from the thing we care
about, what we care about

00:02:45.373 --> 00:02:48.530
is this right here, this
is what we're looking for.

00:02:48.530 --> 00:02:51.108
And so, a good way to start this problem

00:02:51.108 --> 00:02:53.906
is to start at the very
far end of the circuit

00:02:53.906 --> 00:02:56.460
and work our way backwards.

00:02:56.460 --> 00:02:58.596
And this is opposite the
way you read sentences

00:02:58.596 --> 00:03:00.442
and things like that, but in circuits

00:03:00.442 --> 00:03:02.497
it's sometimes what we want to do.

00:03:02.497 --> 00:03:05.876
So what I'm going to do is
start over here on this side

00:03:05.876 --> 00:03:07.617
and we're going to
disassemble this circuit

00:03:07.617 --> 00:03:09.700
and simplify it as we go.

00:03:10.973 --> 00:03:12.958
So we've actually at this
point, we've actually done

00:03:12.958 --> 00:03:15.277
the hard work of this, and
the rest of it's going to be

00:03:15.277 --> 00:03:17.994
relatively straightforward application

00:03:17.994 --> 00:03:19.677
of these two formulas.

00:03:19.677 --> 00:03:22.057
We decided what our
unknown is, and we decided

00:03:22.057 --> 00:03:25.134
where to start and that
strategy is the key

00:03:25.134 --> 00:03:29.174
to making this a simple
straightforward process.

00:03:29.174 --> 00:03:30.497
All right, so let's go after it.

00:03:30.497 --> 00:03:32.375
We're going to, we're
basically now going to do

00:03:32.375 --> 00:03:35.231
the work of simplifying this network.

00:03:35.231 --> 00:03:39.909
So, what I see here is
two resistors in series,

00:03:39.909 --> 00:03:43.242
two ohms plus eight ohms equals 10 ohms.

00:03:45.134 --> 00:03:48.134
So I can remove these resistors here

00:03:53.272 --> 00:03:55.105
and we place them with

00:03:58.300 --> 00:04:01.217
a 10 ohm resistor, that's step one.

00:04:02.259 --> 00:04:04.650
What do we do next? Now
we look at it again.

00:04:04.650 --> 00:04:08.098
Right here we see we have
two parallel resistors.

00:04:08.098 --> 00:04:10.455
So we're actually going to use this form.

00:04:10.455 --> 00:04:13.033
We're going to use this form
of the parallel resistor

00:04:13.033 --> 00:04:16.283
formula, and that says that 10 times 10

00:04:18.071 --> 00:04:19.404
over 10 plus 10,

00:04:22.414 --> 00:04:24.747
this is equal to 100 over 20

00:04:26.779 --> 00:04:27.779
or five ohms

00:04:30.717 --> 00:04:32.342
and I could've done this a little quicker

00:04:32.342 --> 00:04:35.442
if I know that if two
resistors in parallel

00:04:35.442 --> 00:04:38.101
have the same value, 10 and 10.

00:04:38.101 --> 00:04:40.539
I know that the parallel combination

00:04:40.539 --> 00:04:42.617
of those is exactly half.

00:04:42.617 --> 00:04:45.334
So now we have an equivalent
resistor for those two guys.

00:04:45.334 --> 00:04:47.001
We can take them out

00:04:51.963 --> 00:04:53.963
and we replace that with

00:04:57.095 --> 00:04:58.762
a five ohm resistor.

00:05:00.853 --> 00:05:02.096
You can see where this is going,

00:05:02.096 --> 00:05:03.976
we're just basically going
to collapse this circuit

00:05:03.976 --> 00:05:05.335
one step at a time.

00:05:05.335 --> 00:05:09.502
Five ohms plus one ohm is
this series combination here

00:05:11.372 --> 00:05:15.539
and that is equal to six ohms,
so I can replace this now

00:05:17.154 --> 00:05:19.975
with six ohms, we're just
consuming our circuit

00:05:19.975 --> 00:05:22.892
and erasing and rewriting as we go.

00:05:28.764 --> 00:05:33.628
And I said six ohms here,
and what do we have now?

00:05:33.628 --> 00:05:34.963
Now this is a little more challenging.

00:05:34.963 --> 00:05:37.390
I have three resistors in parallel

00:05:37.390 --> 00:05:40.188
and I'm going to use now this formula here

00:05:40.188 --> 00:05:42.812
for resistors in parallel, so

00:05:42.812 --> 00:05:45.812
one over the parallel combination is

00:05:47.189 --> 00:05:48.522
one over 12 plus

00:05:50.707 --> 00:05:52.207
one over four plus

00:05:54.364 --> 00:05:58.114
one over six and the
common denominator is 12

00:05:58.950 --> 00:05:59.783
so that is

00:06:04.232 --> 00:06:07.204
something over 12, which will be

00:06:07.204 --> 00:06:09.037
one plus three over 12

00:06:11.767 --> 00:06:15.517
plus two over 12, and
that equals six over 12

00:06:17.793 --> 00:06:19.210
which is one half

00:06:21.322 --> 00:06:23.725
and one over Rp, that's one over Rp,

00:06:23.725 --> 00:06:26.892
so Rp equals two over one or two ohms.

00:06:29.925 --> 00:06:31.429
And now I can replace all three of these.

00:06:31.429 --> 00:06:33.512
I get to erase all three.

00:06:39.254 --> 00:06:40.754
We're almost done.

00:06:50.492 --> 00:06:53.825
And that was, I said two ohms, two ohms.

00:06:54.672 --> 00:06:58.815
The final step is one
and two ohms in series.

00:06:58.815 --> 00:07:00.898
One plus two equals three

00:07:02.100 --> 00:07:05.017
and finally we get to the last step

00:07:06.814 --> 00:07:07.647
which is

00:07:26.737 --> 00:07:28.200
All right, so we're done.

00:07:28.200 --> 00:07:31.427
What that means is, from the viewpoint,

00:07:31.427 --> 00:07:34.527
as far as that voltage source can tell,

00:07:34.527 --> 00:07:38.115
it's putting out enough
current to drive three ohms.

00:07:38.115 --> 00:07:42.074
And that whole rest of
that complicated network,

00:07:42.074 --> 00:07:44.210
looks like three ohms to that resistor,

00:07:44.210 --> 00:07:46.567
to that voltage source.

00:07:46.567 --> 00:07:50.224
And we have taken a complicated circuit

00:07:50.224 --> 00:07:52.488
and turned it into a really simple one.

00:07:52.488 --> 00:07:54.334
So that's the steps of simplification.

00:07:54.334 --> 00:07:57.364
The key was figuring out what you want

00:07:57.364 --> 00:07:59.454
and then going to the
far end of the circuit

00:07:59.454 --> 00:08:01.486
and working your way backwards

00:08:01.486 --> 00:08:03.653
and simplifying as you go.


Gradient and contour maps

WEBVTT
Kind: captions
Language: en

00:00:00.161 --> 00:00:01.229
- [Voiceover] So here I want to talk about

00:00:01.229 --> 00:00:04.119
the gradient and the
context of a contour map.

00:00:04.119 --> 00:00:06.409
So let's say we have a
multivariable function.

00:00:06.409 --> 00:00:09.169
A two-variable function f of x,y.

00:00:09.169 --> 00:00:12.111
And this one is just
gonna equal x times y.

00:00:13.631 --> 00:00:15.551
So we can visualize
this with a contour map

00:00:15.551 --> 00:00:17.036
just on the xy plane.

00:00:17.036 --> 00:00:17.895
So what I'm gonna do

00:00:17.895 --> 00:00:19.149
is I'm gonna go over here.

00:00:19.149 --> 00:00:21.798
I'm gonna draw a y axis

00:00:22.868 --> 00:00:24.455
and my x axis.

00:00:25.405 --> 00:00:27.084
All right, so

00:00:27.084 --> 00:00:28.981
this right here represents x values.

00:00:28.981 --> 00:00:30.376
And this represents y values.

00:00:30.376 --> 00:00:32.064
And this is entirely the input space.

00:00:32.064 --> 00:00:33.760
And I have a video on contour maps

00:00:33.760 --> 00:00:35.451
if you are unfamiliar with them

00:00:35.451 --> 00:00:36.784
or are feeling uncomfortable.

00:00:36.784 --> 00:00:39.264
And the contour map for x times y

00:00:39.264 --> 00:00:40.643
looks something like this.

00:00:40.643 --> 00:00:43.695
And each one of these lines
represents a constant value.

00:00:43.695 --> 00:00:46.434
So you might be thinking that you have,

00:00:46.434 --> 00:00:47.735
you know, let's say you want a

00:00:47.735 --> 00:00:52.157
the constant value for f of
x times y is equal to two.

00:00:52.157 --> 00:00:53.689
Would be one of these lines.

00:00:53.689 --> 00:00:55.907
That would be what one of
these lines represents.

00:00:55.907 --> 00:00:57.296
And a way you could think about that

00:00:57.296 --> 00:00:58.518
for this specific function

00:00:58.518 --> 00:01:01.548
is you saying hey, when
is x times y equal to two?

00:01:01.548 --> 00:01:03.071
And that's kind of like the graph

00:01:03.071 --> 00:01:05.236
y equals two over x.

00:01:06.636 --> 00:01:09.599
And that's where you would
see something like this.

00:01:09.599 --> 00:01:10.546
So all of these lines,

00:01:10.546 --> 00:01:13.122
they're representing constant
values for the function.

00:01:13.122 --> 00:01:15.440
And now I want to take a
look at the gradient field.

00:01:15.440 --> 00:01:17.850
And the gradient, if you'll remember,

00:01:17.850 --> 00:01:21.576
is just a vector full of the
partial derivatives of f.

00:01:21.576 --> 00:01:23.589
And let's just actually write it out.

00:01:23.589 --> 00:01:27.046
The gradient of f, with
our little del symbol,

00:01:27.046 --> 00:01:28.553
is a function of x and y.

00:01:29.653 --> 00:01:31.938
And it's a vector-valued function

00:01:31.938 --> 00:01:33.929
whose first coordinate is

00:01:33.929 --> 00:01:36.254
the partial derivative
of f with respect to x.

00:01:36.254 --> 00:01:37.969
And the second component

00:01:37.969 --> 00:01:42.125
is the partial derivative
with respect to y.

00:01:44.535 --> 00:01:46.665
So when we actually do
this for our function,

00:01:46.665 --> 00:01:49.017
we take the partial
derivative with respect to x.

00:01:49.017 --> 00:01:49.846
It takes a look.

00:01:49.846 --> 00:01:50.870
X looks like a variable.

00:01:50.870 --> 00:01:52.065
Y looks like a constant.

00:01:52.065 --> 00:01:53.546
The derivative of this whole thing

00:01:53.546 --> 00:01:57.992
is just equal to that constant, y.

00:01:57.992 --> 00:01:59.468
And then kind of the reverse for when you

00:01:59.468 --> 00:02:01.409
take the partial derivative
with respect to y.

00:02:01.409 --> 00:02:02.768
Y looks like a variable.

00:02:02.768 --> 00:02:03.989
X looks like a constant.

00:02:03.989 --> 00:02:07.039
And the derivative is
just that constant, x.

00:02:08.979 --> 00:02:11.131
And this can be visualized
as a vector field

00:02:11.131 --> 00:02:12.316
in the xy plane as well.

00:02:12.316 --> 00:02:14.394
You know, at every given point, xy,

00:02:14.394 --> 00:02:15.377
so you kind of go like

00:02:15.377 --> 00:02:19.597
x equals two, y equals one, let's say.

00:02:19.597 --> 00:02:21.710
So that would be x
equals two, y equals one.

00:02:21.710 --> 00:02:23.344
You would plug in the vector

00:02:23.344 --> 00:02:24.721
and see what should be output.

00:02:24.721 --> 00:02:28.422
And at this point, the point is two, one.

00:02:28.422 --> 00:02:30.423
The desired output kind of swaps those.

00:02:30.423 --> 00:02:33.942
So we're looking somehow to
draw the vector one, two.

00:02:33.942 --> 00:02:36.617
So you would expect to
see the vector that has

00:02:36.617 --> 00:02:39.928
an x component of one
and a y component of two.

00:02:39.928 --> 00:02:41.151
Something like that.

00:02:41.151 --> 00:02:42.769
But it's probably gonna be scaled down

00:02:42.769 --> 00:02:44.724
because of the way we
usually draw vector fields.

00:02:44.724 --> 00:02:47.978
And the entire field looks like this.

00:02:47.978 --> 00:02:51.139
So I'll go ahead and
erase what I had going on.

00:02:51.139 --> 00:02:53.217
Since this is a little bit clearer.

00:02:53.217 --> 00:02:55.181
And remember, we scaled
down all the vectors.

00:02:55.181 --> 00:02:56.309
The color represents length.

00:02:56.309 --> 00:02:57.577
So red here is super-long.

00:02:57.577 --> 00:02:59.802
Blue is gonna be kind of short.

00:02:59.802 --> 00:03:01.619
And one thing worth noticing.

00:03:01.619 --> 00:03:04.690
If you take a look at all
of the given points around,

00:03:04.690 --> 00:03:07.350
if the vector is crossing a contour line,

00:03:07.350 --> 00:03:09.358
it's perpendicular to that contour line.

00:03:09.358 --> 00:03:10.129
Wherever you go.

00:03:10.129 --> 00:03:10.942
You know, you go down here,

00:03:10.942 --> 00:03:12.737
this vector's perpendicular
to the contour line.

00:03:12.737 --> 00:03:15.115
Over here, perpendicular
to the contour line.

00:03:15.115 --> 00:03:16.782
And this happens everywhere.

00:03:16.782 --> 00:03:18.662
And it's for a very good reason.

00:03:18.662 --> 00:03:20.300
And it's also super-useful.

00:03:20.300 --> 00:03:23.094
So let's just think about
what that reason should be.

00:03:23.094 --> 00:03:25.163
Let's zoom in on a point.

00:03:25.163 --> 00:03:27.576
So I'm gonna clear up our function here.

00:03:27.576 --> 00:03:29.350
Clear up all of the information about it.

00:03:29.350 --> 00:03:32.434
And just zoom in on one of those points.

00:03:34.724 --> 00:03:38.166
So let's say like right here.

00:03:38.166 --> 00:03:39.979
We'll take that guy

00:03:39.979 --> 00:03:41.896
and kind of imagine

00:03:41.896 --> 00:03:44.845
zooming in and saying what's
going on in that region?

00:03:44.845 --> 00:03:47.069
So you've got some kind of contour line.

00:03:47.069 --> 00:03:48.899
And it's swooping down like this.

00:03:48.899 --> 00:03:50.596
And that represents some kind of value.

00:03:50.596 --> 00:03:53.071
Let's say that represents
the value f equals two.

00:03:53.071 --> 00:03:56.272
And, you know, it might not
be a perfect straight line.

00:03:56.272 --> 00:03:57.327
But the more you zoom in,

00:03:57.327 --> 00:03:58.687
the more it looks like a straight line.

00:03:58.687 --> 00:04:01.049
And when you want to
interpret the gradient vector.

00:04:01.049 --> 00:04:03.532
If you remember, in the
video about how to interpret

00:04:03.532 --> 00:04:05.303
the gradient in the context of a graph,

00:04:05.303 --> 00:04:08.016
I said it points in the
direction of steepest descent.

00:04:08.016 --> 00:04:10.541
So if you imagine all the possible vectors

00:04:10.541 --> 00:04:12.777
kind of pointing away from this point,

00:04:12.777 --> 00:04:15.352
the question is, which
direction should you move

00:04:15.352 --> 00:04:18.293
to increase the value of f the fastest?

00:04:18.293 --> 00:04:19.952
And there's two ways
of thinking about that.

00:04:19.952 --> 00:04:22.378
One is to look at all of
these different directions

00:04:22.378 --> 00:04:24.532
and say which one increases x the most?

00:04:24.532 --> 00:04:26.063
But another way of doing it

00:04:26.063 --> 00:04:27.683
would be to get rid of them all

00:04:27.683 --> 00:04:29.637
and just take a look
at another contour line

00:04:29.637 --> 00:04:31.234
that represents a slight increase.

00:04:31.234 --> 00:04:32.101
All right, so let's say

00:04:32.101 --> 00:04:34.548
you're taking a look at a contour line,

00:04:34.548 --> 00:04:35.879
another contour line.

00:04:35.879 --> 00:04:36.786
Something like this.

00:04:36.786 --> 00:04:39.549
And maybe that represents
something that's right next to it.

00:04:39.549 --> 00:04:40.783
Like 2.1.

00:04:40.783 --> 00:04:43.876
That represents, you know,
another value that's very close.

00:04:43.876 --> 00:04:45.379
And if I were a better artist,

00:04:45.379 --> 00:04:47.369
and this was more representative,

00:04:47.369 --> 00:04:48.544
it would look like a line

00:04:48.544 --> 00:04:50.091
that's parallel to the original one.

00:04:50.091 --> 00:04:52.084
Because if you change the output

00:04:52.084 --> 00:04:52.993
by just a little bit,

00:04:52.993 --> 00:04:55.935
the set of in points that look like it

00:04:55.935 --> 00:04:58.853
is pretty much the same but
just shifted over a bit.

00:04:58.853 --> 00:05:00.891
So another way we can think
about the gradient here

00:05:00.891 --> 00:05:03.795
is to say of all of the vectors that move

00:05:03.795 --> 00:05:05.915
from this output of two

00:05:05.915 --> 00:05:07.925
up to the value of 2.1.

00:05:07.925 --> 00:05:09.053
You know, you're looking at

00:05:09.053 --> 00:05:10.839
all of the possible different
vectors that do that.

00:05:11.679 --> 00:05:13.914
You know, which one does it the fastest?

00:05:13.914 --> 00:05:15.728
And this time, instead of
thinking of the fastest

00:05:15.728 --> 00:05:17.266
as constant-length vectors,

00:05:17.266 --> 00:05:18.572
what increases it the most,

00:05:18.572 --> 00:05:21.479
we'll be thinking, constant
increase in the output.

00:05:21.479 --> 00:05:24.255
Which one does it with
the shortest distance?

00:05:24.255 --> 00:05:26.906
And if you think of them as
being roughly parallel lines,

00:05:26.906 --> 00:05:28.446
it shouldn't be hard to convince yourself

00:05:28.446 --> 00:05:30.490
that the shortest distance isn't gonna be,

00:05:30.490 --> 00:05:31.927
you know, any of those.

00:05:31.927 --> 00:05:35.570
It's gonna be the one that connects them

00:05:35.570 --> 00:05:38.029
pretty much perpendicular
to the original line.

00:05:38.029 --> 00:05:39.390
Because if you think about these as lines,

00:05:39.390 --> 00:05:40.363
And the more you zoom in,

00:05:40.363 --> 00:05:42.772
the more they pretty much
look like parallel lines,

00:05:42.772 --> 00:05:44.925
the path that connects one to the other

00:05:44.925 --> 00:05:47.392
is gonna be perpendicular to both of them.

00:05:47.392 --> 00:05:49.681
So because of this
interpretation of the gradient

00:05:49.681 --> 00:05:51.236
as the direction of steepest descent,

00:05:51.236 --> 00:05:52.567
it's a natural consequence

00:05:52.567 --> 00:05:54.258
that every time it's on a contour line,

00:05:54.258 --> 00:05:55.475
wherever you're looking

00:05:55.475 --> 00:05:57.671
it's actually perpendicular to that line.

00:05:57.671 --> 00:05:58.675
Because you can think about it

00:05:58.675 --> 00:06:01.348
as getting to the next contour
line as fast as it can.

00:06:01.348 --> 00:06:03.704
Increasing the function as fast as it can.

00:06:03.704 --> 00:06:06.576
And this is actually a
very useful intepretation

00:06:06.576 --> 00:06:08.184
of the gradient in different contexts.

00:06:08.184 --> 00:06:10.438
So it's a good one to keep
in the back of your mind.

00:06:10.438 --> 00:06:13.560
Gradient is always
perpendicular to contour lines.

00:06:13.560 --> 00:06:14.566
Great.

00:06:14.566 --> 00:06:15.936
See you next video.


Directional derivative, formal definition

WEBVTT
Kind: captions
Language: en

00:00:00.094 --> 00:00:02.438
- [Voiceover] So I have written
here the formal definition

00:00:02.438 --> 00:00:04.187
for the partial derivative of

00:00:04.187 --> 00:00:08.014
a two-variable function
with respect to X, and what

00:00:08.014 --> 00:00:10.327
I wanna do is build up
to the formal definition

00:00:10.327 --> 00:00:13.086
of the directional derivative
of that same function

00:00:13.086 --> 00:00:16.121
in the direction of some
vector V, and you know,

00:00:16.121 --> 00:00:17.555
V with the little thing on top,

00:00:17.555 --> 00:00:20.246
this will be some vector in
the input space, and I have

00:00:20.246 --> 00:00:22.086
another video on the formal definition

00:00:22.086 --> 00:00:23.610
of the partial derivative if you want to

00:00:23.610 --> 00:00:25.940
check that out, and just to
really quickly go through here,

00:00:25.940 --> 00:00:27.227
I've drawn this diagram before,

00:00:27.227 --> 00:00:28.933
but it's worth drawing again, if you

00:00:28.933 --> 00:00:33.200
think of your input space,
which is the X Y plane,

00:00:33.200 --> 00:00:34.917
and you think of it
somehow mapping over to

00:00:34.917 --> 00:00:36.461
the real number line, which is where

00:00:36.461 --> 00:00:38.899
your output F lives,
and when you're taking

00:00:38.899 --> 00:00:41.764
the partial derivative at
a point A B, you're looking

00:00:41.764 --> 00:00:43.612
over here and you say,
maybe that's your point,

00:00:43.612 --> 00:00:47.171
some point A B, and you imagine nudging it

00:00:47.171 --> 00:00:49.128
slightly in the X direction, and saying,

00:00:49.128 --> 00:00:51.285
hey, how does that influence the function?

00:00:51.285 --> 00:00:53.407
So, maybe this is where
A B lands, and maybe

00:00:53.407 --> 00:00:55.572
the result is a nudge that's
a little bit negative.

00:00:55.572 --> 00:00:57.782
That would be a negative
partial derivative, and you

00:00:57.782 --> 00:01:01.379
think of the size of that nudge
as partial X, and the size

00:01:01.379 --> 00:01:04.730
of the resulting nudge in the
output space as partial F.

00:01:04.730 --> 00:01:07.434
So, the way that you read
this formal definition

00:01:07.434 --> 00:01:09.616
is you think of this
variable H, you know, people,

00:01:09.616 --> 00:01:12.420
you could say delta X, but H seems to be

00:01:12.420 --> 00:01:15.020
the common variable people
use, you think of it

00:01:15.020 --> 00:01:17.093
as that change in your input space,

00:01:17.093 --> 00:01:21.314
that slight nudge, and you
look at how that influences

00:01:21.314 --> 00:01:24.183
the function when you only
change the X component here,

00:01:24.183 --> 00:01:25.789
you know, you're only
changing the X component

00:01:25.789 --> 00:01:28.288
with that nudge, and you
say what's the change in F?

00:01:28.288 --> 00:01:30.022
What's that partial F?

00:01:30.022 --> 00:01:32.412
So, I'm gonna write this in
a slightly different way,

00:01:32.412 --> 00:01:34.323
using vector notation.

00:01:34.323 --> 00:01:37.241
Instead I'm gonna say,
you know, partial F,

00:01:37.241 --> 00:01:41.566
partial X, and instead of
saying the input is A B,

00:01:41.566 --> 00:01:44.816
I'm gonna say it's a, you know,

00:01:44.816 --> 00:01:46.996
just A, and then make it clear that that's

00:01:46.996 --> 00:01:49.315
a vector, and this will be
a two-dimensional vector,

00:01:49.315 --> 00:01:51.169
so I'll put that little arrow on top

00:01:51.169 --> 00:01:53.948
to indicate that it's a
vector, and if we rewrite

00:01:53.948 --> 00:01:56.211
this definition, we'd be thinking

00:01:56.211 --> 00:01:58.628
the limit, as H goes to zero,

00:02:01.263 --> 00:02:02.096
of

00:02:03.476 --> 00:02:05.711
something divided by H, but that thing,

00:02:05.711 --> 00:02:06.855
now that we're writing in terms of

00:02:06.855 --> 00:02:09.688
vector notation, is gonna be F of,

00:02:11.471 --> 00:02:14.051
so it's gonna be our
original starting point A,

00:02:14.051 --> 00:02:15.636
but plus what?

00:02:15.636 --> 00:02:17.322
I mean, up here, it was
clear we could just add it

00:02:17.322 --> 00:02:19.354
to the first component, but
if I'm not writing in terms

00:02:19.354 --> 00:02:21.403
of components, and I
have to think in terms of

00:02:21.403 --> 00:02:24.566
vector addition, really what I'm adding

00:02:24.566 --> 00:02:28.534
is that H times the
vector, the unit vector

00:02:28.534 --> 00:02:30.411
in the X direction, and it's common

00:02:30.411 --> 00:02:32.605
to use, you know, this little I with a hat

00:02:32.605 --> 00:02:37.371
to represent the unit
vector in the X direction.

00:02:37.371 --> 00:02:39.441
So when I'm adding these,
it's really the same.

00:02:39.441 --> 00:02:40.884
You know, this H is only gonna go to

00:02:40.884 --> 00:02:43.810
that first component, and the
second component is multiplied

00:02:43.810 --> 00:02:46.560
by zero, and what we subtract off

00:02:47.745 --> 00:02:49.101
is the value of the function at

00:02:49.101 --> 00:02:51.496
that original input, that original

00:02:51.496 --> 00:02:53.692
two-dimensional input
that I'm just thinking of

00:02:53.692 --> 00:02:56.746
as a vector here, and
when I write it like this,

00:02:56.746 --> 00:02:59.428
it's actually much clearer
how we might extend

00:02:59.428 --> 00:03:02.357
this idea to moving in
different directions.

00:03:02.357 --> 00:03:04.653
'Cause now, all of the information

00:03:04.653 --> 00:03:06.305
about what direction you're moving

00:03:06.305 --> 00:03:09.505
is captured with this vector here,

00:03:09.505 --> 00:03:13.410
what you multiply your nudge
by as you're adding the input.

00:03:13.410 --> 00:03:15.646
So let's just rewrite that over here

00:03:15.646 --> 00:03:17.615
in the context of directional derivative.

00:03:17.615 --> 00:03:20.051
What you would say is that
the directional derivative

00:03:20.051 --> 00:03:23.407
in the direction of
some vector, any vector,

00:03:23.407 --> 00:03:26.784
of F, evaluated at a point,
and we'll think about

00:03:26.784 --> 00:03:30.064
that input point as
being a vector itself, A.

00:03:30.064 --> 00:03:32.326
Here, I'll get rid of this guy.

00:03:32.326 --> 00:03:35.553
It's also gonna be a limit, and as always,

00:03:35.553 --> 00:03:37.579
with these things, we think of some,

00:03:37.579 --> 00:03:39.420
not, I mean, always, but with derivatives,

00:03:39.420 --> 00:03:43.964
you think of some variable
as going to zero, and then

00:03:43.964 --> 00:03:45.890
that's gonna be on the
denominator, and the change

00:03:45.890 --> 00:03:47.950
in the function that we're looking for

00:03:47.950 --> 00:03:52.639
is gonna be F, evaluated at
that initial input vector

00:03:52.639 --> 00:03:56.806
plus H, that scaling value,
that little nudge of a value,

00:03:57.942 --> 00:04:01.476
multiplied by the vector whose direction

00:04:01.476 --> 00:04:03.691
we care about, and then you subtract off

00:04:03.691 --> 00:04:06.858
the value of F at that original input.

00:04:08.717 --> 00:04:12.130
So, this right here is
the formal definition

00:04:12.130 --> 00:04:14.697
for the directional
derivative, and you see how

00:04:14.697 --> 00:04:16.695
it's much easier to
write in vector notation,

00:04:16.695 --> 00:04:18.381
because you're thinking of your input

00:04:18.381 --> 00:04:22.008
as a vector and your output as
just some nudge by something.

00:04:22.008 --> 00:04:23.226
So, let's take a look at what that

00:04:23.226 --> 00:04:24.567
would feel like over here.

00:04:24.567 --> 00:04:27.066
You know, instead of
thinking of D X and a nudge

00:04:27.066 --> 00:04:31.236
purely in the X direction,
and I'll erase these guys,

00:04:31.236 --> 00:04:34.335
you would think of this point as being A,

00:04:34.335 --> 00:04:36.835
as being a vector valued A, so

00:04:38.148 --> 00:04:40.995
just to make clear how it's a vector,

00:04:40.995 --> 00:04:42.677
you'd be thinking of it starting at

00:04:42.677 --> 00:04:43.835
the origin, and the tip represents

00:04:43.835 --> 00:04:47.052
that point, and then H times V, you know,

00:04:47.052 --> 00:04:49.552
maybe V is some vector, often,

00:04:51.364 --> 00:04:52.673
you know, a direction that's neither

00:04:52.673 --> 00:04:55.386
purely X nor purely Y, but when you

00:04:55.386 --> 00:04:59.874
scale it down, it'll just
be a tiny little nudge

00:04:59.874 --> 00:05:03.417
that's gonna be H, that tiny little value,

00:05:03.417 --> 00:05:06.804
scaling your vector V, so that
tiny little nudge, and what

00:05:06.804 --> 00:05:09.648
you wonder is, hey,
what's the resulting nudge

00:05:09.648 --> 00:05:10.786
to the output?

00:05:10.786 --> 00:05:12.696
And the ratio between the size of that

00:05:12.696 --> 00:05:15.917
resulting nudge to the output
and the original guy there

00:05:15.917 --> 00:05:18.151
is your directional derivative,
and more importantly,

00:05:18.151 --> 00:05:20.207
as you take the limit
for that original nudge

00:05:20.207 --> 00:05:22.163
getting really really
small, that's gonna be

00:05:22.163 --> 00:05:24.679
your directional derivative,
and you can probably anticipate

00:05:24.679 --> 00:05:27.812
there's a way to interpret
this as the slope of a graph.

00:05:27.812 --> 00:05:29.437
That's what I'm gonna
talk about next video,

00:05:29.437 --> 00:05:31.627
but you actually have to
be a little bit careful,

00:05:31.627 --> 00:05:34.294
because we call this the
directional derivative,

00:05:34.294 --> 00:05:37.889
but notice, if you scale
the value V by two,

00:05:37.889 --> 00:05:40.123
you know, if you go over here
and you start plugging in

00:05:40.123 --> 00:05:42.909
two times V and seeing how
that influences things,

00:05:42.909 --> 00:05:47.070
it'll be twice the change, because here,

00:05:47.070 --> 00:05:49.979
even if you're scaling
by the same value H,

00:05:49.979 --> 00:05:54.692
it's gonna double the initial
nudge that you had, and it's

00:05:54.692 --> 00:05:56.623
gonna double the resulting nudge out here,

00:05:56.623 --> 00:05:59.101
even though the denominator
H doesn't stay changed.

00:05:59.101 --> 00:06:01.458
So when you're taking the
ratio, what you're considering

00:06:01.458 --> 00:06:03.063
is the size of your initial nudge

00:06:03.063 --> 00:06:04.923
actually might be influenced.

00:06:04.923 --> 00:06:06.589
So, some authors, they'll actually change

00:06:06.589 --> 00:06:08.702
this definition, and
they'll throw a little

00:06:08.702 --> 00:06:12.214
absolute value of the original
vector, just to make sure

00:06:12.214 --> 00:06:14.660
that when you scale it by
something else, it doesn't

00:06:14.660 --> 00:06:16.984
influence things, and you
only care about the direction.

00:06:16.984 --> 00:06:18.312
But, I actually don't like that.

00:06:18.312 --> 00:06:21.240
I think there's some
usefulness in the definition

00:06:21.240 --> 00:06:23.798
as it is right here,
and that there's kind of

00:06:23.798 --> 00:06:25.128
a good interpretation to be had,

00:06:25.128 --> 00:06:27.982
for when, if you double
the size of your vector,

00:06:27.982 --> 00:06:31.762
why that should double the
size of your derivative,

00:06:31.762 --> 00:06:33.367
but I'll get to that in following videos.

00:06:33.367 --> 00:06:35.419
This right here is the formal definition

00:06:35.419 --> 00:06:37.940
to be thinking about, and
I'll see you next video.


Formal definition of partial derivatives

WEBVTT
Kind: captions
Language: en

00:00:00.002 --> 00:00:00.915
- [Voiceover] So, I've talked about

00:00:00.915 --> 00:00:04.266
the partial derivative
and how you compute it,

00:00:04.266 --> 00:00:06.223
how you interpret in terms of graphs,

00:00:06.223 --> 00:00:09.070
but what I'd like to do here
is give its formal definition.

00:00:09.070 --> 00:00:10.771
So, it's the kind of
thing, just to remind you,

00:00:10.771 --> 00:00:14.294
that applies to a function that
has a multi-variable input.

00:00:14.294 --> 00:00:16.401
So, X, Y, and you know,
I'll emphasize that

00:00:16.401 --> 00:00:19.153
it could actually be a
number of other inputs.

00:00:19.153 --> 00:00:21.177
You could have 100 inputs, or something

00:00:21.177 --> 00:00:24.006
like that, and as with
a lot of things here,

00:00:24.006 --> 00:00:25.233
I think it's helpful to take a look at

00:00:25.233 --> 00:00:28.058
the one dimensional
analogy and think about

00:00:28.058 --> 00:00:32.169
how we define the derivative,
just the ordinary derivative,

00:00:32.169 --> 00:00:34.252
when you have a function
that's just one variable.

00:00:34.252 --> 00:00:36.455
You know, this would be
just something simple,

00:00:36.455 --> 00:00:38.967
F of X, and you know, if you're thinking

00:00:38.967 --> 00:00:42.437
in the back of your mind that
it's a function like F of X

00:00:42.437 --> 00:00:45.212
equals X squared, and
the way to think about

00:00:45.212 --> 00:00:47.647
the definition of this is
to just actually spell out

00:00:47.647 --> 00:00:49.832
how we interpret this
D F and D X, and then

00:00:49.832 --> 00:00:52.851
slowly start to tighten it
up into a formalization.

00:00:52.851 --> 00:00:55.126
So, you might be thinking of
the graph of this function.

00:00:55.126 --> 00:00:57.655
You know, maybe it's some
kind of curve, and when you

00:00:57.655 --> 00:00:59.739
think of evaluating it
at some point, you know,

00:00:59.739 --> 00:01:02.793
let's say you're
evaluating it at a point A,

00:01:02.793 --> 00:01:07.103
you're imagining D X here as
representing a slight nudge,

00:01:07.103 --> 00:01:09.621
just a slight nudge in the input value.

00:01:09.621 --> 00:01:11.369
So, this is in the X direction.

00:01:11.369 --> 00:01:14.079
Got your X coordinate, your output,

00:01:14.079 --> 00:01:17.687
F of X is what the Y axis
represents here, and then

00:01:17.687 --> 00:01:21.568
you're thinking of D F as
being the resulting nudge here,

00:01:21.568 --> 00:01:23.832
the resulting change to the function.

00:01:23.832 --> 00:01:26.281
So, when we formalize this,
we're gonna be thinking of

00:01:26.281 --> 00:01:30.434
a fraction that's gonna
represent D F over D X, and I'll

00:01:30.434 --> 00:01:31.594
leave myself some room.

00:01:31.594 --> 00:01:34.328
You can probably anticipate
why if you know where

00:01:34.328 --> 00:01:38.370
this is going, and instead
of saying D X, I'll say H.

00:01:38.370 --> 00:01:41.284
So instead of thinking
D X is that tiny nudge,

00:01:41.284 --> 00:01:45.243
you'll think H, and I'm not
sure why H is used necessarily,

00:01:45.243 --> 00:01:47.379
but just having some
kind of variable that you

00:01:47.379 --> 00:01:49.637
think of as getting
small, maybe all the other

00:01:49.637 --> 00:01:51.224
letters in the alphabet were taken.

00:01:51.224 --> 00:01:53.024
Now, when you actually
say, what do we mean by

00:01:53.024 --> 00:01:56.565
the resulting change in
F, we should be writing,

00:01:56.565 --> 00:02:00.108
as well, where does it go after you nudge?

00:02:00.108 --> 00:02:02.590
So, when you take, you
know, from that input point,

00:02:02.590 --> 00:02:05.077
plus that nudge, plus that little H,

00:02:05.077 --> 00:02:07.353
what's the difference
between that and the original

00:02:07.353 --> 00:02:10.003
function, or the original value of

00:02:10.003 --> 00:02:11.855
the function, at that point?

00:02:11.855 --> 00:02:15.201
So, this top part is really
what's representing D F.

00:02:15.201 --> 00:02:19.194
You know, this is what's
representing D F over here,

00:02:19.194 --> 00:02:21.633
but you don't do this for
any actual value of H.

00:02:21.633 --> 00:02:24.656
You don't do it for any specific nudge.

00:02:24.656 --> 00:02:27.074
Largely, the whole point
of calculus is that you're

00:02:27.074 --> 00:02:31.702
considering the limit as
H goes to zero of this,

00:02:31.702 --> 00:02:35.078
and this is what makes
concrete the idea of, you know,

00:02:35.078 --> 00:02:37.496
a tiny little nudge or a
tiny little resulting change.

00:02:37.496 --> 00:02:39.548
It's not that it's any specific one.

00:02:39.548 --> 00:02:41.586
You're taking the limit,
and you know, you could

00:02:41.586 --> 00:02:43.428
get into the formal definition of a limit,

00:02:43.428 --> 00:02:45.602
but it gives you room for rigor as soon as

00:02:45.602 --> 00:02:47.219
you start writing something like this.

00:02:47.219 --> 00:02:52.004
Now, over in the multi-variable
world, very similar story.

00:02:52.004 --> 00:02:54.491
We can pretty much do the
same thing, and we're gonna

00:02:54.491 --> 00:02:56.585
look at our original
fraction, and just start

00:02:56.585 --> 00:02:59.165
to formalize what we think of each of

00:02:59.165 --> 00:03:01.212
these variables as representing.

00:03:01.212 --> 00:03:03.940
That partial X, still it's
common to use the letter H,

00:03:03.940 --> 00:03:05.269
just to represent a tiny nudge in

00:03:05.269 --> 00:03:07.459
the X direction, and now if we think about

00:03:07.459 --> 00:03:09.881
what is that nudge, and here, let me

00:03:09.881 --> 00:03:11.303
draw it out, actually.

00:03:11.303 --> 00:03:13.607
The way that I kinda
like to draw this out is

00:03:13.607 --> 00:03:16.601
you think of your entire input space as,

00:03:16.601 --> 00:03:17.639
you know, the X Y plane.

00:03:17.639 --> 00:03:18.927
If it was more variables, this would be

00:03:18.927 --> 00:03:20.606
a high dimensional space,
and you're thinking

00:03:20.606 --> 00:03:22.905
at some point, you know,
maybe you're thinking of it

00:03:22.905 --> 00:03:26.044
as A B, or maybe I should specify that,

00:03:26.044 --> 00:03:28.172
actually, where we're doing this

00:03:28.172 --> 00:03:31.339
at a specific point how you define it.

00:03:32.320 --> 00:03:34.756
We're doing this at a very specific point,

00:03:34.756 --> 00:03:38.128
A B, and when you're
thinking of your tiny little

00:03:38.128 --> 00:03:40.505
change in X, you'd be thinking, you know,

00:03:40.505 --> 00:03:43.258
a tiny little nudge in the X direction,

00:03:43.258 --> 00:03:46.307
a tiny little shift there,
and the entire function

00:03:46.307 --> 00:03:49.119
maps that input space, whatever it is, to

00:03:49.119 --> 00:03:50.501
the real number line.

00:03:50.501 --> 00:03:52.975
This is your output
space, and you're saying,

00:03:52.975 --> 00:03:55.726
hey, how does that tiny
nudge influence the output?

00:03:55.726 --> 00:03:58.376
I've drawn this diagram
a lot, this loose sketch.

00:03:58.376 --> 00:03:59.446
I think it's actually a pretty good model,

00:03:59.446 --> 00:04:00.710
because once we start thinking of

00:04:00.710 --> 00:04:02.577
higher dimensional outputs
or things like that,

00:04:02.577 --> 00:04:04.594
it's pretty flexible, and
you're thinking of this

00:04:04.594 --> 00:04:06.849
as your partial X, your changing

00:04:06.849 --> 00:04:09.127
the X direction, and
this is that resulting

00:04:09.127 --> 00:04:10.953
change for the function.

00:04:10.953 --> 00:04:12.367
But, we go back up here, and we say,

00:04:12.367 --> 00:04:13.595
well what does that mean, right?

00:04:13.595 --> 00:04:16.879
If H represents that tiny
change to your X value,

00:04:16.879 --> 00:04:21.729
well then you have to evaluate
the function at the point A,

00:04:21.729 --> 00:04:24.853
but plus that H, and you're adding it

00:04:24.853 --> 00:04:26.811
to the X value, that first component,

00:04:26.811 --> 00:04:28.736
just because this is
the partial derivative

00:04:28.736 --> 00:04:31.629
with respect to X, and the point B

00:04:31.629 --> 00:04:33.689
just remains unchanged, right?

00:04:33.689 --> 00:04:36.396
So, this is you evaluating it, kind of, at

00:04:36.396 --> 00:04:37.971
the new point, and you have to say,

00:04:37.971 --> 00:04:40.073
what's the difference
between that and the old

00:04:40.073 --> 00:04:44.233
evaluation, where it was just at A and B.

00:04:44.233 --> 00:04:45.191
And that's it.

00:04:45.191 --> 00:04:49.623
That's the formal definition
of your partial derivative,

00:04:49.623 --> 00:04:51.767
except, oh, the most
important part, right?

00:04:51.767 --> 00:04:53.305
The most important part, given that

00:04:53.305 --> 00:04:55.743
this is calculus is that
we're not doing this

00:04:55.743 --> 00:04:59.743
for any specific value
of H, but we're actually,

00:05:01.415 --> 00:05:02.853
let me just move this guy.

00:05:02.853 --> 00:05:05.482
Give a little bit of room here.

00:05:05.482 --> 00:05:06.515
Yes.

00:05:06.515 --> 00:05:07.973
But, we're actually taking the limit

00:05:07.973 --> 00:05:10.556
here, limit, as H goes to zero,

00:05:13.459 --> 00:05:15.612
and what this means is you're
not considering any specific

00:05:15.612 --> 00:05:17.981
size of D X, any specific size of this.

00:05:17.981 --> 00:05:20.703
Really, this is H, considering
the notation up here,

00:05:20.703 --> 00:05:22.398
but any size for that partial X.

00:05:22.398 --> 00:05:24.392
You're imagining that nudge shrinking

00:05:24.392 --> 00:05:26.706
more and more and more,
and the resulting change

00:05:26.706 --> 00:05:28.123
shrinks more and more and more, and you're

00:05:28.123 --> 00:05:31.390
wondering what the ratio
between them approaches.

00:05:31.390 --> 00:05:32.829
So, that would be the partial derivative

00:05:32.829 --> 00:05:35.028
with respect to X, and just for practice,

00:05:35.028 --> 00:05:36.872
let's actually write out
what the partial derivative

00:05:36.872 --> 00:05:38.367
with respect to Y would be.

00:05:38.367 --> 00:05:39.660
So, we'll get rid of some of this

00:05:39.660 --> 00:05:41.952
one dimensional analogy stuff here.

00:05:41.952 --> 00:05:44.349
Don't need that anymore,
and let's just think about

00:05:44.349 --> 00:05:47.660
what the partial
derivative with respect to

00:05:47.660 --> 00:05:49.298
a different variable would be.

00:05:49.298 --> 00:05:51.553
So, if we were doing it
as partial derivative

00:05:51.553 --> 00:05:55.800
of F with respect to Y,
now we're nudging slightly

00:05:55.800 --> 00:05:57.000
in the other direction, right?

00:05:57.000 --> 00:05:59.413
We're nudging in the Y
direction, and you'd be thinking,

00:05:59.413 --> 00:06:01.842
okay, so we're still
gonna divide something

00:06:01.842 --> 00:06:03.610
by that nudge, and again I'm
just using the same variable.

00:06:03.610 --> 00:06:05.892
Maybe it would be clearer
to write something like

00:06:05.892 --> 00:06:08.364
the change in Y, or to
go up here and write

00:06:08.364 --> 00:06:09.897
something like, you know, the change

00:06:09.897 --> 00:06:12.912
in X, and people will do
that, but it's less common.

00:06:12.912 --> 00:06:14.477
I think people just kinda want

00:06:14.477 --> 00:06:17.600
the standard go-to limiting variable.

00:06:17.600 --> 00:06:18.923
But, this time when you're considering

00:06:18.923 --> 00:06:21.840
what is the resulting
change, oh, and again,

00:06:21.840 --> 00:06:24.998
I always forget to write
in we're evaluating this

00:06:24.998 --> 00:06:28.748
at a specific point, at
a specific point A B,

00:06:34.542 --> 00:06:35.694
and as a result, maybe I'll give myself

00:06:35.694 --> 00:06:38.335
a little bit more room here.

00:06:38.335 --> 00:06:41.847
So, we're taking this whole thing,

00:06:41.847 --> 00:06:45.040
dividing by H, but what is
the resulting change in F?

00:06:45.040 --> 00:06:49.310
This time you say F, the new
value is still gonna be at A,

00:06:49.310 --> 00:06:51.941
but the change happens
for that second variable.

00:06:51.941 --> 00:06:54.508
It's gonna be that B, B plus H.

00:06:54.508 --> 00:06:56.060
So, you're adding that nudge to

00:06:56.060 --> 00:06:58.835
the Y value, and as
before, you subtract off.

00:06:58.835 --> 00:07:00.774
You see the difference
between that and how you

00:07:00.774 --> 00:07:04.095
evaluate it at the
original point, and again,

00:07:04.095 --> 00:07:06.194
the whole reason I move
this over and give myself

00:07:06.194 --> 00:07:09.218
some room is because
we're taking the limit,

00:07:09.218 --> 00:07:12.637
as this H goes to zero, and the way that

00:07:12.637 --> 00:07:14.058
you're thinking about
this is very similar.

00:07:14.058 --> 00:07:15.746
It's just that when you change the input

00:07:15.746 --> 00:07:19.301
by adding H to the Y value,
you're shifting it upwards.

00:07:19.301 --> 00:07:22.532
So, again, this is the partial derivative,

00:07:22.532 --> 00:07:24.797
the formal definition of
the partial derivative.

00:07:24.797 --> 00:07:26.624
Looks very similar to
the formal definition of

00:07:26.624 --> 00:07:30.267
the derivative, but I just
always think about this

00:07:30.267 --> 00:07:32.325
as spelling out what we mean by

00:07:32.325 --> 00:07:35.031
partial Y and partial F,
and kinda spelling out

00:07:35.031 --> 00:07:36.876
why it is that the Leibniz's came up

00:07:36.876 --> 00:07:38.757
with this notation in the first place.

00:07:38.757 --> 00:07:40.858
Well, I don't know if Leibniz
came up with the partials,

00:07:40.858 --> 00:07:44.548
but the D F, D X portion, and this is good

00:07:44.548 --> 00:07:45.829
to keep in the back of your mind,

00:07:45.829 --> 00:07:47.697
especially as we introduce new notions,

00:07:47.697 --> 00:07:50.155
new types of multi-variable derivatives,

00:07:50.155 --> 00:07:51.627
like the directional derivative.

00:07:51.627 --> 00:07:54.227
 I think it helps clarify
what's really go on

00:07:54.227 --> 00:07:55.454
in certain contexts.

00:07:55.454 --> 00:07:56.380
Great.

00:07:56.380 --> 00:07:57.213
See ya next video.


Partial derivatives and graphs

WEBVTT
Kind: captions
Language: en

00:00:01.139 --> 00:00:02.370
- [Voiceover] Hello everyone.

00:00:02.370 --> 00:00:05.504
So I have here the graph
of a two-variable function

00:00:05.504 --> 00:00:07.385
and I'd like to talk about
how you can interpret

00:00:07.385 --> 00:00:09.428
the partial derivative of that function.

00:00:09.428 --> 00:00:11.253
So specifically, the function

00:00:11.253 --> 00:00:14.170
that you're looking at is f of x, y

00:00:15.654 --> 00:00:18.786
is equal to x squared times y

00:00:18.786 --> 00:00:20.036
plus sine of y.

00:00:21.804 --> 00:00:23.732
And the question is, if I
take the partial derivative

00:00:23.732 --> 00:00:26.263
of this function, so maybe I'm looking at

00:00:26.263 --> 00:00:30.024
the partial derivative
of f with respect to x,

00:00:30.024 --> 00:00:32.927
and let's say I want to do
this at negative one, one

00:00:32.927 --> 00:00:34.599
so I'll be looking at
the partial derivative

00:00:34.599 --> 00:00:35.899
at a specific point.

00:00:35.899 --> 00:00:38.425
How do you interpret
that on this whole graph?

00:00:38.425 --> 00:00:41.495
First, let's consider where
the point negative one, one is.

00:00:41.495 --> 00:00:42.702
If we're looking above,

00:00:42.702 --> 00:00:45.234
this is our x-axis, this is our y-axis

00:00:45.234 --> 00:00:46.557
the point negative one, one

00:00:46.557 --> 00:00:47.579
is sitting right there.

00:00:47.579 --> 00:00:49.367
So negative one, move up one

00:00:49.367 --> 00:00:52.037
and it's the point that's
sitting on the graph.

00:00:52.037 --> 00:00:53.580
And the first thing you might do

00:00:53.580 --> 00:00:55.636
is you say well, when we're
taking the partial derivative

00:00:55.636 --> 00:00:57.552
with respect to x, we're going to pretend

00:00:57.552 --> 00:00:59.096
that y is a constant

00:00:59.096 --> 00:01:02.207
so let's actually just go
ahead and evaluate that.

00:01:02.207 --> 00:01:04.761
When you're doing this,

00:01:04.761 --> 00:01:06.898
x squared looks like a variable,

00:01:06.898 --> 00:01:08.267
y looks like a constant,

00:01:08.267 --> 00:01:10.856
sine of y also looks like a constant.

00:01:10.856 --> 00:01:12.470
So this is going to be...

00:01:12.470 --> 00:01:14.560
We differentiate x squared

00:01:14.560 --> 00:01:17.927
and that's two times x times
y which is like a constant,

00:01:17.927 --> 00:01:20.551
and then the derivative of
a constant there is zero

00:01:20.551 --> 00:01:22.966
and we're evaluating this whole thing

00:01:22.966 --> 00:01:27.133
at x is equal to negative
one and y is equal to one.

00:01:28.198 --> 00:01:30.153
So when we actually plug that in,

00:01:30.153 --> 00:01:34.320
it would be two times negative
one multiplied by one,

00:01:35.528 --> 00:01:39.150
which is two... Negative two, excuse me.

00:01:39.150 --> 00:01:40.218
But what does that mean, right?

00:01:40.218 --> 00:01:42.517
We evaluate this, and
maybe you're thinking

00:01:42.517 --> 00:01:44.375
this is kind of slight
nudge in the x direction,

00:01:44.375 --> 00:01:45.930
this is the resulting nudge of f.

00:01:45.930 --> 00:01:47.370
What does that mean for the graph?

00:01:47.370 --> 00:01:49.553
Well first of all,
treating y as a constant

00:01:49.553 --> 00:01:52.084
is basically like slicing the whole graph

00:01:52.084 --> 00:01:54.800
with a plane that represents
a constant y value.

00:01:54.800 --> 00:01:56.124
So this is the y-axis,

00:01:56.124 --> 00:01:58.190
and the plane that cuts it perpendicularly

00:01:58.190 --> 00:02:00.559
that represents a constant y value.

00:02:00.559 --> 00:02:02.835
This one represents the
constant y value one

00:02:02.835 --> 00:02:06.875
but you could imagine sliding
the plane back and forth

00:02:06.875 --> 00:02:10.237
and that would represent
various different y values.

00:02:10.237 --> 00:02:12.378
So for the general partial derivative,

00:02:12.378 --> 00:02:14.479
you can imagine whichever one you want

00:02:14.479 --> 00:02:16.465
but this one is y equals one

00:02:16.465 --> 00:02:19.344
and I'll go ahead and slice
the actual graph at that point

00:02:19.344 --> 00:02:21.011
and draw a red line.

00:02:22.241 --> 00:02:23.241
And this red line is basically

00:02:23.241 --> 00:02:26.941
all the points on the graph
where y is equal to one.

00:02:26.941 --> 00:02:31.820
So I'll emphasize that...
where y is equal to one.

00:02:31.820 --> 00:02:33.987
This is y is equal to one.

00:02:35.808 --> 00:02:37.371
So, when we're looking at that

00:02:37.371 --> 00:02:40.102
we can actually interpret the
partial derivative as a slope

00:02:40.102 --> 00:02:42.703
because we're looking at the point here,

00:02:42.703 --> 00:02:44.267
we're asking how the function changes

00:02:44.267 --> 00:02:46.279
as we move in the x direction.

00:02:46.279 --> 00:02:48.353
From single variable calculus,
you might be familiar

00:02:48.353 --> 00:02:50.621
with thinking of that
as the slope of a line

00:02:50.621 --> 00:02:52.734
and to be a little more
concrete about this,

00:02:52.734 --> 00:02:54.411
I could say you're starting here,

00:02:54.411 --> 00:02:58.075
you consider some nudge over
there, just some tiny step.

00:02:58.075 --> 00:02:59.538
I'm drawing it as a sizable one

00:02:59.538 --> 00:03:02.835
but you imagine it as a
really small step, as your dx,

00:03:02.835 --> 00:03:05.250
and then the distance
to your function here

00:03:05.250 --> 00:03:07.850
the change in the value
of your function...

00:03:07.850 --> 00:03:12.332
I said dx, but I should say
partial x or del x... Partial f.

00:03:12.332 --> 00:03:14.765
And as that tiny nudge
gets smaller and smaller,

00:03:14.765 --> 00:03:16.720
this change here is going to correspond

00:03:16.720 --> 00:03:19.544
with what the tangent
line does, and that's why

00:03:19.544 --> 00:03:22.665
you have this rise over
run feeling for the slope.

00:03:22.665 --> 00:03:24.831
And you look at that
value, and the line itself

00:03:24.831 --> 00:03:26.937
looks like it has a slope
of about negative two

00:03:26.937 --> 00:03:28.728
so it should actually
make sense that we get

00:03:28.728 --> 00:03:32.742
negative two over here
given what we're looking at.

00:03:32.742 --> 00:03:33.883
But let's do this

00:03:33.883 --> 00:03:35.933
with the partial derivative
with respect to y.

00:03:35.933 --> 00:03:39.266
Let's erase what we've got going on here

00:03:40.520 --> 00:03:43.795
and I'll go ahead and move
the graph back to what it was,

00:03:43.795 --> 00:03:45.931
get rid of these guys,

00:03:45.931 --> 00:03:48.439
so now we're no longer
slicing with respect to y,

00:03:48.439 --> 00:03:52.606
but instead let's say we slice
it with a constant x value.

00:03:53.570 --> 00:03:55.778
So this here is the x-axis;
this plane represents

00:03:55.778 --> 00:03:58.795
the constant value x equals negative one

00:03:58.795 --> 00:04:01.581
and we can slice the graph there.

00:04:01.581 --> 00:04:03.787
Kind of slice it, I'll
draw the red line again

00:04:03.787 --> 00:04:05.529
that represents the curve

00:04:05.529 --> 00:04:08.245
and this time, that curve represents

00:04:08.245 --> 00:04:11.245
that value x equals negative one.

00:04:12.541 --> 00:04:13.702
It's all the points on the graph

00:04:13.702 --> 00:04:15.583
where x equals negative one.

00:04:15.583 --> 00:04:17.464
And now when we take
the partial derivative,

00:04:17.464 --> 00:04:19.391
we're going to interpret it as a slice...

00:04:19.391 --> 00:04:22.474
As the slope of this resulting curve.

00:04:24.170 --> 00:04:26.148
So that slope ends up looking like this,

00:04:26.148 --> 00:04:28.097
that's our blue line, and
let's go ahead and evaluate

00:04:28.097 --> 00:04:31.442
the partial derivative
of f with respect to y.

00:04:31.442 --> 00:04:34.692
So I'll go over here,
use a different color

00:04:34.692 --> 00:04:38.859
so the partial derivative of f
with respect to y, partial y.

00:04:39.871 --> 00:04:42.732
So we go up here, and it says,
okay. So x squared times y.

00:04:42.732 --> 00:04:45.305
It's considering x squared
to be a constant now.

00:04:45.305 --> 00:04:47.162
So it looks at that and
says x, you're a constant,

00:04:47.162 --> 00:04:49.600
y, you're the variable,
constant times a variable,

00:04:49.600 --> 00:04:53.083
the derivative is just
equal to that constant.

00:04:53.083 --> 00:04:54.833
So that x squared.

00:04:56.040 --> 00:04:58.029
And over here, sine of y,

00:04:58.029 --> 00:05:00.003
the derivative of that with respect to y

00:05:00.003 --> 00:05:01.003
is cosine y.

00:05:02.111 --> 00:05:02.944
Cosine y.

00:05:05.065 --> 00:05:06.388
And if we actually want to evaluate this

00:05:06.388 --> 00:05:08.037
at our point negative one, one

00:05:08.037 --> 00:05:11.120
what we'd get is negative one squared

00:05:12.053 --> 00:05:13.636
plus cosine of one.

00:05:15.676 --> 00:05:17.789
And I'm not sure what the cosine of one is

00:05:17.789 --> 00:05:19.461
but it's something a little bit positive,

00:05:19.461 --> 00:05:21.551
and the ultimate result that we see here

00:05:21.551 --> 00:05:24.105
is going to be one plus something,

00:05:24.105 --> 00:05:26.566
I don't know what it is,
but it's something positive,

00:05:26.566 --> 00:05:28.019
and that should make sense
'cause we look at the slope here

00:05:28.019 --> 00:05:29.604
and it's a little bit more than one.

00:05:29.604 --> 00:05:31.829
I'm not sure exactly, but it's
a little bit more than one.

00:05:31.829 --> 00:05:33.068
So you often hear about people

00:05:33.068 --> 00:05:34.693
talking about the partial derivative

00:05:34.693 --> 00:05:37.456
as being the slope of
the slice of a graph.

00:05:37.456 --> 00:05:38.647
Which is great,

00:05:38.647 --> 00:05:40.064
if you're looking at a function that has

00:05:40.064 --> 00:05:42.395
a two-variable input and
a one-variable output,

00:05:42.395 --> 00:05:44.399
so that we can think about its graph.

00:05:44.399 --> 00:05:47.116
And in other contexts,
that might not be the case.

00:05:47.116 --> 00:05:49.926
Maybe it's something that has
a multidimensional output,

00:05:49.926 --> 00:05:50.947
we'll talk about that later,

00:05:50.947 --> 00:05:52.341
when you have a vector-valued function,

00:05:52.341 --> 00:05:54.245
what its partial derivative looks like,

00:05:54.245 --> 00:05:57.449
but maybe it's also something
that has a hundred inputs

00:05:57.449 --> 00:05:59.539
and you certainly can't
visualize the graph

00:05:59.539 --> 00:06:01.187
but the general idea of saying,

00:06:01.187 --> 00:06:03.403
"Well, if you take a tiny
step in a direction"--

00:06:03.403 --> 00:06:04.438
here, I'll actually walk through it

00:06:04.438 --> 00:06:06.272
in this graph's context again.

00:06:06.272 --> 00:06:08.571
You're looking at your point here

00:06:08.571 --> 00:06:10.173
and you say we're going
to take a tiny step

00:06:10.173 --> 00:06:11.903
in the y direction.

00:06:11.903 --> 00:06:14.167
And I'll call that partial y.

00:06:14.167 --> 00:06:15.375
And you say that makes
some kind of change,

00:06:15.375 --> 00:06:17.836
it causes a change in the function

00:06:17.836 --> 00:06:19.856
which you'll call partial f.

00:06:19.856 --> 00:06:21.876
And as you imagine this
getting really really small,

00:06:21.876 --> 00:06:24.531
and the resulting change
also getting really small

00:06:24.531 --> 00:06:26.431
the rise over run of
that is going to give you

00:06:26.431 --> 00:06:28.355
the slope of the tangent line.

00:06:28.355 --> 00:06:30.375
So this is just one way of
interpreting that ratio,

00:06:30.375 --> 00:06:31.606
the change in the output

00:06:31.606 --> 00:06:33.974
that corresponds to a
little nudge in the input.

00:06:33.974 --> 00:06:35.282
But later on we'll talk
about different ways

00:06:35.282 --> 00:06:36.575
that you can do that.

00:06:36.575 --> 00:06:38.920
So I think graphs are
very useful (laughs)...

00:06:38.920 --> 00:06:41.936
When I move that, the text doesn't move.

00:06:41.936 --> 00:06:43.273
I think graphs are very
useful for thinking

00:06:43.273 --> 00:06:45.445
about these things, but
they're not the only way

00:06:45.445 --> 00:06:47.534
and I don't want you to
get too attached to graphs

00:06:47.534 --> 00:06:49.088
even though they can be handy

00:06:49.088 --> 00:06:53.036
in the context of two-variable
input, one-variable output.

00:06:53.036 --> 00:06:53.869
See you next video!