 
Hello Students, I Dr. Gajendra Purohit
today we are going to discuss Gauss Divergence Theorem
so students, before we have discussed
what is stoke theorem, how we solve
green theorem,and then we discussed about line integral
and in previous videos we discussed
what is curl, divergence and
the basic concept of a vector. You can click on the Information Tab
and check out the videos
and after that, you will understand this topic more perfectly
I have uploaded more videos on engineering mathematics like complex analysis
on that, I have put on good content
with the following topics
 
you can check out all the videos in the playlist
 
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so students go to the youtube setting
and improve the video quality
 
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so for that, you need to change your settings
now we talk about Gauss Divergence Theorem
so it states that
in green and stoke theorem we convert
line integral to the surface intgeral
but in Gauss Divergence Theorem
it converts surface integral
to volume integral
 
 
 
it is saying that
the value of the surface integral is equal to
the divergence of volume integral
Another statement can be
we can derive it in this form also
but there is no need to derive
if any question is in this form then you have to put this statement
or else this one
 
 
 
 
so, students, there are two statements of Gauss Divergence Theorem
one this and other is this
both are almost same only
based on this statement we will solve one question
that how we will solve it
 
you have to prove
 
 
we will use this statement here
 
 
 
 
 
 
 
 
 
 
 
we will differentiate it with respect to x
 
 
 
 
 
solve as follows
 
 
so students let me tell you
the formula for the area is
and formula for volume is
 
so here as you can see it is volume
and whose volume is it? The surface you have given
surface is sphere
and the volume of the sphere is
 
 
this is the way you will solve it
 
now we will discuss this equation and will solve questions based on it
 
so students, check out one more question
on gauss divergence theorem
 
 
 
 
 
we will calculate the divergence of F
 
solve as follows
 
you have to put this here
 
 
 
 
 
 
 
 
students if you have problem calculating the divergent
you can click on the information tab and check out the video
 
 
 
 
now according to theorem
 
 
 
 
 
 
 
and this formula is for volume
and volume is of the sphere whose radius is 3
so its volume will be
 
 
 
 
 
we will see one or two more questions
to clear your concept
so students check out one more question
which is an important one and here we will prove
we are given F here
and calculate the value of f.nds
 
first of all we will calculate the value
by surface integral
and this by volume integral and will prove it equal
but here we will do something opposite
 
 
 
 
 
we have to calculate its divergence
 
 
 
 
 
 
 
 
then we will get the value as
 
 
 
divergent of F is
 
now you have to put a limit here
you need to understand the concept of limit
in triple integration
understand carefully
 
first, we will put the limit of z
see here z is going from where to where
it is going from 0 to 1
 
now you have to calculate limit here
so what we will do here
we will take the limit of y
the value of y will be
calculating value of y
 
 
 
so what you will do here is
 
 
now we will calculate the limit of x
we will put y as 0
so the value of x will be
 
first, we will put the value of z
then we will put the limit of y
 
 
 
 
 
 
then we will calculate the limit of x
so this is how we calculate the limit here
now we will integrate it
 
 
 
 
 
 
first, we will integrate it with respect to z
 
 
solve as follows
 
 
 
 
 
 
as you can see this is the formula for the area
and the area is of circle
and we know the area of a circle
 
what you will do here is
 
y is an even function
 
 
 
 
integrate it with respect to y
 
 
 
we will integrate it again
we will get
 
 
 
 
put the limits and solve
 
 
so our final answer is
this is the way we will solve it, but if you dont want to
from here you can directly write, this is the formula for the area and its value is
and we have the value of r so we will get our required answer
 
so this we discussed
by the help of Gauss Divergence theorem
now we will discuss using surface integral how we can get the value
if both are same then
our gauss divergence theorem will be verified
so students, as we discussed earlier this question
and we solved it using the gauss divergence theorem
and now we will do it using surface integral
and we will prove the answer as same
you have to verify gauss divergence theorem
so students observe carefully
here you are given
so it is a cylinder, whose circle is
 
 
here you have to calculate its surface integral
here there are 3 phases of it
there is a lower phase and top phase
and this curved surface is the third one
we will calculate the value of 3 and solve
so I have written
 
 
 
 
now we have to calculate the value of
 
 
 
we will calculate 3 surfaces differently
 
 
first, we will talk about S1
 
 
 
 
 
 
 
 
now we will talk about normal vector
here in xy plane
z=0 is also in xy plane
z=1 is also in xy plane
so here we will take the value of n as
this needs to be kept in mind
as in xy plane z is 0
and this is it's perpendicular vector
so the value of n will be
 
 
 
 
 
 
 
 
solve as follows
 
 
 
 
 
 
we know this is the formula for the area
so it will be
 
 
now we will calculate S2
 
 
we will keep the value as
 
 
 
 
solve as follows
 
 
 
 
 
 
as you can see there is no K here
so here its value will be
now we will take S3
as we know
 
 
as this is a curved surface area, you have to calculate its normal vector
so the formula for normal vector is
let me clean it first
so students here we are talking about S3
first, we want the value of the normal vector
and here what we are given as a normal vector
the surface is given here
 
with this, we will calculate the normal vector
I have uploaded a video on vector calculus
 
the concept of the gradient where you can see the concept of the normal vector
so what we will do here is
 
 
you can check out the video by clicking on the information tab
 
 
 
 
 
now we want the value of S3
 
 
 
 
 
 
 
 
now here we have to solve it
 
 
 
with that, we will solve our question
so look here how we will do it
 
 
 
now what we will do is
 
 
 
 
 
and here the value of z is
 
and theta is going from
 
 
in this whole surface what all changes are happening
x and y is changing and z is also changing
x and y we have converted to theta
 
now what we will do
 
 
 
 
we will integrate it with respect to z
 
 
 
 
 
 
 
after solving we will get the value as
so we have got the value of all the 3
substituting the values
 
 
This is how the gauss divergence theorem is verified
so, students, I hope you have understood the whole question
and there is no doubt
so here we have discussed three types of question
first, we discussed
how through Gauss divergence theorem statement
we solve a particular question
after that, we discussed a question
to solve through gauss divergence
and one more question
on how we verify the gauss divergence theorem
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