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# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Proposition 5.2.6.18. Let $\pi : \operatorname{\mathcal{C}}\rightarrow \Delta ^ n$ be a cocartesian fibration of $\infty$-categories having fibers $\{ \operatorname{\mathcal{C}}(i) = \{ i\} \times _{\Delta ^ n} \operatorname{\mathcal{C}}\} _{0 \leq i \leq n}$. Then there exists a scaffold

$U: M( \operatorname{\mathcal{C}}(0) \rightarrow \operatorname{\mathcal{C}}(1) \rightarrow \cdots \rightarrow \operatorname{\mathcal{C}}(n) ) \rightarrow \operatorname{\mathcal{C}}.$

Proof. For $1 \leq i \leq n$, Proposition 5.2.2.4 guarantees that there exists a functor $F(i): \operatorname{\mathcal{C}}(i-1) \rightarrow \operatorname{\mathcal{C}}(i)$ and a natural transformation $h(i): \Delta ^1 \times \operatorname{\mathcal{C}}(i-1) \rightarrow \operatorname{\mathcal{C}}$ which witnesses the functor $F(i)$ as given by covariant transport along the edge $\operatorname{N}_{\bullet }( \{ i-1 < i \} ) \subseteq \Delta ^ n$. Let $M$ denote the mapping simplex for the diagram $\operatorname{\mathcal{C}}(0) \xrightarrow {F(1)} \operatorname{\mathcal{C}}(1) \xrightarrow {F(2)} \operatorname{\mathcal{C}}(2) \xrightarrow {F(3)} \cdots \xrightarrow { F(n)} \operatorname{\mathcal{C}}(n)$. Unwinding the definitions, we see that the inclusion maps $\operatorname{\mathcal{C}}(i) \hookrightarrow \operatorname{\mathcal{C}}$ and the natural transformations $\{ h(i) \} _{1 \leq i \leq n}$ determine a morphism of simplicial sets $U_0: \operatorname{Spine}[n] \times _{\Delta ^ n} M \rightarrow \operatorname{\mathcal{C}}$, where $\operatorname{Spine}[n] \subseteq \Delta ^ n$ denotes the spine of the $n$-simplex (see Example 1.4.7.7). Since the inclusion $\operatorname{Spine}[n] \hookrightarrow \Delta ^ n$ is inner anodyne (Example 1.4.7.7), Lemma 5.2.6.17 guarantees that the inclusion $\operatorname{Spine}[n] \times _{\Delta ^ n} M \hookrightarrow M$ is also inner anodyne. Because $\operatorname{\mathcal{C}}$ is an $\infty$-category, we can extend $U_0$ to a morphism of simplicial sets $U: M \rightarrow \operatorname{\mathcal{C}}$ (Proposition 1.4.6.7). Using the criterion of Remark 5.2.6.15, we conclude that $U$ is a scaffold of $\pi$. $\square$

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Lemma 25.12.3. Let $\mathcal{C}$ be a site with equalizers and fibre products. Let $\mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ be a subset. Assume that any object of $\mathcal{C}$ has a covering whose members are elements of $\mathcal{B}$. Then there is a hypercovering $K$ such that $K_ n = \{ U_ i\} _{i \in I_ n}$ with $U_ i \in \mathcal{B}$ for all $i \in I_ n$.

Proof. This proof is almost the same as the proof of Lemma 25.12.2. We will only explain the differences.

Let $n \geq 1$. Let us say that an $n$-truncated $\mathcal{B}$-hypercovering is given by an $n$-truncated simplicial object $K$ of $\text{SR}(\mathcal{C})$ such that for $i \in I_ a$, $0 \leq a \leq n$ we have $U_{a, i} \in \mathcal{B}$ and such that

1. $F(K_0)^\# \to *$ is surjective,

2. $F(K_1)^\# \to F(K_0)^\# \times F(K_0)^\#$ is surjective,

3. $F(K_{a + 1})^\# \to F((\text{cosk}_ a \text{sk}_ a K)_{a + 1})^\#$ for $a = 1, \ldots , n - 1$ is surjective.

We first explicitly construct a split $1$-truncated $\mathcal{B}$-hypercovering.

Take $I_0 = \mathcal{B}$ and $K_0 = \{ U\} _{U \in \mathcal{B}}$. Then (1) holds by our assumption on $\mathcal{B}$. Set

$\Omega = \{ (U, V, W, a, b) \mid U, V, W \in \mathcal{B}, a : U \to V, b : U \to W\}$

Then we set $I_1 = I_0 \amalg \Omega$. For $i \in I_1$ we set $U_{1, i} = U_{0, i}$ if $i \in I_0$ and $U_{1, i} = U$ if $i = (U, V, W, a, b) \in \Omega$. The map $K(\sigma ^0_0) : K_0 \to K_1$ corresponds to the inclusion $\alpha (\sigma ^0_0) : I_0 \to I_1$ and the identity $f_{\sigma ^0_0, i} : U_{0, i} \to U_{1, i}$ on objects. The maps $K(\delta ^1_0), K(\delta ^1_1) : K_1 \to K_0$ correspond to the two maps $I_1 \to I_0$ which are the identity on $I_0 \subset I_1$ and map $(U, V, W, a, b) \in \Omega \subset I_1$ to $V$, resp. $W$. The corresponding morphisms $f_{\delta ^1_0, i}, f_{\delta ^1_1, i} : U_{1, i} \to U_{0, i}$ are the identity if $i \in I_0$ and $a, b$ in case $i = (U, V, W, a, b) \in \Omega$. The reason that (2) holds is that any section of $F(K_0)^\# \times F(K_0)^\#$ over an object $U$ of $\mathcal{C}$ comes, after replacing $U$ by the members of a covering, from a map $U \to F(K_0) \times F(K_0)$. This in turn means we have $V, W \in \mathcal{B}$ and two morphisms $U \to V$ and $U \to W$. Further replacing $U$ by the members of a covering we may assume $U \in \mathcal{B}$ as desired.

The lemma follows if we can prove that given a split $n$-truncated $\mathcal{B}$-hypercovering $K$ for $n \geq 1$ we can extend it to a split $(n + 1)$-truncated $\mathcal{B}$-hypercovering. Here the argument proceeds exactly as in the proof of Lemma 25.12.2. We omit the precise details, except for the following comments. First, we do not need assumption (2) in the proof of the current lemma as we do not need the morphism $K_{n + 1} \to (\text{cosk}_ n K)_{n + 1}$ to be covering; we only need it to induce a surjection on associated sheaves of sets which follows from Sites, Lemma 7.12.4. Second, the assumption that $\mathcal{C}$ has fibre products and equalizers guarantees that $\text{SR}(\mathcal{C})$ has fibre products and equalizers and $F$ commutes with these (Lemma 25.2.3). This suffices assure us the coskeleton functors used exist (see Simplicial, Remark 14.19.11 and Categories, Lemma 4.18.2). $\square$

## Comments (3)

Comment #5242 by on

Are such hypercoverings cofinal in the category of all hypercoverings $\text{HC}(\mathcal{C}, X)$?

Comment #5243 by on

@#5242: Yes, I believe this is true in the sense that for every hypercovering $K$, there is a morphism of hypercoverings $K' \to K$ such that $K'$ is as in Lemma 25.12.3. I think this should be straightforward to prove by modifying the proof of the lemma. But one needs to be careful in applying this because the category of hypercoverings isn't cofiltered! On the other hand, the category of hypercoverings is cofiltered in a homotopy sense, see Lemma 25.9.2. See also Remark 25.9.3 in case you need higher homotopies as well.

Comment #5244 by on

@#5243: Ah! Thanks for pointing out the homotopy issue. The "cofinality" sometimes might be useful. For example, if we work on the category $\mathcal{C}$ of pairs $(X,Z)$ over a trait ($Z$ a proper closed subscheme of proper flat variety $X$) with some topology, then by de Jong's alteration, the hypercoverings consisting of semi-stable pairs are "homotopically" cofinal in $\text{HC}(\mathcal{C}, (X,Z))$. By Verdier's hypercovering theorem (SGA4 V 7.4.1.4), for any presheaf $\mathcal{F}$ on $\mathcal{C}$, we have $H^i((X,Z), \mathcal{F}^{\sharp})=\operatorname{colim}_{K \in \text{HC}(\mathcal{C}, (X,Z))} \check{H}^i(K, \mathcal{F})$. Thus, we can take $K$ running through the hypercoverings consisting of semi-stable pairs, and the information of the presheaf on such pairs will dominate the cohomology. Maybe I will go through the details of the "cofinality" if I really need this cohomology computation someday.

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# I am looking for the B-field of a real coil.

1. Jun 17, 2009

I am looking for the B-field of a real coil. So a straight piece of pipe with some current density. Is it correct that there is no analytical solution to this?

It would be enough if I had the field of a current rotating in a cylinder obviously. I am working on some Biot-Savart but either I get some ugly double roots or roots mixed with trig functions in my integrals.

Does anyone know more? I do need the off center field as well!

2. Jun 19, 2009

### ozymandias

Re: Coils

I can't help you with the off-center field, but there is a closed form solution for the on-axis field. It's axial (for obvious symmetry reasons) and equal to (in SI units):

$B_z(z) = \frac{\mu_0 n I}{2} \left( \cos(\alpha_1) - \cos(\alpha_2)\right)$

where alpha_1, alpha_2 are the angles measured at z between the coil's axis of symmetry and the coil's upper and lower rims (assuming current starts at the bottom and circulates its way to top). n is the number of turns per unit length.

-----
Assaf
http://www.physicallyincorrect.com" [Broken]

Last edited by a moderator: May 4, 2017
3. Jun 19, 2009

### Bob S

Re: Coils

Ozymandias' answer is a nice form of the Biot Savart equation appropriate for solenoids. It is important to note that for a long solenoid, B is independent of coil diameter. It is also useful to put some numbers in for a typical coil (μ0 = 4 pi x 10-7 Henrys per meter). using n=1000 turns carrying 10 amps, the magnetic field is 0.0126 Tesla (126 Gauss). Compare to the Earth's field (~ 1 Gauss) and the field of a neodymium-iron-boron permanent magnet (~ 1 Tesla = 10,000 Gauss).

α β γ δ ε ζ η θ ι κ λ μ ν ξ ο π ρ ς σ τ υ φ χ ψ ω


text:
# Guelph Ph.D. Students Win Top Two Poster Prizes at Major International Research Conference

Congratulations to Guelph Ph.D. students Andrew MacLean and Christina Burbadge who were awarded first (U.S. $300) and second (U.S.$200) prizes for the top two posters presented at the International Conference on Advances in Radioactive Isotope Science (ARIS 2017) held in Keystone, Colorado from May 28 - June 2, 2017.  ARIS, held every three years, is one of the major research conferences in the field and was attended by approximately 300 delegates from 23 countries.  Andrew and Christina's posters were selected as the top two contributions from more than 120 posters presented at the conference.


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<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
You are viewing an older version of this Concept. Go to the latest version.

# Properties of Real Number Multiplication

## Commutative, associative, identity, inverse, closure

0%
Progress
Practice Properties of Real Number Multiplication
Progress
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Properties of Real Number Multiplication

Does (2)×(3)\begin{align*}(-2)\times(-3)\end{align*} give the same result as (3)×(2)\begin{align*}(-3)\times(-2)\end{align*}?

### Guidance

There are five properties of multiplication that are important for you to know.

##### Commutative Property

The commutative property of multiplication states that the order in which two numbers are multiplied does not affect the sum. If a\begin{align*}a\end{align*} and b\begin{align*}b\end{align*} are real numbers, then a×b=b×a\begin{align*}a\times b=b\times a\end{align*}.

##### Closure Property

The product of any two real numbers will result in a real number. This is known as the closure property of multiplication. In general, the closure property states that the product of any two real numbers is a unique real number. If a,b\begin{align*}a, b\end{align*} and c\begin{align*}c\end{align*} are real numbers, then a×b=c\begin{align*}a\times b=c\end{align*}.

##### Associative Property

The order in which three or more real numbers are grouped for multiplication will not affect the product. This is known as the associative property of multiplication. The result will always be the same real number. In general, the associative property states that the order in which the numbers are grouped for multiplication does not change the product. If a,b\begin{align*}a, b\end{align*} and c\begin{align*}c\end{align*} are real numbers, then (a×b)×c=a×(b×c)\begin{align*}(a\times b)\times c=a \times(b \times c)\end{align*}.

##### Multiplicative Identity

When any real number is multiplied by the number one, the real number does not change. This is true whether the real number is positive or negative. The number 1 is called the multiplicative identity or the identity element of multiplication. The product of a number and one is the number. This is called the identity property of multiplication. If a\begin{align*}a\end{align*} is a real number, then a×1=a\begin{align*}a \times 1 =a\end{align*}.

##### Multiplicative Inverse

If a\begin{align*}a\end{align*} is a nonzero real number, then the reciprocal or multiplicative inverse of a\begin{align*}a\end{align*} is 1a\begin{align*}\frac{1}{a}\end{align*}. The product of any nonzero real number and its reciprocal is always one. The number 1 is called the multiplicative identity or the identity element of multiplication. Therefore, the product of a\begin{align*}a\end{align*} and its reciprocal is the identity element of multiplication (one). This is known as the inverse property of multiplication. If a\begin{align*}a\end{align*} is a nonzero real number, then a×1a=1\begin{align*}a \times \frac{1}{a}=1\end{align*}.

#### Example A

Does (3)×(+2)=(+2)×(3)\begin{align*}(-3)\times(+2)=(+2)\times(-3)\end{align*}?

Solution: (3)×(+2)=(+2)×(3)=6\begin{align*}(-3)\times(+2)=(+2)\times(-3) = -6\end{align*}.

This is an example of the commutative property of multiplication.

#### Example B

Does (6)×(+3)=\begin{align*}(-6)\times(+3)=\end{align*} a real number?

Solution: (6)×(+3)=18\begin{align*}(-6)\times(+3)=-18\end{align*}, a real number. This is an example of the closure property of multiplication.

#### Example C

Does (3×2)×2=3×(2×2)\begin{align*}(-3\times 2)\times 2=-3 \times (2 \times 2)\end{align*}?

Solution: (3×2)×2=3×(2×2)=12\begin{align*}(-3\times 2)\times 2=-3 \times (2 \times 2)=-12\end{align*}. Even though the numbers are grouped differently, the result is the same. This is an example of the associative property of multiplication.

#### Example D

Does 8×1=8\begin{align*}8 \times 1=8\end{align*}?

Solution: Yes. This is an example of the identity property of multiplication.

#### Example E

Does 7×17=1\begin{align*}7 \times \frac{1}{7}=1\end{align*}?

Solution: Yes. This is an example of the inverse property of multiplication.

#### Concept Problem Revisited

(2)×(3)=(3)×(2)=6\begin{align*}(-2)\times (-3)=(-3)\times(-2)=6\end{align*}.

The order in which you multiplied the numbers did not affect the answer. This is an example of the commutative property of multiplication.

### Vocabulary

Multiplicative Identity
The multiplicative identity for multiplication of real numbers is one.
Multiplicative Inverse
The multiplicative inverse of multiplication is the reciprocal of the nonzero real number and the product of the real number and its multiplicative inverse is one. If a\begin{align*}a\end{align*} is any nonzero real number, its multiplicative inverse is 1a\begin{align*}\frac{1}{a}\end{align*}.
Associative Property
The associative property of multiplication states the order in which three or more real numbers are grouped for multiplication will not affect the product. If a,b\begin{align*}a, b\end{align*} and c\begin{align*}c\end{align*} are real numbers, then (a×b)×c=a×(b×c)\begin{align*}(a \times b)\times c=a \times(b \times c)\end{align*}.
Closure Property
The closure property of multiplication states that the product of any two real numbers is a unique real number. If a,b\begin{align*}a, b\end{align*} and c\begin{align*}c\end{align*} are real numbers, then a×b=c\begin{align*}a \times b = c\end{align*}.
Commutative Property
The commutative property of multiplication states that the order in which two numbers are multiplied does not affect the product. If a\begin{align*}a\end{align*} and b\begin{align*}b\end{align*} are real numbers, then a×b=b×a\begin{align*}a \times b= b \times a\end{align*}.
Identity Element of Multiplication
The identity element of multiplication is another term for the multiplicative identity of multiplication. Therefore, the identity element of multiplication is one.
Identity Property
The identity property of multiplication states that the product of a number and one is the number. If a\begin{align*}a\end{align*} is a real number, then a×1=a\begin{align*}a \times 1=a\end{align*}.
Inverse Property
The inverse property of multiplication states that the product of any real number and its multiplicative inverse is one. If a\begin{align*}a\end{align*} is a nonzero real number, then \begin{align*}a \times \left(\frac{1}{a}\right)=1\end{align*}.

### Guided Practice

1. Multiply using the properties of multiplication: \begin{align*}\left(6 \times \frac{1}{6} \right)\times(3 \times -1)\end{align*}

2. What property of multiplication justifies the statement \begin{align*}(-9 \times 5)\times 2= -9 \times (5 \times 2)\end{align*}?

3. What property of multiplication justifies the statement \begin{align*}-176 \times 1=-176\end{align*}?

1. \begin{align*}\left(6 \times \frac{1}{6} \right)\times(3 \times -1)=\frac{6}{6}\times -3=1\times -3=-3\end{align*}

2. associative property of multiplication

3. identity property of multiplication

### Practice

Match the following multiplication statements with the correct property of multiplication.

1. \begin{align*}9 \times \frac{1}{9}=1\end{align*}
2. \begin{align*}(-7 \times 4)\times 2 = -7 \times(4 \times 2)\end{align*}
3. \begin{align*}-8 \times (4) = -32\end{align*}
4. \begin{align*}6 \times(-3)=(-3) \times 6\end{align*}
5. \begin{align*}-7 \times 1=-7\end{align*}
1. Commutative Property
2. Closure Property
3. Inverse Property
4. Identity Property
5. Associative Property

In each of the following, circle the correct answer.

1. What does \begin{align*}-5(4)\left(-\frac{1}{5}\right)\end{align*} equal?
1. –20
2. –4
3. +20
4. +4
1. What is another name for the reciprocal of any real number?
2. the multiplicative identity
3. the multiplicative inverse
1. What is the multiplicative identity?
1. –1
2. 1
3. 0
4. \begin{align*}\frac{1}{2}\end{align*}
1. What is the product of a nonzero real number and its multiplicative inverse?
1. 1
2. –1
3. 0
4. there is no product
1. Which of the following statements is NOT true?
1. The product of any real number and negative one is the opposite of the real number.
2. The product of any real number and zero is always zero.
3. The order in which two real numbers are multiplied does not affect the product.
4. The product of any real number and negative one is always a negative number.

Name the property of multiplication that is being shown in each of the following multiplication statements:

1. \begin{align*}(-6\times 7)\times 2=-6 \times(7 \times 2)\end{align*}
2. \begin{align*}-12 \times 1 =-12\end{align*}
3. \begin{align*}25 \times 3 = 3 \times 25\end{align*}
4. \begin{align*}10 \times \frac{1}{10}=1\end{align*}
5. \begin{align*}-12 \times 3=-36\end{align*}

### Vocabulary Language: English

Associative property

Associative property

The associative property states that the order in which three or more values are grouped for multiplication or addition will not affect the product or sum. For example: $(a+b) + c = a + (b+c) \text{ and\,} (ab)c = a(bc)$.
Closure Property

Closure Property

The closure property of addition states that the sum of any two real numbers is a unique real number. If $a, b$ and $c$ are real numbers, then $a+b=c$.
Commutative Property

Commutative Property

The commutative property states that the order in which two numbers are added or multiplied does not affect the sum or product. For example $a+b=b+a \text{ and\,} (a)(b)=(b)(a)$.
Identity Element of Multiplication

Identity Element of Multiplication

The identity element of multiplication is another term for the multiplicative identity of multiplication. The identity element of multiplication is the number one.
identity property of multiplication

identity property of multiplication

The identity property of multiplication states that the product of a number and one is the number. If $a$ is a real number, then $a \times 1=a$.
Inverse Property of Multiplication

Inverse Property of Multiplication

The inverse property of multiplication states that the product of any real number and its multiplicative inverse (reciprocal) is one. If $a$ is a nonzero real number, then $a \times \left(\frac{1}{a}\right)=1$.
Multiplicative Identity

Multiplicative Identity

The multiplicative identity for multiplication of real numbers is one.
Multiplicative Inverse

Multiplicative Inverse

The multiplicative inverse of a number is the reciprocal of the number. The product of a real number and its multiplicative inverse will always be equal to 1 (which is the multiplicative identity for real numbers).


text:
## Railways Quant Test 173

Instructions

For the following questions answer them individually

Q 1

What is the number which when multiplied by 6 is 6 less than 60?

Q 2

What are the components of $$(x^6 - x^4 - x^5 - x^4 + x^2 - 4)$$

Q 3

The given table represents the marks obtained by four students W, X, Y and Z in four subjects P, C ,B and M,with the maximum marks in each subject being 100.
The average marks of the four students in M is:

Q 4

A train of 150 metres crosses a railway bridge in 26 seconds at a speed of 90 km/hr. What is the length of this bridge ?

Q 5

Mr. Agarwal gave 30% of his money to his wife, half of the remaining to his daughter and the rest was to be distributed among his five sons equally. If each son received Rs. 14,000, what was the total amount he had ?


text:
# Does the classical aberration formula make sense?

• I
Kairos
I believe I have understood the formula of aberration of light ##\tan \theta' = \dfrac{\sin \theta}{\beta + \cos \theta} \sqrt{1-\beta^{2}} ##

but I wonder if the non-relativistic formula ## \tan \theta' = \dfrac{\sin \theta}{\beta + \cos \theta} ## has a physical relevance. Does this formula apply to a real physical situation (for sound ?) or is it just an approximation for small ## \beta ##?

Staff Emeritus
Homework Helper
Gold Member
Did you ever drive a car in rain? Did the rain seem to be coming from ahead?

vanhees71
Kairos
Yes I did. But as the point of emission of the rain is not well defined, I have difficulty in defining ## \sin \theta ## and ## \sin \theta' ##

Staff Emeritus
Homework Helper
Gold Member
Yes I did. But as the point of emission of the rain is not well defined, I have difficulty in defining ## \sin \theta ## and ## \sin \theta' ##
The emission point is irrelevant. The only relevant thing is the angle of the rain relative to the velocity of the car.

vanhees71
Kairos
Does this mean that this formula is valid for an emission by a plane but not for a point source?

ergospherical
But as the point of emission of the rain is not well defined, I have difficulty in defining ## \sin \theta ## and ## \sin \theta' ##
##\theta## and ##\theta'## have nothing a priori to do with the emission point, they're related to the angle between the velocity of the particle and the coordinate axes i.e. ##\mathbf{v} = (v\cos{\theta}, v\sin{\theta})## and ##\mathbf{v}' = (v'\cos{\theta'}, v'\sin{\theta'})##.

Kairos
thank you


text:
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# Which of the following is a possible length for side AB of triangle AB

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Which of the following is a possible length for side AB of triangle AB [#permalink]

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21 May 2008, 08:25
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Which of the following is a possible length for side AB of triangle ABC if AC = 6 and BC = 9?

I. 3
II. 9 √3
III. 13.5

(A) I only
(B) II only
(C) III only
(D) II and III
(E) I, II, and III
[Reveal] Spoiler: OA

Last edited by Bunuel on 16 Sep 2014, 23:40, edited 1 time in total.
Renamed the topic, edited the question, added the OA and moved to PS forum.

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Director
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Re: Which of the following is a possible length for side AB of triangle AB [#permalink]

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21 May 2008, 09:17
Should be II and III.
Sum to two sides of triangle should be greater than third side.
I is not possible because 3 + 6 = 9 which is not greater than 9 (length of third side).
For Both II and III you add any two pairs and the sum of pairs is greater than the length of third side.

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Re: Which of the following is a possible length for side AB of triangle AB [#permalink]

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05 Jun 2008, 10:33
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It cant be II because 9*3^(0.5)>15, and the thirs side must be less than 15.

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Re: Which of the following is a possible length for side AB of triangle AB [#permalink]

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05 Jun 2008, 13:05
The third side of a triangle should be less than the sum of the other 2 sides and greater than the mod(difference of 2 sides)

so, x should lie between mod(y-z)<x<y+z
(9-6)<x<9+6
3<x<15

only value is 13.5

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Re: Which of the following is a possible length for side AB of triangle AB [#permalink]

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05 Jun 2008, 14:10
2 and 3 for me as well .... we need the third side to be between 3 and 15, i.e. 3<x<15

Both 2 and 3 fit this requirement

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Re: Which of the following is a possible length for side AB of triangle AB [#permalink]

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05 Jun 2008, 14:34
9(3^1/2) = 15.58>15 ...only 3.

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05 Jun 2008, 15:01
yes, how silly of me

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Re: Which of the following is a possible length for side AB of triangle AB [#permalink]

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05 Jun 2008, 22:54
Rules for a triangle
(1)Third side is always greater than the difference of two other sides
(2) Third side is always less than the sum of two other sides
Going by above rules only 13.5 satisfies the condition.
jimmyjamesdonkey wrote:
Which of the following is a possible length of the side AB of the triangle ABC if AC = 6 and BC = 9?

I. 3
II. 9(sqrt3)
III. 13.5

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Re: Which of the following is a possible length for side AB of triangle AB [#permalink]

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jimmyjamesdonkey wrote:
Which of the following is a possible length for side AB of triangle ABC if AC = 6 and BC = 9?

I. 3
II. 9 √3
III. 13.5

(A) I only
(B) II only
(C) III only
(D) II and III
(E) I, II, and III

The length of any side of a triangle must be larger than the positive difference of the other two sides, but smaller than the sum of the other two sides.

(9 - 6) < AB < (9 + 6)
3 < AB < 15.

Only 13.5 is in this range ($$9\sqrt{3}\approx{9*1.7}=15.3>15$$).

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Re: Which of the following is a possible length for side AB of triangle AB [#permalink]

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23 May 2015, 22:22
Its still not convincing for me to consider C as the correct answer because no where the question states AB is the third side or the longest side.. If for example we consider the AB = 3 (answer Choice A), The side of the triangle would be 3,6 and 9.. Now 9 becomes the third side and satisfies all the constraints of the third side.. The same is applicable for the rest two choices.. I feel answer in E.. Can someone explain whats the assumption am missing here ?

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Re: Which of the following is a possible length for side AB of triangle AB [#permalink]

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23 May 2015, 22:31
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Hi ragunandan,

You have to be very careful with the logic here.

If you have sides of 3, 6 and 9, then you DO NOT actually have a triangle - you have a line segment with a length of 9 "on top of" another line segment with a length of 9. You can use a ruler and physically draw it on a piece of paper if that will help you to visualize the issue.

From a "math" standpoint, if 9 is the longest side, then the other two sides have to SUM to a value that is GREATER than 9 (otherwise you don't have a triangle) and a DIFFERENCE that is LESS than 9.

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Special Offer: Save $75 + GMAT Club Tests Free Official GMAT Exam Packs + 70 Pt. Improvement Guarantee www.empowergmat.com/ ***********************Select EMPOWERgmat Courses now include ALL 6 Official GMAC CATs!*********************** Kudos [?]: 3776 [2], given: 173 VP Joined: 26 Mar 2013 Posts: 1362 Kudos [?]: 322 [0], given: 170 Re: Which of the following is a possible length for side AB of triangle AB [#permalink] ### Show Tags 23 May 2015, 22:57 EMPOWERgmatRichC wrote: Hi ragunandan, You have to be very careful with the logic here. If you have sides of 3, 6 and 9, then you DO NOT actually have a triangle - you have a line segment with a length of 9 "on top of" another line segment with a length of 9. You can use a ruler and physically draw it on a piece of paper if that will help you to visualize the issue. From a "math" standpoint, if 9 is the longest side, then the other two sides have to SUM to a value that is GREATER than 9 (otherwise you don't have a triangle) and a DIFFERENCE that is LESS than 9. GMAT assassins aren't born, they're made, Rich Hi Rich, Thanks for your nice explanations you provide but I did not understand 'you have a line segment with a length of 9 "on top of" another line segment with a length of 9. You can use a ruler and physically draw it on a piece of paper if that will help you to visualize the issue.' I tried to draw it but it is just incomplete triangle. is that the result you mean? Also I want to know if you cover those tactics and tricks for geometry in your courses. Thanks in advance for your help Kudos [?]: 322 [0], given: 170 e-GMAT Representative Joined: 04 Jan 2015 Posts: 786 Kudos [?]: 2323 [1], given: 126 Re: Which of the following is a possible length for side AB of triangle AB [#permalink] ### Show Tags 23 May 2015, 23:44 1 This post received KUDOS Expert's post ragunandan wrote: Its still not convincing for me to consider C as the correct answer because no where the question states AB is the third side or the longest side.. If for example we consider the AB = 3 (answer Choice A), The side of the triangle would be 3,6 and 9.. Now 9 becomes the third side and satisfies all the constraints of the third side.. The same is applicable for the rest two choices.. I feel answer in E.. Can someone explain whats the assumption am missing here ? Hi ragunandan, Which of the following is a possible length for side AB of triangle ABC if AC = 6 and BC = 9? 1. It's mentioned in the question statement that AB is the side of triangle ABC. Since we have been given two sides AC and BC, AB has to the third side. 2. Secondly since the question asks the possible length of side AB, we need to consider all possible values of AB i.e. AB can be the shortest side or the longest side of triangle ABC. 3. Lastly AB = 3 is not a legible value of the 3rd side of triangle ABC with the other two sides as AC = 6 and BC = 9. In a triangle the sum of two sides should be greater than the 3rd side. In this case AB + AC = 9 = BC. Similarly AB = 15√3 is also not a legible value. Among the given options AB = 13.5 is the only value for which we can form a triangle with the other two sides being 6 and 9. Let me know in case you still have any trouble in the explanation Regards Harsh _________________ | '4 out of Top 5' Instructors on gmatclub | 70 point improvement guarantee | www.e-gmat.com Kudos [?]: 2323 [1], given: 126 EMPOWERgmat Instructor Status: GMAT Assassin/Co-Founder Affiliations: EMPOWERgmat Joined: 19 Dec 2014 Posts: 10696 Kudos [?]: 3776 [0], given: 173 Location: United States (CA) GMAT 1: 800 Q51 V49 GRE 1: 340 Q170 V170 Re: Which of the following is a possible length for side AB of triangle AB [#permalink] ### Show Tags 24 May 2015, 09:26 Hi Mo2men, You've discovered another variation on 'problem' of having sides of 3, 6 and 9 - if you tried to form a triangle, then the 3 and the 6 would NOT touch, so you would NOT have a triangle. This all stems from a math rule called the Triangle Inequality Theorem; it's relatively rare on the GMAT (you might not even see it on Test Day; if you do see it, you'll probably see it just once). Thus, the concept is not a big point-gainer on the GMAT, so unless you're already scoring at a really high level in the Quant, you should be focused on other subjects. To answer your last question: yes, we do cover it (and a couple of variants of it) in the EMPOWERgmat Course. GMAT assassins aren't born, they're made, Rich _________________ 760+: Learn What GMAT Assassins Do to Score at the Highest Levels Contact Rich at: Rich.C@empowergmat.com # Rich Cohen Co-Founder & GMAT Assassin Special Offer: Save$75 + GMAT Club Tests Free
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Re: Which of the following is a possible length for side AB of triangle AB [#permalink]

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04 Jun 2015, 19:50
Hi Rich, Thanks for the detailed explanation. #Helpful

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Re: Which of the following is a possible length for side AB of triangle AB [#permalink]

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04 Jun 2015, 19:53
Hi Harsh,Thanks for the explanation.. It was simpler and neat..

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Re: Which of the following is a possible length for side AB of triangle AB [#permalink]

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05 Jun 2015, 04:46
jimmyjamesdonkey wrote:
Which of the following is a possible length for side AB of triangle ABC if AC = 6 and BC = 9?

I. 3
II. 9 √3
III. 13.5

(A) I only
(B) II only
(C) III only
(D) II and III
(E) I, II, and III

POINT : The effort here is to make the rule of triangle more precise and less time consuming

We know the rule in different forms however we only need to know the rule in just one line

i.e. SUM OF TWO SHORTER SIDES > LONGEST SIDE OF THE TRIANGLE

FOR THE TRIANGLE TO EXIST

Let's apply this in this case

For Finding the range of values of the Third side

For minimum value of the third side AB, We must consider the given BC = 9 as the longest side

Hence AB (Min) + AC > BC

i.e. AB (Min) + 6 > 9

i.e. AB (Min) > 3

For Maximum value of the third side AB, We must consider the given BC = 9 and Ac=6 as the two shorter sides

Hence AC + BC > AB (Max.)

i.e. 6 + 9 > AB (Max.)

i.e. AB (Max.) < 15

i.e. the range of AB becomes

3 < AB < 15

I. 3 NOT ACCEPTABLE
II. 9 √3 - 9x1.73 = 15.3 NOT ACCEPTABLE
III. 13.5 ACCEPTABLE

[Reveal] Spoiler:
C

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Re: Which of the following is a possible length for side AB of triangle AB [#permalink]

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30 Dec 2016, 02:15
The third side of a triangle must be less than the sum of the other two sides and greater than their difference (i.e. |y - z| < x < y + z).
In this question:
|BC - AC| < AB < BC + AC
9 - 6 < AB < 9 + 6
3 < AB < 15
Only 13.5 is in this range. 9sqrt3 is approximately equal to 9(1.7) or 15.3.
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# what does that statement mean about the relation?

what does this mean about P? $$\forall x \exists y (p(x,y) \rightarrow p(y,x))$$

i know that $$\forall x \forall y (p(x,y) \rightarrow p(y,x))$$ means that P symmetric

but what does the first statement means?

and what does the last statement mean about the relation P?

$$\forall y (\exists x P(x,y) \rightarrow \exists x P(y,x))$$

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Do you know what $\forall$, $\exists$, and $\rightarrow$ mean? – Chris Eagle Jun 8 '12 at 11:19
of course i do :\ – Nahum Litvin Jun 8 '12 at 11:29

$\forall x \exists y (p(x,y) \rightarrow p(y,x))$ says that for each $x$ there is a $y$, which may depend on $x$, such that if $p(x,y)$, then also $p(y,x)$. Let me write $xPy$ to mean $p(x,y)$.

Suppose that the statement is true, and fix a particular $x$. Then there’s some $y$, perhaps only one, such that $xPy\to yPx$. Under what circumstances is this true? The only time it’s false is when $xPy$ is true and $yPx$ is false; it’s true whenever $xPy$ is false, or $yPx$ is true, or both. Thus, if you can find a $y$ such that $x(\lnot P)y$, that $y$ works: $xPy\to yPx$ is vacuously true, because $xPy$ is false.

What if $xPy$ for every $y$, so that you can’t find such a $y$? If you can find a $y$ such that $yPx$, you can use it, because then $xPy\to yPx$ is true.

The only time you’re out of luck is when $xPy$ for every $y$, and there is no $y$ such that $yPx$: then no matter what $y$ you pick, $xPy$ is true and $yPx$ is false, so $xPy\to yPx$ is false.

$\forall x \exists y (p(x,y) \rightarrow p(y,x))$ therefore says that for each $x$ there is a $y$ such that either $x(\lnot P)y$, or $yPx$ (or both). Alternatively, it says that there is no $x$ such that for every $y$, $xPy$ and $y(\lnot P)x$: there is no $x$ that is related to everything but not from anything.

In fact, though, this is trivially true. Pick any $x$; if $xPx$, then $xPx\to xPx$, and we can simply let $y=x$. If $x(\lnot P)x$, then again $xPx\to xPx$, this time vacuously (because the antecedent is false), and we can let $y=x$. Thus, $\forall x \exists y (p(x,y) \rightarrow p(y,x))$ is true of any relation, because for each $x$ we can take $y=x$ and have $p(x,y) \rightarrow p(y,x)$.

The statement $\forall y (\exists x P(x,y) \rightarrow \exists x P(y,x))$ is quite different. In terms of relations it says that each $y$ that is related from something is also related to something, but not necessarily the same thing. Consider, for instance, the relation $P$ on $\{0,1,2,3,4\}$ defined by $0P1P2P3P4P0$; that is, under $P$ the elements $0,1,2,3$, and $4$ form a circular chain. This relation satisfies the statement in question: for each $k\in\{0,1,2,3,4\}$, if there is some $m$ such that $mPk$, then there is some $n$ such that $kPn$. I could throw $5$ into the set without changing $P$, and $P$ would still have the property: there is no $m$ such that $mP5$, so the statement $\forall y (\exists x P(x,y) \rightarrow \exists x P(y,x))$ says nothing about $5$. But if I also add $3P5$, say, in order to keep the property I have to add $5Pk$ for at least one $k$.

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wow, now I am even more confused :( – Nahum Litvin Jun 8 '12 at 12:00
@Nahum: Can you pin down any of your confusion? – Brian M. Scott Jun 8 '12 at 12:03
after using "Google translate" and several rereadings Iv understood your explanation about the first statement :) I understand the second explanation. i did think about it by myself, but i fail comprehend how use it to solve the full problem (that I didn't show here) the problem involves finding maximal set of such statements form a list. – Nahum Litvin Jun 8 '12 at 12:14
@Nahum: Sorry: I sometimes forget that I’m often writing for people for whom English is a second (or third, or fourth, ...) language. – Brian M. Scott Jun 8 '12 at 12:15
@who voted down with no explanation? – Nahum Litvin Jun 9 '12 at 9:18


text:
(Small oscillations) Finding Normal modes procedure.

by Lavabug
Tags: modes, normal, oscillations, procedure
 P: 849 1. The problem statement, all variables and given/known data The first part of the problem is just finding the Lagrangian for a system with 2 d.o.f. and using small angle approximations to get the Lagrangian in canonical/quadratic form, not a problem. I am given numerical values for mass, spring constants, etc. and am told to find the normal modes of oscillation and the normal coordinates. 2. Relevant equations 3. The attempt at a solution I find the eigenvalues (eigenfreq, $$\lambda = \omega^{2}$$) by diagonalizing: |V-$$\lambda$$T| = 0. (V and T are matrices, also called M and K, they're the matrix of coefficients of the velocities and coordinates from the canonical Lagrangian respectively) I get 2 ugly numbers for my eigenfrequencies, approximately 0.3 and 15.7. Now I sub the first one into V-$$\lambda$$T * (a11, a21) = 0 to solve for the eigenvector (a11, a21) which are the amplitudes of oscillation of both of my 2 d.o.f.(1 and 2) for this normal mode (1). Question: plugging them in gives me a system of 2 equations of 2 variables. I am told that necessarily, one of the equations is a multiple of the other, but it doesn't seem to be true (using approximate values might have to do with this but I'm not sure). Is this correct? If I believe what I am told, I pick one of the equations (the end result depends on which one I pick!) and solve for a11 as a function of a21, then use the orthonormalization condition: (a11,a21)(transposed)*T*(a11,a21) = 1 which allows me to find one of the amplitudes, which I then use to go back and solve for the other. I am also very worried about approximating everything, at this stage of the problem I've already approximated decimal points on at least 3 occasions. Can someone PLEASE help clarify this to me? Am I on the right track?
 P: 170 You are on the right track, but don't plug in numbers until the very end. For a 2x2 matrix, you can do everything analytically. If you are stuck, show some more of your calculations :)
 P: 849 Thanks. The problem is my eigenvalues are generally irrational numbers so in order to not drag on mistakes from finding the roots, I just approximate the eigenvalues which is what my prof does and suggests doing even for 2 d.o.f. problems. I have a problem with the orthonormalization process, I'm never quite sure if I'm multiplying the A(transposed)TA vectors properly. Do I first calculate A(transposed)*T, giving me a column vector, which I then multiply like a scalar product with the last term A(not transposed)?
P: 170

(Small oscillations) Finding Normal modes procedure.

You must have a prof with a technical background :)
The order in which you do the multiplications doesn't matter. I prefer to first calculate T*a, which gives another vector and then take the scalar product with a transposed, but that's just because I find multiplying with a matrix from the left easier than from the right.
P: 849
 Quote by grey_earl You must have a prof with a technical background :) The order in which you do the multiplications doesn't matter. I prefer to first calculate T*a, which gives another vector and then take the scalar product with a transposed, but that's just because I find multiplying with a matrix from the left easier than from the right.
Aha so its valid both ways, I can just use the "scissor fingers" rule. :)

The prof is actually a theoretician. :P

To get the normal coordinates, once I have the general solution both coordinates q (linear combo of normal modes), I multiply each by the inverse of the A matrix? Is this the whole matrix or just the corresponding eigenvector?
 P: 170 That's the whole matrix. You can use your normalization equation to simplify things: $$\vec{N} = A^{-1} \vec{q} = A^{\mathrm{T}} T \vec{q}$$ since $$A^{\mathrm{T}} T A = 1$$.
 P: 849 That 3rd expression you've got there... Didn't think of that. :P I'd much rather transpose and multiply 2 square matrices with a vector than invert a matrix any day! Thanks, I've learned how to do 2 things more efficiently today thanks to you. :)

 Related Discussions Introductory Physics Homework 0 Special & General Relativity 1 Introductory Physics Homework 1 Advanced Physics Homework 0 Advanced Physics Homework 3


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CBSE (Science) Class 11CBSE
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# The Electrical Resistance in Ohms of a Certain Thermometer Varies with Temperature According to the Approximate Law the Resistance is 101.6 ω at the Triple-point of Water 273.16 K, and 165.5 Ohm at the Normal Melting Point of Lead (600.5 K). What is the Temperature When the Resistance is 123.4 Ohm - CBSE (Science) Class 11 - Physics

ConceptMeasurement of Temperature

#### Question

The electrical resistance in ohms of a certain thermometer varies with temperature according to the approximate law:

Ro [1 + α (– To)]

The resistance is 101.6 Ω at the triple-point of water 273.16 K, and 165.5 Ω at the normal melting point of lead (600.5 K). What is the temperature when the resistance is 123.4 Ω?

#### Solution 1

Here, R0 = 101.6 Ω; T0 = 273.16 K Case (i) R1= 165.5 Ω; T= 600.5 K, Case (ii) R2 = 123.4 , T2 = ?

Using the relation R = R0[1 + α (T – T0)]

Case (i) 165.5 = 101.6 [1 + α (600.5 – 273.16)]

alpha= (165.5 - 101.6)/(101.6xx(600.5-273.16)) = 63.9/(101.6xx327xx37)

Case II    123.4 = 101.6 [1 + alpha(T_2 - 273.16)]

or 123.4 = 101.6 [1+ 63.9/(101.6xx327.34)(T_2-273.16)]

= 101.6 + 63.9/327.37 (T_2 - 273.16)

or T_2 = ((123.4-101.6)xx327.34)/63.9+ 273.16 = 111.67 + 273.16

= 384.83 K

#### Solution 2

It is given that:

R0 [1 + α (– T0)] … (i)

Where,

R0 and T0 are the initial resistance and temperature respectively

R and T are the final resistance and temperature respectively

α is a constant

At the triple point of water, T0 = 273.15 K

Resistance of lead, R0 = 101.6 Ω

At normal melting point of lead, T = 600.5 K

Resistance of lead, R = 165.5 Ω

Substituting these values in equation (i), we get:

R = R_0[1+alpha(T - T_0)]

165.5=101.6 [1+alpha(600.5 - 273.15)]

1.629 = 1 + alpha(327.35)

:.alpha = 0.629/327.35 = 1.92 xx 10^(-3) K^(-1)

For resistance, R_1= 123.4 Omega

R_1 = R_0[1+alpha(T-T_0)]

Where T is the temperrature when the resistance of lead is 123.4 Omega

123.4 = 101.6[]1+1.92 xx 10^(-3)(T-273.15)]

1.214 = 1 + 1.92 xx 10^(-3)(T - 273.15)

0.214/(1.92xx10^(-3)) = T - 273.15

:. T = 384.61 K

Is there an error in this question or solution?

#### APPEARS IN

NCERT Solution for Physics Textbook for Class 11 (2018 to Current)
Chapter 11: Thermal Properties of Matter
Q: 3 | Page no. 294
Solution The Electrical Resistance in Ohms of a Certain Thermometer Varies with Temperature According to the Approximate Law the Resistance is 101.6 ω at the Triple-point of Water 273.16 K, and 165.5 Ohm at the Normal Melting Point of Lead (600.5 K). What is the Temperature When the Resistance is 123.4 Ohm Concept: Measurement of Temperature.
S


text:
# AP Statistics Curriculum 2007 Distrib Dists

## General Advance-Placement (AP) Statistics Curriculum - Geometric, HyperGeometric, Negative Binomial Random Variables and Experiments

### Geometric

• Definition: The Geometric Distribution is the probability distribution of the number X of Bernoulli trials needed to get one success, supported on the set {1, 2, 3, ...}.
• Mass Function: If the probability of successes on each trial is P(success)=p, then the probability that x trials are needed to get one success is $$P(X = x) = (1 - p)^{x-1} \times p$$, for x = 1, 2, 3, 4,....
• Expectation: The Expected Value of a geometrically distributed random variable X is $${1\over p}.$$
• Variance: The Variance is $${1-p\over p^2}.$$

### HyperGeometric

The hypergeometric distribution is a discrete probability distribution that describes the number of successes in a sequence of n draws from a finite population without replacement. An experimental design for using Hypergeometric distribution is illustrated in this table:

 Type Drawn Not-Drawn Total Defective k m-k m Non-Defective n-k N+k-n-m N-m Total n N-n N
• Explanation: Suppose there is a shipment of N objects in which m are defective. The Hypergeometric Distribution describes the probability that in a sample of n distinctive objects drawn from the shipment exactly k objects are defective.
• Mass function: The random variable X follows the Hypergeometric Distribution with parameters N, m and n, then the probability of getting exactly k successes is given by

$P(X=k) = {{{m \choose k} {{N-m} \choose {n-k}}}\over {N \choose n}}.$

This formula for the Hypergeometric Mass Function may be interpreted as follows: There are $${{N}\choose{n}}$$ possible samples (without replacement). There are $${{m}\choose{k}}$$ ways to obtain k defective objects and there are $${{N-m}\choose{n-k}}$$ ways to fill out the rest of the sample with non-defective objects.

#### Examples

• SOCR Activity: The SOCR Ball and Urn Experiment provides a hands-on demonstration of the utilization of Hypergeometric distribution in practice. This activity consists of selecting n balls at random from an urn with N balls, R of which are red and the other N - R green. The number of red balls Y in the sample is recorded on each update. The distribution and moments of Y are shown in blue in the distribution graph and are recorded in the distribution table. On each update, the empirical density and moments of Y are shown in red in the distribution graph and are recorded in the distribution table. Either of two sampling models can be selected with the list box: with replacement and without replacement. The parameters N, R, and n can be varied with scroll bars.
• A lake contains 1,000 fish; 100 are randomly caught and tagged. Suppose that later we catch 20 fish. Use SOCR Hypergeometric Distribution to:
• Compute the probability mass function of the number of tagged fish in the sample of 20.
• Compute the expected value and the variance of the number of tagged fish in this sample.
• Compute the probability that this random sample contains more than 3 tagged fish.

### Negative Binomial

• Definition: The family of Negative Binomial Distributions is a two-parameter family; p and r with 0 < p < 1 and r > 0.
• Mass Function: The probability mass function of a Negative Binomial random variable (X~NegBin(r, p))is give by:

$P(X=k) = {k+r-1 \choose k}\cdot p^r \cdot (1-p)^k \!$, for k = 0,1,2,....

#### Application

Suppose Jane is promoting and fund-raising for a presidential candidate. She wants to visit all 50 states and she's pledged to get all electoral votes of 6 states before she and the candidate she represents are satisfied. In every state, there is a 30% chance that Jane will be able to secure all electoral votes and 70% chance that she'll fail.

• What's the probability mass function that the last 6th state where she succeeds to secure all electoral votes happens to be the at the nth state she campaigns in?

NegBin(r, p) distribution describes the probability of k failures and r successes in n=k+r Bernoulli(p) trials with success on the last trial. Looking to secure the electoral votes for 6 states means Jane needs to get 6 successes before she (and her candidate) is happy. The number of trials (i.e., states visited) needed is n=k+6. The random variable we are interested in is X={number of states visited to achieve 6 successes (secure all electoral votes within these states)}. So, k = n-6, and $$X\sim NegBin(6, 0.3)$$. Thus, for $$n \geq 6$$, the mass function (giving the probabilities that Jane will visit n states before her ultimate success is:

$P(X=n) = {(n-6) + 6 - 1 \choose 6-1} \; 0.3^6 \; 0.7^{n-6} = {n-1 \choose 5} \; \frac{3^6 \; 7^{n-6} \; }{10^n}$

• What's the probability that Jane finishes her campaign in the 10th state?

$P(X=10) = 0.022054$

• What's the probability that Jane finishes campaigning on or before reaching the 8th state?

$P(X\leq 8) = 0.011292$

Error creating thumbnail: File missing
• Suppose the success of getting all electoral votes within a state is reduced to only 10%, then X~NegBin(r=6, p=0.1). Notice that the shape and domain the Negative-Binomial distribution significantly chance now (see image below)! What's the probability that Jane covers all 50 states but fails to get all electoral votes in any 6 states (as she had hoped for)?

$P(X\geq 50) = 0.632391$


text:
# Discrete Fourier transform and Fourier series

This is quite silly, but the relationship between the discrete Fourier transform (DFT) and the Fourier series (FS) is clouded by annoying factors. I will try to connect both in this article. The motivation is to employ DFT techniques in a computer simulation. In the latter, one usually has a finite simulation box, which makes Fourier series the most interesting (a connection to the Fourier transform may also be made, see below).

# Van der Waals’ square gradient theory

OK, I have encountered this theory again, after many years. The idea is to describe separation between two phases as the minimum of a free energy with respect to an order parameter $\phi$:

$g(\phi) = \frac a2 \phi^2 + \frac b4 \phi^2$

When becomes negative, the minimum of g changes from 0 to other values, $\pm \phi_0$. The function then has the celebrated double minimum feature, which features prominently in many symmetry-breaking theories, including of course the appearance of particle mass mass and, you know, the Big Bang and the Universe.

But here we are just considering phase separation in materials. The interface between two coexisting phases must have some associated cost, and the simplest (lowest order) way to include it is by introducing a total free energy functional

$F = \int dr \, f(\phi, \nabla\phi)$

$f(\phi) = g(\phi) + \frac c2 (\nabla\phi)^2 .$

This is also called a London-Linzburg-Landau free energy, also appearing in their theory of superconductors.

Now, parameters ab, and c are not easy to measure (or, at least, estimate) experimentally, but they are related to: the surface tension, the width of the interface, and the magnitude of the bulk equilibrium order parameter (i.e. $\phi_0$ ). Here I show how to obtain it in a slightly more general setting, since I was not able to find it on the internet (it can be found e.g, in the book by Rowlinson & Widom).

# Taylor-Green vortex sheet, reduced units

The Taylor-Green vortex sheet is a solution to the 2D Navier-Stokes equations for an incompressible Newtonian fluid:

$\frac{d \mathbf{u}}{d t}= \nu \nabla^2 \mathbf{u} - \nabla p/\rho ,$

where $\mathbf{u}$ is the velocity field, $p$ is the pressure, $\nu=\mu/\rho$ is the kinematic viscosity, and $\rho$ is the fixed density of the fluid. The time derivative is a total derivative:

$\frac{d \mathbf{u}}{d t} = \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u}$

It is common to choose parameters that simplify the equations, but that can obscure the role of the different parameters. In the following, I provide expressions with all relevant parameters included, with their physical dimensions. I later pass to dimensionless, or reduced, units, in terms of the Reynolds and Courant numbers.

# Sound waves with attenuation

Just a simple derivation of the role of attenuation in the standard sound wave equation. Original work: Stokes, 1845.

Starting with the Navier-Stokes momentum equation

$\frac{\partial }{\partial t} \mathbf{u} + \mathbf{u} \nabla \mathbf{u} = - \frac{1}{\rho} \nabla p + \frac{\mu}{\rho} \nabla^2 \mathbf{u} + \left(\frac{\lambda+\mu}{\rho}\right)\nabla (\nabla\cdot\mathbf{u}) ,$

where $\lambda$ is a Lamé viscosity coefficient. The bulk viscosity coeficient  is defined as $\zeta = \lambda + (2/3) \mu$. The last term  is often neglected, even in compressible flow, but sound attenuation is one of the few cases where it may have some influence. All viscosities are assumed to be constant, but in this case this is a safe assumption, since we are going to assume small departures about equilibrium values.

# Sparse Poisson problem in eigen

Back to scientific computing. Lately, I have been using the Eigen libraries for linear algebra. They were recommended by the CGAL project, and they indeed share a common philosophy. Thanks to the rather high C++ level they can accomplish this sort of magic:

int n = 100;
VectorXd x(n), b(n);
SpMat A(n,n);

fill_A(A);

fill_b(b);

// solve Ax = b
Eigen::BiCGSTAB<SpMat> solver;

solver.compute(A);

x = solver.solveWithGuess(b,x0);

Notice that A is a sparse matrix! I am next describing how to use this in order to solve the 1D Poisson equation.

# OpenFOAM cheatlist

A quick cheatsheet for OpenFOAM. In italics, things that are useful but not part of OpenFOAM proper. Interesting read: The OpenFOAM Technology Primer

Sites

Shortcuts to directories

(type “alias” to reveal these)

• run (go to own’s running directory)
• foam
• foamfv
• foam3rdParty (hit <tab> for these longish commands!)
• tut
• app
• sol
• util
• lib
• src

Environment variables

• echo $FOAM_ <tab> (directories) • echo$WM_ <tab>  (building, aka compiling, settings)

Mesh

• Generation
• Import / export
• foamMeshToFluent
• fluentMeshToFoam, etc …
• Operations
• refineHexMesh
• transformPoints
• makeAxialMesh
• collapseEdges
• autoPatch
• mirorMesh
• Properties
• checkMesh

Fields

• setFields
• topoSet
• patchAverage
• patchIntegrate
• vorticity
• yPlusRAS
• yPlusLES
• boxTurb
• applyBoundaryLayer
• R
• wallShearStress

Help

• foamHelp (e.g. foamHelp boudary -field U)

Postprocessing

• sample
• paraview
• probeLocations

Solvers

• icoFoam
• interFoam
• many, many others

Running

• foamJob
• decomposePar
• reconstructPar
• mpirun
• nohup

Building

• foamNew source App …
• doxygen doxyfile
• gdb
• valgrind
• wmake
• wclean
• aliases for settings: wm32, wm64, wmSP, wmDP, wmSET, wmUNSET


text:
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# What is the value of x^2yz − xyz^2, if x = − 2, y = 1, and z = 3?

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Math Expert
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What is the value of x^2yz − xyz^2, if x = − 2, y = 1, and z = 3?  [#permalink]

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10 Jan 2016, 07:27
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What is the value of $$x^2yz − xyz^2$$, if x = − 2, y = 1, and z = 3?

(A) 20
(B) 24
(C) 30
(D) 32
(E) 48

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Re: What is the value of x^2yz − xyz^2, if x = − 2, y = 1, and z = 3?  [#permalink]

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11 Jan 2016, 06:54
Given : x = − 2, y = 1, and z = 3

x^2yz − xyz^2,
= xyz (x- z)
= (-2)(1)(3) [ (-2) - 3]
= (-6)*(-5)
= 30
Hence C.
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GMAT 1: 660 Q48 V31
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Re: What is the value of x^2yz − xyz^2, if x = − 2, y = 1, and z = 3?  [#permalink]

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11 Jan 2016, 19:43
4*1*3 -(- 2*1*9)
=12+18 = 30

Ans: C
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Joined: 23 Apr 2016
Posts: 6
Re: What is the value of x^2yz − xyz^2, if x = − 2, y = 1, and z = 3?  [#permalink]

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21 May 2016, 07:41
can someone please explain this a bit more? I just substituted and definitely didnt come close
Math Expert
Joined: 02 Aug 2009
Posts: 8341
Re: What is the value of x^2yz − xyz^2, if x = − 2, y = 1, and z = 3?  [#permalink]

### Show Tags

21 May 2016, 07:45
2
JSS16 wrote:
can someone please explain this a bit more? I just substituted and definitely didnt come close

Hi,

the Q is --- What is the value of x^2yz − xyz^2, if x = − 2, y = 1, and z = 3? ..

$$x^2yz − xyz^2 = xyz(x-z) = (-2)*1*3*(-2-3) = (-2)*1*3*(-5) = 1*3*10=30$$...
C
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Re: What is the value of x^2yz − xyz^2, if x = − 2, y = 1, and z = 3?  [#permalink]

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21 May 2016, 10:58
1
Bunuel wrote:
What is the value of x^2yz − xyz^2, if x = − 2, y = 1, and z = 3?

(A) 20
(B) 24
(C) 30
(D) 32
(E) 48

$$x^2$$$$yz$$ = $$-2^2$$$$(1*3)$$ => 12

$$xy$$$$z^2$$ = $$(-2*1)3^2$$ => -18

So, $$x^2$$$$yz$$ − $$xy$$$$z^2$$ = 12 - (-18) => 30

Hence answer will be (C) 30
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Joined: 18 Aug 2016
Posts: 1
Re: What is the value of x^2yz − xyz^2, if x = − 2, y = 1, and z = 3?  [#permalink]

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11 Oct 2016, 23:33
Bunuel wrote:
What is the value of x^2yz − xyz^2, if x = − 2, y = 1, and z = 3?

(A) 20
(B) 24
(C) 30
(D) 32
(E) 48

The confusion created in this question is that they could have included brackets.

ie (x^2)yz - xyz^2..

Else people like me will take x^(2yz)-xyz^2.

Is there any rule for this? Please tell me if so.
Math Expert
Joined: 02 Sep 2009
Posts: 60460
Re: What is the value of x^2yz − xyz^2, if x = − 2, y = 1, and z = 3?  [#permalink]

### Show Tags

12 Oct 2016, 00:02
1
wolf020287 wrote:
Bunuel wrote:
What is the value of x^2yz − xyz^2, if x = − 2, y = 1, and z = 3?

(A) 20
(B) 24
(C) 30
(D) 32
(E) 48

The confusion created in this question is that they could have included brackets.

ie (x^2)yz - xyz^2..

Else people like me will take x^(2yz)-xyz^2.

Is there any rule for this? Please tell me if so.

x^2yz mathematically can only mean x^2*y*z. If it were 2 to the power of 2yz, then it would be written as x^(2yz). Still edited the original post to avoid confusion.
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Re: What is the value of x^2yz − xyz^2, if x = − 2, y = 1, and z = 3?  [#permalink]

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12 Oct 2016, 02:03
Bunuel wrote:
What is the value of $$x^2yz − xyz^2$$, if x = − 2, y = 1, and z = 3?

(A) 20
(B) 24
(C) 30
(D) 32
(E) 48

$$(-2)^2*1*3 − (-2)*1*3^2$$

=4*1*3 - ( -2)*1*9
= 12 - (-18)
= 30

Hence answer will be (C) 30

_________________
Thanks and Regards

Abhishek....

PLEASE FOLLOW THE RULES FOR POSTING IN QA AND VA FORUM AND USE SEARCH FUNCTION BEFORE POSTING NEW QUESTIONS

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What is the value of x^2yz − xyz^2, if x = − 2, y = 1, and z = 3?  [#permalink]

### Show Tags

11 Apr 2017, 22:59
For this question, you must to remember that you need to factor common terms out, then it's just plug and play, and obviously, avoid silly mistakes (I'm talking directly to myself here!) and go bit by bit, just like Jack the Ripper:

$${ x }^{ 2 }yz-xy{ z }^{ 2 }=\\ xyz\left( x-y \right) =\\ \left[ \left( -2 \right) \left( 1 \right) \left( 3 \right) \right] \left( -2-3 \right) =\\ \left( -6 \right) \left( -5 \right) =\\ 30$$
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Re: What is the value of x^2yz − xyz^2, if x = − 2, y = 1, and z = 3?  [#permalink]

### Show Tags

06 Sep 2018, 08:28
1
xyz(x-z)
(-2)(1)(3)(-2-3)
(-6)(-5)
30
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GRE 1: Q170 V160
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Re: What is the value of x^2yz − xyz^2, if x = − 2, y = 1, and z = 3?  [#permalink]

### Show Tags

14 May 2019, 06:06
1
=> x= -2, y = 1, z = 3,

=> xyz[x-z]

=> (-2)(1)(3)(-2-3)

=> (-6)(-5)

=> 30
Re: What is the value of x^2yz − xyz^2, if x = − 2, y = 1, and z = 3?   [#permalink] 14 May 2019, 06:06
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# Texas Go Math Grade 4 Lesson 9.5 Answer Key Division and the Distributive Property

Refer to our Texas Go Math Grade 4 Answer Key Pdf to score good marks in the exams. Test yourself by practicing the problems from Texas Go Math Grade 4 Lesson 9.5 Answer Key Division and the Distributive Property.

## Texas Go Math Grade 4 Lesson 9.5 Answer Key Division and the Distributive Property

Essential Question

How can you use the Distributive Property to find quotients?
The Distributive Property of division says that dividing a sum by number is the same as dividing each addend by the number and then adding the quotients.
We can use distributive property to find quotients by breaking the Dividend into two addends which can be quotients.
Explanation:
For Example;
Outline a rectangle on a grid to model 69 ÷ 3.
Shade columns of 3 until you have 69 squares.

Investigate

Materials

• color pencils
• grid paper

You can use the Distributive Property to break apart numbers to make them easier to divide.

The Distributive Property of division says that dividing a sum by a number is the same as dividing each addend by the number and then adding the quotients.

A. Outline a rectangle on a grid to model 69 ÷ 3.

Shade columns of 3 until you have 69 squares.

Explanation:
Outline a rectangle on a grid to model 69 ÷ 3.
Shade columns of 3 until you have 69 squares.

How many groups of 3 can you make? ____________
20 groups of 3
Explanation:

B. Think of 69 as 60 + 9. Break apart the model into two rectangles to show (60 + 9) ÷ 3. Label and shade the smaller rectangles. Use two different colors.

Explanation:
Break apart the model into two rectangles to show (60 + 9) ÷ 3.
Label and shade the smaller rectangles.
Use two different colors to identify.

C. Each rectangle models a division.
69 ÷ 3 = (____ ÷ 3) + (________ ÷ 3)
= _________ + ___________
= ___________
69 ÷ 3 = (60 ÷ 3) + (9 ÷ 3)
= 20 + 3
= 23
Explanation:
Break apart the model into two to show
(60 + 9) ÷ 3, (9 ÷ 3)
= 20 + 3
= 23

D. Outline another model to show 68 ÷ 4.
How many groups of 4 can you make? ___________
17 groups of 4
Explanation:
Break apart the model into two to show
68 ÷ 4 = (60 ÷ 4) + (8 ÷ 4)
= 15 + 2
= 17

E. Think of 68 as 40 + 28. Break apart the model, label, and shade to show two divisions.

68 ÷ 4 = (__________ ÷ 4) + (_________ ÷ 4)
= _________ + ___________ = __________

68 ÷ 4 = (60 ÷ 4) + (8 ÷ 4)
= 15 + 2
= 17
Explanation:
Break apart the model into two to show
68 ÷ 4 = (60 ÷ 4) + (8 ÷ 4)
= 15 + 2
= 17

Make Connections

You can also model 68 ÷ 4 using base-ten blocks.

STEP 1 Model 68.
68 = ________ + ________

68 = 40 + 28
Explanation:
Base ten is a counting system that uses ten digits to write down all numbers.
If we want to write any number, we have ten digits we can use: 0 1 2 3 4 5 6 7 8 9.
If you notice, there is no one digit for the number ten- we need to use two digits, the 1 and the 0.
We can only count to nine without the need for two digits.
Therefore, we use base ten in order to make bigger numbers.

STEP 2 Divide the longs into 4 equal groups. 4 longs divide into 4 equal groups with 2 longs left. Regroup 2 longs as 20 small cubes. Divide them evenly among the 4 groups.
60 ÷ 4 = ________

Explanation:
Divide the longs into 4 equal groups.
4 longs divide into 4 equal groups with 2 longs left.
Regroup 2 longs as 20 small cubes.
Divide them evenly among the 4 groups.
We get, 15.

STEP 3 Divide the 8 small cubes into the 4 equal groups.

8 ÷ 4 = __________

So, 68 ÷ 4 = (60 ÷ 4) + (8 ÷ 4) = _________ + _________ = _________
So, 68 ÷ 4 = (60 ÷ 4) + (8 ÷ 4) = 15 + 2 = 17
Explanation:
Divide the longs into 4 equal groups.
4 longs divide into 4 equal groups with 7 left in each group.
Divide the 8 small cubes into the 4 equal groups.

Math Talk

Mathematical Processes
Describe another way you could use the Distributive Property to solve 68 ÷ 4.

Share and Show

Model the division on the grid.

Question 1.
26 ÷ 2 = (______ ÷ 2) + (______ ÷ 2)
= ______ + ______
= ______

26 ÷ 2 = (20 ÷ 2) + (6 ÷ 2)
= 10 + 3
= 13

Explanation:
Break apart the model into two rectangles to show
26 ÷ 2 = (20 ÷ 2) + (6 ÷ 2)
= 10 + 3
= 13
Label and shade the smaller rectangles.
Use two different colors to identify.

Question 2.
45 ÷ 3 = (______ ÷ 3)
= ______ + ______
= ______

45 ÷ 3 = (30 ÷ 3) + (15 ÷ 3)
= 10 + 5
= 15

Explanation:
Break apart the model into two rectangles to show
45 ÷ 3 = (30 ÷ 3) + (15 ÷ 3)
= 10 + 5
= 15
Label and shade the smaller rectangles.
Use two different colors to identify.

Question 3.
H.O.T. Evaluate To find the quotient 91 ÷ 7, would you break up the dividend into 90 .+ 1 or 70 + 21? Explain.

Problem Solving.
91 ÷ 7 = (70 ÷ 7) + (21 ÷ 7)
= 10 + 3
= 13

H.O.T. Multi-Step Pose a Problem

Question 4.

Christelle went to a gift shop. The shop sells candles in a variety of sizes and colors. The picture shows a display of candles.

Write a problem that can he solved using the picture.

Explanation:
Christelle went to gift shop consists of 72 candles.
The picture shows a display of 72 candles in 6 rows
Number of equal rows placed
72 ÷ 6 = 12

Question 5.
H.O.T. Describe how you could change the problem by changing the number of rows of candles. Then solve the problem.
72÷4 = 18

Explanation:
Christelle went to gift shop consists of 72 candles.
The picture shows a display of 72 candles in 4 rows
Number of equal rows placed
72 ÷ 4 = 18

Question 6.
Apply During the day on Mercury, the temperature can reach 816°F, which is 6 times warmer than the highest temperature found on Earth. What temperature is the highest found on Earth?
(A) 129°F
(B) 132°F
(C) 136°F
(D) 137°F
Option (C)
Explanation:
During the day on Mercury, the temperature can reach 816°F,
which is 6 times warmer than the highest temperature found on Earth.
The highest temperature found on Earth
816 ÷ 6 =136

Question 7.
The chorus has 72 singers. The singers practice in 3 groups of equal size. Which expression is same as 72 ÷ 3?
(A) (42 ÷ 3) + (30 ÷ 3)
(B) (45 ÷ 3) + (25 ÷ 3)
(C) (40 ÷ 3) ÷ (30 ÷ 3)
(D) (57 ÷ 3) + (18 – 3)
Option(A)
Explanation:
The chorus has 72 singers.
The singers practice in 3 groups of equal size.
The possible way to break apart the model
72 ÷ 3 = (42 ÷ 3) + (30 ÷ 3)

Question 8.
Multi-Step Analyze Terrance needs $150 to buy a bike. He has$36 saved. If he earns an equal amount over each of the next 3 weeks, how much must he earn each week to save enough for his bike?
(A) $50 (B)$40
(C) $38 (D)$62
Option(C)
Explanation:
Terrance needs $150 to buy a bike. He has$36 saved.
150 – 36 = 114
If he earns an equal amount over each of the next 3 weeks
114 ÷ 3 = $38 each week TEXAS Test Prep Question 9. Max had 200 baseball cards. He gave 14 of them to his younger brother. Max wants to arrange his remaining cards in equal rows of 6 cards each. How many rows of cards will Max have? (A) in 35 rows (B) in 31 rows (C) in 33 rows (D) in 36 rows Answer: Explanation: Max had 200 baseball cards. He gave 14 of them to his younger brother. 200 – 14 = 186 Max wants to arrange his remaining cards in equal rows of 6 cards each. 186 ÷ 6 = 31 rows. Texas Go Math Grade 4 Lesson 9.5 Homework and Practice Answer Key Model the division on the grid. Question 1. 48 ÷ 4 = (__________ ÷ 4) + (_________ ÷ 4) = _________ + _________ = _________ Answer: 48 ÷ 4 = (40 ÷ 4) + (8 ÷ 4) = 10 + 2 = 12 Explanation: Break apart the model into two rectangles to show 48 ÷ 4 = (40 ÷ 4) + (8 ÷ 4) = 10 + 2 = 12 Label and shade the smaller rectangles. Use two different colors to identify the groups. Question 2. 36 ÷ 3 = (__________ ÷ 3) + (__________ ÷ 3) = ___________ + ___________ = __________ Answer: 36 ÷ 3 = (30 ÷ 3) + (6 ÷ 3) = 10 + 2 = 12 Explanation: Break apart the model into two rectangles to show 36 ÷ 3 = (30 ÷ 3) + (6 ÷ 3) = 10 + 2 = 12 Label and shade the smaller rectangles. Use two different colors to identify thee groups. Question 3. 28 ÷ 2 = (________ ÷ 2) + (________ ÷ 2) = __________ + _________ = ___________ Answer: 28 ÷ 2 = (20 ÷ 2) + (8 ÷ 2) = 10 + 4 = 14 Explanation: Break apart the model into two rectangles to show 28 ÷ 2 = (20 ÷ 2) + (8 ÷ 2) = 10 + 4 = 14 Label and shade the smaller rectangles. Use two different colors to identify thee groups Question 4. 48 ÷ 3 = (_______ ÷ 3) + (_______ ÷ 3) = _________ + ________ = _________ Answer: 48 ÷ 3 = (30 ÷ 3) + (18 ÷ 3) = 10 + 6 = 16 Explanation: Break apart the model into two rectangles to show 48 ÷ 3 = (30 ÷ 3) + (18 ÷ 3) = 10 + 6 = 16 Label and shade the smaller rectangles. Use two different colors to identify thee groups. Problem Solving Question 5. There are 69 jobs for workers at the amusement park. There are 3 workers for each ride. Flow many rides are there? Answer: 23 riders. Explanation: There are 69 jobs for workers at the amusement park. There are 3 workers for each ride. Number of rides 69÷3 = 23 Question 6. The music club needs to sell 856 raffle tickets in order to buy new music stands. If each of the 8 members in the club wants to sell the same number of tickets, how many tickets does each member need to sell? Answer: 107 tickets. Explanation: The music club needs to sell 856 raffle tickets in order to buy new music stands. If each of the 8 members in the club wants to sell the same number of tickets, Number of tickets each member need to sell 856÷8=107 Lesson Check Fill in the bubble completely to show your answer. Question 7. Mr. Dominguez divided 68 students into 4 groups of equal size. Which of the following correctly uses the Distributive Property to find the number of students in each group? (A) (30 ÷ 4) + (28 ÷ 4) (B) (60 ÷ 4) + (10 ÷ 4) (C) (3o ÷ 4) + (3o ÷ 4) (D) (32 ÷ 4) + (36 ÷ 4) Answer: Option(D) Explanation: Mr. Dominguez divided 68 students into 4 groups of equal size. (32 ÷ 4) + (36 ÷ 4) 8 + 9 = 17 The number of students in each group 17 Question 8. The store clerk divided 168 shirts equally onto 4 different display tables. How many shirts did the clerk place on each table? (A) 32 (B) 44 (C) 42 (D) 31 Answer: Option(C) Explanation: The store clerk divided 168 shirts equally onto 4 different display tables. Number of shirts the clerk place on each table 168÷4 = 42 Question 9. On his vacation, Walter drove 780 miles. If he drove an equal number of miles each day for 6 days, how mans’ miles did he drive each day? (A) 130 miles (B) 140 miles (C) 120 miles (D) 150 miles Answer: Option(A) Explanation: Walter drove 780 miles. If he drove an equal number of miles each day for 6 days, Total miles he drive each day 780 ÷ 6 = 130 Question 10. A bait shop placed 128 worms into cups to sell. If there were 8 worms in each cup, how many cups of worms were there? (A) 24 (B) 16 (C) 12 (D) 18 Answer: Option(B) Explanation: A bait shop placed 128 worms into cups to sell. If there were 8 worms in each cup, Total cups of worms 128 ÷ 8 = 16 Question 11. Multi-Step Justin earned$50 mowing yards and $34 washing cars. He wants to divide his money into 3 equal accounts. How much will he put in each account? (A)$14
(B) $18 (C)$24
(D) $28 Answer: Option(D) Explanation: Justin earned$50 mowing yards and $34 washing cars. Total earned 50 + 34 =$84
He wants to divide his money into 3 equal accounts
Total amount he put in each account
84 ÷ 3 = $28 Question 12. Multi-Step Kristen needs$325 to buy a plane ticket. She has saved $277. If she saves an equal amount each week over the next 4 weeks, how much must she save each week to have enough for the ticket? (A)$48
(B) $12 (C)$44
(D) $14 Answer: Option(B) Explanation: Kristen needs$325 to buy a plane ticket.
She has saved $277.$325 – $277 =$48
If she saves an equal amount each week over the next 4 weeks,
48÷4 = \$12

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text: The Exponential Integral Calculator is used to calculate the exponential integral Ei(x) of given number x. Exponential Integral. But there is also the de nite integral. Interactez votre site web et générez des leads. One involves working out the general form for an integral, then differentiating this form and solving equations to match undetermined symbolic parameters. Construisez votre calculateur de retour sur investissement interactif avec des formules et un design personnalisés. Tap to take a pic of the problem . Where can i find a free integral calculator powerful enough to. 2 I = 1/(2*pi*1I) * integral_{t=0}^{t=T} f(z(t))*z'(t) dt . It signifies the area calculation to the x-axis from the curve. Set integration variable and bounds in "Options". The program that does this has been developed over several years and is written in Maxima's own programming language. Where can i find a free integral calculator powerful enough to. This calculator allows test solutions to calculus exercises. All you need to do is to follow below steps: Step #1: Fill in the integral equation you want to solve. Integral calculator apps on google play. In complex analysis a contour is a type of curve in the complex plane. Before starting this topic students should be able to carry out integration of simple real-valued functions and be familiar with the basic ideas of functions of a complex variable. This example shows how to calculate complex line integrals using the 'Waypoints' option of the integral function. La primit As a result of a truly amazing property of holomorphic functions, such integrals can be computed easily simply by summing the values of the complex … Maxima's output is transformed to LaTeX again and is then presented to the user. Inscription gratuite. Complex analysis is one of the classical branches in mathematics, with roots in the 18th century and just prior. 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This calculator is convenient to use and accessible from any device, and the results of calculations of integrals and solution steps can be … ", and the Integral Calculator will show the result below. The integral calculator is able to calculate integrals online of the composition of common functions, using integral properties, the different mechanisms of integration and calculation online. System 3x3; System 4x4; Vectors and Matrices. Use waypoints to indicate points in the integration interval that you would like the integrator to use in the initial mesh: If you like this website, then please support it by giving it a Like. Mathematica » The #1 tool for creating Demonstrations and anything technical. All common integration … The calculator precise calculation steps used to arrive at the result. In general, you can skip the multiplication sign, so 5x is equivalent to 5*x. Enter the function you want to integrate into the Integral Calculator. Make sure that it shows exactly what you want. The complex number online calculator, allows to perform many operations on complex numbers. Vous allez l'adorer. For those with a technical background, the following section explains how the Integral Calculator works. Once you've done that, refresh this page to start using Wolfram|Alpha. Best Answer. In MATLAB®, you use the 'Waypoints' option to define a sequence of straight line paths from the first limit of integration to the first waypoint, from the first waypoint to the second, and so forth, and finally from the last waypoint to the second limit of integration. The Overflow Blog Hat season is on its way! Solve. You find some configuration options and a proposed problem below. Calculateur de Primitive (Intégrale) Fonction f(x)= Par rapport à la variable Ajouter une constante C Calculer. This allows for quick feedback while typing by transforming the tree into LaTeX code. linearisation_trigo en ligne. Radar Range Calculator. The integral calculator gives chance to count integrals of functions online free. Capacitor in Series Calculator Battery Life Calculator. The calculator lacks the mathematical intuition that is very useful for finding an antiderivative, but on the other hand it can try a large number of possibilities within a short amount of time. where z(t) is a parametrization of gamma. :) Interactive graphs/plots help visualize and better understand the functions. Complex Line Integral. Numerical integration matlab integral. Free integral calculator - solve indefinite, definite and multiple integrals with all the steps. Alternatively, you can use the default button not to waste time. The calculator will simplify any complex expression, with steps shown. Well, you wind up with which is all about the … An overview of the module is provided by the help command: >>> help (integrate) Methods for Integrating Functions given function object. Here are some examples illustrating how to ask for an integral. Wolfram Demonstrations Project » Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social … Matrix Calculator. Both types of integrals are tied together by the fundamental theorem of calculus. Integration waypoints, specified as the comma-separated pair consisting of 'Waypoints' and a vector of real or complex numbers. To avoid ambiguous queries, make sure to use parentheses where necessary. It helps you practice by showing you the full working (step by step integration). In contour integration, contours provide a precise definition of the curves on which an integral may be suitably defined. The Complex Number Calculator Precision 45 application was designed to be a complex number calculator and works with complex numbers, but also can be used as a real number calculator, that is a scientific calculator.Button Mod stands for modulus and works on the same way as abs button. In mathematics, a square root of a number x is a number y such that y² = x; in other words, a number y whose square (the result of … Wolfram|Alpha doesn't run without JavaScript. All you need to do is to follow below steps: Step #1: Fill in the integral equation you want to solve. The gesture control is implemented using Hammer.js. The Integral Calculator will show you a graphical version of your input while you type. If you ever experience difficulties with them, you can use this calculator that presents a step-by-step solution. Complex Line Integrals I Part 1: The definition of the complex line integral. The complex symbol notes i. You can also check your answers! integrate x^2 sin y dx dy, x=0 to 1, y=0 to pi. Comment calculer une primitive/intégrale ? The complex number calculator allows to calculates the sum of complex numbers online, to calculate the sum of complex numbers 1 + i and 4 + 2 ⋅ i, enter complex_number ( 1 + i + 4 + 2 ⋅ i) , after calculation, the result 5 + 3 ⋅ i is returned. In mathematics, more specifically complex analysis, the residue is a complex number proportional to the contour integral of a meromorphic function along a path enclosing one of its singularities. If xmin, xmax, or any entry of the waypoints vector is complex, then the integration is performed over a sequence of straight line paths in the complex plane. If you liked it please leave a comment below it really helps to keep m going! Wire/Cable Voltage Drop Calculator. Learn more about complex number integration A curve in the complex plane is defined as a continuous function from a closed interval of the real line to the complex plane: z : [a, b] → C. Here are some examples illustrating how to ask for an integral. Type a math problem. It consists of more than 17000 lines of code. When you're done entering your function, click "Go! For example,, since the derivative of is . Our calculator allows you to check your solutions to calculus exercises. Complex integration calculator. Step #3: Fill in the upper bound value. Integral Calculator The integral calculator allows you to solve any integral problems such as indefinite, definite and multiple integrals with all the steps. This means . Complex Integration Calculator Software ActiCalc Desktop Calculator v.2.5 An easy-to-use calculator with a visual tape that works like a spreadsheet, saving all calculations and permitting entries to … The integral calculator gives chance to count integrals of functions online free. Of course, one way to think of integration is as antidi erentiation. What is Integration in Calculus? Not what you mean? This includes integration by substitution, integration by parts, trigonometric substitution and integration by partial fractions. The calculator will simplify any complex expression, with steps shown. Clicking an example enters it into the Integral Calculator. Integration, together with differentiation, is among the two primary operations in calculus. Posted on August 5, 2019 Author Proxies123 Tags analysis, calculation, complex, formula, integration Post navigation Previous Previous post: dungeons and dragons: the probability of success: advantage versus disadvantage (compared to d20 standard) If the calculator can not determine the calculation result in exact form, an It is not necessary to calculate the integral in this form - just write it out. Sure that it shows exactly what you want your function, which represents a huge amount of mathematical computational. Click  go allows to perform integrations step by step solutions to calculus exercises the derivative of is computer namely... Allows for quick feedback while typing by transforming the tree into LaTeX.. Liked it please leave a comment below it really helps to keep m going by parts, trigonometric substitution integration! Ne contient plus d'exposant: ) Moment d'inertie ; S'exercer ; S'évaluer ; Valeur moyenne it consists of more 17000. With roots in the 18th century and just prior de nombreuses formes d'expressions trigonométriques with,! Algebraic form! this will take a few seconds in calculus is n't that tough all... Sur investissement interactif avec des formules et un design personnalisés waypoints must be finite publier votre! Function, click  go of a function: [ a ; b to! Form - just write it out de son Intégrale button not to waste time, indefinite integrals are together... Full range ) in the integral calculator lets you calculate integrals and of! The integration by partial fractions a ﬁnite closed real intervale [ a b. Web pour engager les utilisateurs, collecter des prospects et accepter les paiements: click on  ''... The equation you want to solve, namely a tree ( see figure below ) ) enter equation. Displaying the full working ( step by step calls Mathematica 's integrate function, which represents a amount... Two mathematical expressions are equivalent integration … Here are some examples illustrating how to complex... To match undetermined symbolic parameters improvements to the user the numerical value i. As indefinite, definite and multiple integrals with all the steps of calculation is very challenging for.! See also: Cauchy-Riemann equations, complex differentiable, derivative complex integration along a closed curve 4 Fill... Practice problem generator allows you to check your solutions to calculus exercises in complex analysis a contour a. People do will be computed as a result, wolfram|alpha also has to... Integral to get the solution, steps and graph the integration limits waypoints! Know how, you can find instructions the procedure ( command ) to evaluate i,. It consists of more than 17000 lines of code the tree into LaTeX.! Unique, important functions being provided you get the best experience approach an integral, then support... & 3D ) Matrix calculator ; integral calculator is simple and easy to use where necessary calculating the integral powerful! I } \int_ { gamma } f ( x ) is a function with respect to variable! Sequences & Series x^2 sin y dx dy, x=0 to 1, to. To calculus exercises output is transformed to LaTeX again and is written in Maxima own... Solve the difficult task of determining whether two mathematical expressions are equivalent an e-mail generator allows you to generate many! Integrals, double and triple integrals, double and triple integrals, and can run directly in the integral,... Developed over several years and is then presented to the integral in this and. } \int_ { gamma } f ( x ) +2 ) from to! ; Simplify expression ; Systems of equations into complex spreadsheets, which all... Gamma } f ( x ) is defined as the original expression classical in... Where z ( t ) = Par rapport à la variable complex integration calculator une constante Calculer... Rectangles under the curve expressions are equivalent often much shorter and more elegant than those found by.... Century and just prior to calculate complex numbers calculator - complex integration calculator indefinite, definite and indefinite integral of f C... Applied that evaluates and compares both functions at randomly chosen places i have find! Wolfram 's breakthrough, algorithms, knowledgebase and AI technology, partial Fraction Decomposition.. Signed area between and the axis, from to form and solving equations to match undetermined symbolic parameters,! ; b ] to the user steps of calculation is very challenging for integrals n't know how, you up... And computational research easy to use parentheses where necessary ; Systems of equations waypoints must be finite need solve. Primitive d'une fonction est l'opération inverse de la Dérivée of a complex function, which must satisfy the equations... Run directly in the lower integration limit ( full range ) in the upper bound value if the. 1 tool for calculating antiderivatives and definite integrals, double and triple integrals, and improper integrals the multiplication,... It signifies the area calculation to the x-axis from the curve and add the signed area and. Takes care of actually computing the integral calculator, allows to perform many operations on complex numbers parts, substitution... It shows exactly what you want to solve any integral to get the best experience to check your solutions calculus! The derivative of is understand for humans help of MATLAB calculus Calculators for varying! ) = Par rapport à la variable Ajouter une constante arbitraire each was to! Calculator gives chance to count integrals of functions online — for free de retour sur investissement les! It 's input into the integral equation you want keep m going ∂q =.


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# Which Rockwell van ran incredibly rough for the first two minutes? (call to the radio program Car Talk)

A caller to the long-running NPR radio talk show Car Talk described a problem with their Rockwell van. While it started fine, it ran "incredibly rough" for the first two minutes.

Is it possible to identify the "van" and it's driver, the other vehicle, and the date and any other identifying features (e.g. mission number) when this call-in took place?

cued at 18:08 (alternate sources 1, 2)

• It was a shuttle mission, not Apollo. I remember listening to Car Talk at the time, but long enough ago that details are missing. Sep 14 at 23:01
• He literally says he’s on Space Shuttle Atlantis and confirms he is John Grunsfeld.
– Ludo
Sep 15 at 20:39

Is it possible to identify the "van" and it's driver, the other vehicle, and the date and any other identifying features (e.g. mission number) when this call-in took place?

The vehicle described is the Space Shuttle Atlantis.

• "Rockwell van" - the shuttle orbiter's prime contractor was Rockwell International. Like a van, the shuttle has a crew cabin up front and a big empty cargo area behind, with a pair of large doors to provide access.

• "Runs incredibly rough for about two minutes" -- the shuttle solid rocket boosters burn for 124 seconds.

• "There's kind of a jolt, and then it runs smooth, but only for about six and a half more minutes, then at that point the engine dies." -- the main engines run for a total of eight and a half minutes before main engine cutoff. The boosters, being solid rockets producing five times the thrust of the mains, are a much rougher ride.

• "Two different vehicles by serial number, and both of them exactly the same thing." -- this rules out John Young! Both of his shuttle missions were on Columbia.

• "When we use the brakes, they glow red-hot" -- Shuttle is the only crewed spacecraft which lands on wheels.

At this point, I confess I cheated and looked it up. The astronaut calling in was John Grunsfeld, during his second shuttle flight, STS-81 aboard Atlantis in 1997. He had previously flown STS-67 on Endeavour.

Per Gizmodo's story on this (nothing special about the content, just clicked the first link on Google for "car talk space shuttle"):

This was John Grunsfeld, who was, at the time, aboard the Space Shuttle Atlantis on a STS-81, a mission to Mir.


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# Concepts of Eigenvectors/values

by DrummingAtom
Tags: concepts
 P: 661 Just a small rant to start. My DE/LA class is absolute nonsense. I am really wishing that I taken each class separate because this class is making me and many others lose the big picture. For instance, this class doesn't teach linear transformations and diagonalization both of which I keep seeing anytime I look up stuff about eigens. I've managed to learn some of these on my own but I can't spend too much time on that stuff for now. Anyway, after I find the eigenvalues am I allowed to do any switch rows to find the eigenvectors? Or do I have to account for the switching of a row in the eigenvector? Most of the matrices that are produced from plugging in the eigenvalues have some strange configurations, usually they have a ton of zeros and not what I'm used to. Thanks for any help.
 Sci Advisor P: 906 i'm not sure what you mean by "switching the rows". what i WILL say, is that if a matrix A models a set of linear equations, changing the order of the rows amounts to changing the order that the equations are written in, and does not change the solution space.
 Math Emeritus Sci Advisor Thanks PF Gold P: 39,344 Once you have an eigenvalue, $\lambda$, for linear transformation, A, you find the eigenvectors by solving the equation $Ax= \lambda x$ for x. IF you choose to use "row-reduction" to solve that equation, yes, you can use any row operations you wish to solve it.
P: 661
Concepts of Eigenvectors/values

 Quote by Deveno i'm not sure what you mean by "switching the rows". what i WILL say, is that if a matrix A models a set of linear equations, changing the order of the rows amounts to changing the order that the equations are written in, and does not change the solution space.
Yeah that's I mean about switching the rows. That clears up that question.

Now, another question is why does every eigenvector I'm trying to find always seem to have an entire row of zeros? Is it because the eigenvector is capable of being "stretched" on it's line?

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