# Questions tagged [vector-analysis]

Questions related to understanding line integrals, vector fields, surface integrals, the theorems of Gauss, Green and Stokes. Some related tags are (multivariable-calculus) and (differential-geometry).

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### About the conservative and not conservative vector field

I was wondering why (3) is conservative while (4) is not. I thought a conservative vector field should be a closed loop, and the two vector fields should have similar results since they both excluded ...
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### If two functions are equal, why is it not a contradiction for one to be at a minimum while another is at a maximum?

“Consider the PDE $u=u_{xx}+u_{yy}$ in dimension 2, where $u:\mathbb{D}\to\mathbb{R}$ is a scalar function on the unit disk $\mathbb{D}=\{(x,y)|x^2+y^2\leq1\}$. Show that if $u$ is a solution that is ...
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### Line integral of a Vector Field around a Closed path

This is just a little question. Suppose you want to evaluate an integral around a closed path formed by a curve $C(t)$(only one curve), I suspect that the result would be $0$, because you will do an ...
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### Proof of Vector Triple Product by Directions and Magnitudes [duplicate]

I'm trying to prove the vector triple product expansion by magnitude and direction: $$\vec{a} \times(\vec{b}\times \vec{c})=(\vec a\cdot \vec{c})\vec{b} - (\vec{a} \cdot \vec{b})\vec{c}$$ The ...
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### Line integral counterclockwise.

let $F$ be a vector field, such that $$F(x,y)=\langle x^2y^2,xy^2\rangle$$ Calculate the integral of $F$ along the closed formed by the line $x=1$ and the parabola $y^2=x$. first, we have to ...
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### Does Gauss' theorem and Stokes' theorem still hold for distributions?

We can define the derivative of distributions and then we can define their grad, div and curl, so do the Stokes' theorem and Gauss' theorem still hold for distributions? if not, under which condition ...
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### How to generalize the various vector calculus theorems to distributions?

https://en.wikipedia.org/wiki/Vector_calculus_identities Here are the list of vector calculus identities, in the proof of these identities, we all assume that these functions are $C^k$ in an open set, ...
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### Curl of cross product of two vector-valued functions

In Madsen and Tornehave's From Calculus to Cohomology Page 4, there's a formula which the text says can be obtained by straightforward calculations but I don't know how, I hoped to use vector calculus ...
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### Can we obtain the one parameter function that represents the arguments progression on gradient progression of a multiple parameters function?

Assume that $(a_1,a_2,...,a_n) \in \Bbb R^n$ and the $F: \Bbb R^n \rightarrow \Bbb R$ $(x_1,x_2,...,x_n) \longmapsto F(x_1,x_2,...,x_n)$ is differentiable function at all parameters. Is there any math ...
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### Stokes' Theorem - cylindrical coordinates

I'm currently having an issue with verifying the validity of Stokes' Theorem on a particular problem. I can solve the problem by using Stokes' theorem to turn a surface integral of the curl of a ...
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### Taylor expansion of $\frac {1}{|x-y|}$with x and y two vectors

This equation comes from a physics script on electrodynamics, saying that this equation comes from a Taylor series expansion. I understand the first equality, but not the second one. It is really not ...
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### To find the angle a particle makes with the horizontal at any time 't'

Should you vector sum the position vectors at this time 't' or vector sum the velocity vector at this time 't' to find the angle a particle makes with the horizontal at any time 't'
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### Proving $(\nabla \times \mathbf{v}) \cdot \mathbf{c} = \nabla \cdot (\mathbf{v} \times \mathbf{c})$ using cylindrical coordinates

Assuming the form of divergence in polar coordinates is known, I am attempting to use the following definition of the curl of a vector field to determine the form of the curl in cylindrical ...
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### Relation between a scalar function and its “inverse” gradient

I have a surface defined by the implicit formula: $$F(\boldsymbol{x}) = 0$$ where $\boldsymbol{x} \in \mathbb{R}^n$ and $F : \mathbb{R}^n \to \mathbb{R}$ (actually I have $n=3$, but if the result ...
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### The proof of the multivariable chain rule (Vector form)

Let $f$ be a function defined on an open set $U$, and let $X(t)$ be a curve such that $X(t)$ is contained in $U$ for all $t$. We define the function $g(t)=f(X(t))$, I want to know what is $dg/dt$, ...
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### What is the relationship between $\mathbb{E}(\|\mathbf{X}\|)$ and $\|\mathbf{Y}\|$?

Given that we have two vectors $\mathbf{X}\in\mathbb{R}^N$ and $\mathbf{Y}\in\mathbb{R}^N$, where $\mathbf{X}$ is a random variable with $\mathbb{E}(\mathbf{X}) = \mathbf{Y}$. Here $\mathbb{E}$ ...
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### Let $f(X)=\|X\|$ and $g(X)=\|X\|^2$, $X\in \mathbb{R}^n$.

Let $f(X)=\|X\|$ and $g(X)=\|X\|^2$, $X\in \mathbb{R}^n$. Give an example of points $X$ and $Y$ at distance one unit, such that. $a)$ $|g(Y) - g(X)|> 10^{60}$ $b)$ Show that $f$ is uniformly ...
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### Use the chain's rule to find $f'(x,y,z)$ if $f(x,y,z)=\int_{\sin x}^{yz} g(t)dt$

Let $g:\mathbb{R} \rightarrow \mathbb{R}$ continuous and $f:\mathbb{R}^{3} \rightarrow \mathbb{R}$ defined for $f(x,y,z)=\int_{\sin x}^{yz} g(t)dt$. Express $f$ like the composition of differentiable ...
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### Evaluate the line integral $\int_\gamma \frac{-y}{x^2+y^2}dx + \frac{x}{x^2+y^2}dy + \ln{(z^4+z^2+1)}dz$ using Stokes' Theorem

I found the following problem in a textbook (translated): Evaluate $$\int_\gamma \frac{-y}{x^2+y^2}dx + \frac{x}{x^2+y^2}dy + \ln{(z^4+z^2+1)}dz$$ where $\;\gamma\;$ is given by the intersection ...
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### A vector-product formula

Let $\mathbf a, \mathbf b,$ be vectors in $\mathbb R^3$ and let $R$ be a $3\times 3$ matrix. Then we have $$^t\!R\bigl(R\mathbf a\times R\mathbf b\bigr)=(\det R)(\mathbf a\times \mathbf b). \tag 1$$ ...
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### Is there a way to unify curl, gradient, and divergence operators as special cases of a more general operator?

I realize there's probably an answer to this question somewhere on this site, but it would seem I'm having trouble picking the right search terms. In my multivariable calculus class, the professor ...
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### Vena contracta effect, why can't streamlines change direction abruptly?

I am curious about the common explanation for the vena contracta effect that occurs as a flow moves around a sharp corner, or within a free jet of liquid issuing from a nozzle. The explanation goes ...