# 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).

4,091 questions
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### Prove the diagonals of a parallelpiped bisect each other

I am stuck on how to Prove the diagonals of a parallelpiped bisect each other I have been given the hint to make one of the corners O. If possible I would just like a push in the right direction. ...
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### Vector calculus problem involving planes

I'm working on a vector calculus problem (provided below) and the issue is I'm getting two different answers, and I'm not sure which is right. The question is as follows: Calculate the surface ...
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### How to find counter example to show that a vector field is not conservative? [on hold]

I know that to show it is not a conservation vector field I could find a closed parameterized path and then calculate the line integral such that it is not equal to zero, I've tried using a unit ...
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### How to find a scalar field, given its gradient?

Given that a vector field $\mathbf v$ satisfies $$\nabla \cdot\mathbf v =0.$$ How can I find $\phi(\mathbf r)$ such that $\mathbf v= \nabla \phi$?
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### Is The Dirac Delta Communitive Over a Cross Product?

Is the statement, $\delta(\vec{x}) \vec{A}\times \vec{B} \equiv \vec{A}\times \vec{B}\ \delta(\vec{x})$ True? If so, how would I go about proving this?
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### Proof of Dirac Delta Sifting Property With Volume Integral

The Dirac delta function possess the sifting property which states, $\int _{a}^{b} f'( x) \delta ( x-x') dx'=\begin{cases} f( x) & a< x< b\\ 0 & otherwise \end{cases}$ I suspect by ...
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### Is Leibniz integral rule (basic form) allowed in this (physics) improper integral? Why?

Electric potential at a point inside the charge distribution is: $\displaystyle \psi (\mathbf{r})=\lim\limits_{\delta \to 0} \int_{V'-\delta} \dfrac{\rho (\mathbf{r'})}{|\mathbf{r}-\mathbf{r'}|} dV'$ ...
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### Finding gradient of a function

https://i.stack.imgur.com/1EIpR.jpg https://i.stack.imgur.com/O7OCP.jpg I have no idea how the answer is calculated, because there are some transpose matrix. I don't know i should treat them as ...
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### Gradient field integrable

My question is, for a gradient field $G$ defined on $B_1 \setminus \{0\}$, can we show that $f \in L^1_{loc}(B_1)$ for $f$, s.t. $\nabla f =G$?
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### How to solve the derivatives of the compound functions in vector form？

for example： $f(x)=(xx^T)^{-\frac{1}{2}}x$, where $x \in \mathbb R_{+}^{1\times d}$ is a row vector. It is hoped that there will be specific theoretical basis (formula derivation and origin) (...
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### Let $G$ be a vector field defined in $A = \mathbb{R}^n -{0}$ with $\operatorname{div}G = 0$ in $A$.

Let $G$ be a vector field defined in $A = \mathbb{R}^n -{0}$ with $\operatorname{div} G = 0$ in $A$. let $M_1$ and $M_2$ be compact n-manifolds in $\mathbb{R}^n$, such that the origin is contained in ...
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### Value of $a,b,c$ that makes the vector field a gradient of a function

I am given vector field $F = (ay^2 + 2cxz)i + (ybx+ycz)j + (ay^2+cx^2)k$ and I am supposed to find values $a,b,c$ that would make $f(x,y,z)$ a function. I first took partial derivatives and I get the ...
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### in 2D dimensional plane, is it problematic to have Frenet-Serret frame with zero curvature?

I have a Frenet-Serret frame moving on a 2-D plane. As of now, I do not care about the binormal vector. So my equations are given by, \begin{align} \dot{T} = v\kappa N \\ \dot{N} = -v\kappa T \end{...
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### Is it “Valid” to prove Stokes' Theorem with Green's Theorem?

In my Vector Calculus course, the professor is rigorous enough that we do a decent number of proofs, but not rigorous enough to go all the way with manifolds/differential forms/etc. One proof in ...
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### General Solution to Laplace's Equation in $\mathbb{R^3}$

I am trying to find the Green's function for $\ \nabla^2\phi=S(x)\$ for $\ x\in\mathbb{R^3}$ and express the general solution to Laplace's equation in $\mathbb{R^3}$. To find the Green's function, I ...