# 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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### Equation of a surface from its flux and divergence. [closed]

I have been trying to find a equation for the equipotential surface of a dipole , so I started with a simpler system of a singular charged particle , here are few things I know about the equipotential ...
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### Question on a body rotating about axis [closed]

A rigid Body is rotating at the rate of $3$ rads about an axis AB,Where A and B Point are ($1,-2,1$) and ($3,-4,2$) Then velocity of point P at $(5,-1,-1)$ of body is??
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### Domain $U$ for "if $\nabla\times \mathbf{F}=\mathbf{0}$, then $\mathbf{F}$ is conservative"

$U$ is the union of two disjoint open simply connected sets. $\mathbf{F}:U\to\mathbb{R}^3$ is $C^1$. Then is it true that if $\nabla\times \mathbf{F}=\mathbf{0}$, then $\mathbf{F}$ is conservative? ...
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### Stokes theorem to calculate line integral

Let $\gamma$ be the intersection between $z=x^2+y^2$ and the plane $z=1+2x$. Calculate the work done by the field $F=(0,x,-y)$ when the curve $\gamma$ traverses on lap in positive direction seen from ...
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### Calculus Identities

I am trying to write an expression to $\partial_t \|\nabla u\|_{L^p(\Omega)}^p.$ Here $\Omega$ is a smooth domain, the function $u$ has no regularity problems (you can take it smooth) and the normal ...
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### Problem with a proof on a vector calculus book

I have seen a proof that concludes this: $\iiint_{V} \nabla \times \mathbf{B} \, dV = \iint_{S} \mathbf{n} \times \mathbf{B}\,dS$ My question is: if is it possible to take the volume integral of a ...
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### Determining condition of coplanarity

Determine the value of $\lambda$ such that the vectors $$5\vec{a}+6\vec{b}+7\vec{c},7\vec{a}+\lambda\vec{b}+9\vec{c},3\vec{a}+20\vec{b}+5\vec{c}$$ are coplanar given that $\vec{a},\vec{b},\vec{c}$ are ...
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### Is the vector function $\mathbf r(t) = \langle t^3, t^3 \rangle$ smooth at $t = 0$?

This was confusing me when learning about curvature and smoothness. The condition for smoothness on interval $I$ is given as: $\mathbf r'$ is continuous; $\mathbf r'(t) \neq \mathbf 0$. In this ...
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### Jacobian vs Gradient in polar coordinates [duplicate]

According to the wikipedia definition of Jacobian, if $f:\mathbb{R}^{n}\to \mathbb{R}$, then $Jf=\nabla f^{T}$, where $Jf$ is the jacobian of $f$ and $\nabla f$ is its gradient. Note: I simply took ...
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### What are $r(s)$ and $\theta(s)$ for an ellipse?

Book: https://www.amazon.com/Vector-Calculus-Dover-Books-Mathematics/dp/0486466205 We have this problem here on page 89. I think I was able to solve a) and c). I know that b) is an ellipse. And I ...
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