# Tagged Questions

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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This is a question from classical electrodynamics, but it's the maths of it I need some help in. I have fields $\rho\left(\vec{r},t\right)$, $\vec{J}\left(\vec{r},t\right)$, $\vec{E}\left(\vec{r},t\... 0answers 48 views ### Surface Integral of a Vector Field Consider the vector field$\vec{F}(x,y,z)=(x,y,z)$, and the surface parametrized by$\Phi(u,v)=(uv,\frac{u^{2}+v^{2}}{2},\frac{u^{2}-v^{2}}{2})$where$0\leq u\leq 2 $and$0\leq v \leq 4$. Evaluate ... 0answers 27 views ### Vector Calculus: solution to Poisson equation This is problem 8.4.17. from Marsden Vector Calculus book. Let$\rho$be a continuous function which vanishes outside a 3D region$W$. Define$\phi(\textbf{p})=\displaystyle\iiint_W\frac{\rho(\...
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Let $f:\mathbb R^n \to \mathbb R$ be a function that satisfies the equation: $$\nabla^T f = - (\nabla^T \phi ) f$$ where $\phi:\mathbb R^n \to \mathbb R$ is some given function and $\nabla^T$ is the ...
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### Determinant of non-square Jacobian

Suppose I have a 3d solid in ${\bf R}^4$ which can be parametrized by the function $F:W\subset{\bf R}^3\rightarrow{\bf R}^4$. Now suppose I want to calculate the volume of this solid. Then naively I ...
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### Gradient control by div and curl

For any $u\in W^{1,p}(\mathbb R^3)$ how to prove: $$\|\nabla u\|_p \leq C(\|\operatorname{div}u\|_p+\|\operatorname{curl}u\|_p)$$ Any suggestions? Thanks!
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### Gauss´s law proof “details”

I know that this question has already been asked multiple times but I´m still not getting on the mathematical details behind the answers... So I hope that this question doesn´t get closed; also I ...