For questions on the vector operators: gradient, curl and divergence.

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### Error in translating the Laplacian in polar coordinates

I am translating the Laplacian into polar coordinates and somewhere in this process something goes wrong. I wish someone of you might help. Premise: some notation might not be the best, if this ...
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1 vote
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### Curl in non-orthogonal coordinates

How can I transform the curl operator into general non-orthogonal coordinates? I have tried to transform its orthogonal expression using the determinant but to no avail. I can't get the same results ...
1 vote
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### Who are the divergence free vector fields of a compact Lie group?

Let $G$ be a compact Lie group and $X\in\mathfrak{X}(G)$ a divergence free vector field. Is there a characterization of such fields? For example, if $G=S^1$, from the fact that it is parallelizable ...
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### Regularity conditions needed for divergence to equal "flux density"? (I.e. for "coordinate-free definition" to be valid?)

Background: Let $\vec{V} : \mathbb{R}^n \to \mathbb{R}^n$ be treated as a vector field, $V_i$ denote the corresponding scalar coordinate functions $\mathbb{R}^n \to \mathbb{R}$, and then if these ...
113 views

### Proof that $\frac{n^6+n^2}{n^7+3}$ converges

I'm looking at a problem that follows: Test if $$\sum_{n=1}^\infty \frac{n^6+n^2}{n^7+3}$$ converges or diverges. I think I have a proof but it seems a bit awkward. You take out the 3 in the ...
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• 4,470
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### Divergence of two orthogonal vector fields [closed]

I have the following question. Let $b_1, b_2$ two suitably regular vector fields such that $\operatorname{div}b_1=0$ and $\langle b_1, b_2\rangle=0$. What can we say about $\operatorname{div}b_2$? Is ...
• 681
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### What is the Divergence of a Spherically Symmetric Vector Fields?

A vector field is spherically symmetric about the origin if, on every sphere centered at the origin, it has constant magnitude and points either away from or toward the origin. A vector field that is ...
• 4,470
1 vote
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### How to solve this kind of surface integral with Hamilton Operator?

In $\mathbb{R}^3$, $f=\left(\frac{x}{2}\right)^2+\left(\frac{y}{2}\right)^2+\left(\frac{z}{4}\right)^2$, Surface $S$ is defined by $S=\{(x,y,z)|f(x,y,z)=1, z>0\}$, and the vector field $A$ is ...
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