# 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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### $(\nabla\cdot\mathbf{n})^{2}+(\mathbf{n}\times\nabla\times\mathbf{n})^{2}+(\mathbf{n}\cdot\nabla\times\mathbf{n})^{2}=|\nabla\bf n|^2$

As in the title, I would like to show this: $$(\nabla\cdot\mathbf{n})^{2}+(\mathbf{n}\times\nabla\times\mathbf{n})^{2}+(\mathbf{n}\cdot\nabla\times\mathbf{n})^{2}=|\nabla\bf n|^2$$ where $|\bf n|=1$....
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### Find the surface area of that part of the cylinder

Find the surface area of that part of the cylinder given by $\mathbf{r}(u,v) = 3\cos u \mathbf{i} + 3\sin u\mathbf{j} + v\mathbf{k}$ over the region where $0\leq u \leq2\pi$ and $0\leq v \leq2$. The ...
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### Angular Velocity cross product with Vorticity

I am trying to show that the following identity is true. $$\Omega\times\omega=(\omega\cdot\nabla)(\Omega\times r)$$ My professor assures me that this is correct, I've tried using vector algebra and ...
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### What is gradient operator in geometric algebra with some null basis vectors?

Consider a geometric algebra with the orthogonal basis $\{e_x,e_y\}$ where $e_x\cdot e_x = 1$ and $e_y\cdot e_y = 0$. Then define the vector function $f(r) = r$ where $r = xe_x+ye_y$. Then the ...
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### Domain Definition for Gradient Operator

I want to define to gradient operator $\nabla f$ on an $n-$variable function $f\left(x_1,\cdots,x_n\right)$. For this purpose I want to well define the spaces that the domain and the range of the ...
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### Basis Vectors in a General Curvilinear Coordinate System

I'm confused as to how does one find out the basis vectors of a curvilinear co-ordinate system. In the context of a general, arbitrary curvilinear co-ordinate system, the textbook I'm reading states ...
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### Luminosity and Apparent flux

The stars in our Galaxy have luminosities ranging from $L_{\text{min}}$ to $L_{\text{max}}$. Suppose that the number of stars per unit volume with luminosities in the range of $L$, $L+dL$ is $n(L)dL$. ...