# Tagged Questions

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### Intuition behind divergence?

$\overrightarrow f = 3x\overrightarrow i - 3y\overrightarrow j$ $\overrightarrow g = 3x\overrightarrow i + 3y\overrightarrow j$ If I calculate the divergence of $f$ I get $0$. If I calculate the ...
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### Quick Question: dot product with del

Is $(v \cdot \nabla)F = (\nabla F) \cdot v$? I'm not quite sure.
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### Curl and gradient properties for $f ( r)\vec r$

I need to show that the curl of $f( r) \vec{r}$ is $0$. I think I can use this property: $$\operatorname{curl}(Av) = \operatorname{grad}(A)\times v+A \operatorname{curl}(v)$$ I have started ...
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### Integration against divergence free vector fields

Let $\chi:\Omega\to \mathbb{R}^n$ be a vector field on a bounded, smooth domain $\Omega \subset \mathbb{R}^n$. Assume that for any divergence free vector field $\eta:\Omega \to \mathbb{R}^n$ we have ...
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### Flow of fluid through a really tricky closed surface S (divergence theorem)

Considering a fluid whose velocity field is $\vec{v}(x,y,z)= (y^{3}e^{-z^{2}} + x)\vec{i} + (ze^{x} + y^{2})\vec{j} + (cos(x^{2}+ y^{2}) +2z)\vec{k}$ Calculate the flow of fluid through the closed ...
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### Finding a closed line integral using Stokes' Theorem

Find the line integral $\int_C \vec{F} \cdot \vec{dS}$, where $C$ is the circle of radius 3 in the $xz$-plane oriented counter-clockwise when looking from the points $(0, 1, 0)$ into the plane and ...
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### is it necessary that curl of 2d vector is perpendicular to the plane.

I am just confused, help me guys. The question comes up, because we say that curl is either clockwise or anti-clockwise at a point.
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### Integral and area in a plane [closed]

Show that the value of the following integral is proportional to the area included in a curve $C$: $$\oint_C3y\,dx+3z\,dy-x\, dz,$$ where $C$ is a smooth, closed curve upon the $2x+2y+z=2$ plain.
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