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The question is

If $\vec { F } = x \hat { i } + y \hat { j } + z \hat { k }$ then find the value of $\int \int _ { S } \vec { F } \cdot \hat { n } d s$ where $S$ is the sphere $\{(x,y,z)\in\mathbb{R}^3 \vert x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4\}$.

Here if I'll calculate the answer using gauss divergence theorem, then I have to find the volume integral of the function, but how will I take the limits of the integration. Can someone help please? Thank you.

saz
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Hawkingo
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    What about $$0\le \rho\le2;,;;0\le\theta\le2\pi;,;;0\le\phi\le\pi;?$$ These is the parametrization of the given sphere in spherical coordinates... – DonAntonio Jan 20 '19 at 15:30

1 Answers1

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You have ${\bf F}({\bf x}):={\bf x}$, and on the sphere $S$ of radius $2$ you have ${\bf n}({\bf x})={1\over2}{\bf x}$. Furthermore $$\int_S{\rm d}\omega=4\cdot4\pi\ .$$ It follows that $$\int_S{\bf F}({\bf x})\cdot{\bf n}({\bf x})\>{\rm d}\omega=\int_S{\bf x}\cdot{1\over2}{\bf x}\>{\rm d}\omega=2\cdot16\pi=32\pi\ .$$

Christian Blatter
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  • sir,can you provide some method in which one can derive the limits directly from the given equation of the surface without knowing the geometry of the surface or plotting a graph? – Hawkingo Jan 21 '19 at 18:40