# Questions tagged [stokes-theorem]

For questions about Stokes' theorem. Stokes' theorem relates the integral of a differential form over the boundary of some orientable manifold M is equal to the integral of its exterior derivative over the whole of M.

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### Stokes Theorem for the intersection of $z=x^2+y^2$ and $z=x+2$ and parameterizing the portion of the plane inside of $z=x^2+y^2$

I have been trying to solve the surface integral $$\int \int_S curl(\vec{F}) \cdot \vec{dS}$$ using stokes theorem for the vector field $$F(x,y,z)=\left(xz,yz,xy\right)$$ and where $S$ is the ...
• 190
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### Confirmation of calculation: $\displaystyle \iint_S (yz\,\hat{\imath}+zx\,\hat{\jmath}+xy\,\hat{k})dS$

The question goes like this: Evaluate $\displaystyle \iint_S (yz\,\hat{\imath}+zx\,\hat{\jmath}+xy\,\hat{k})\,dS$ where $S$ is the surface of the sphere $x^2+y^2+z^2=a^2$ in the first octant. I saw ...
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### Gradient and differential surface element

In my textbook There is the following step made when dealing with a certaion surface integral over a random surface: $$\int_S(\nabla\otimes\vec{v})\cdot d\vec{S}=\int_S\nabla(\vec{v}\cdot d\vec{S})$$ ...
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### Integral of a 2-form on a smoothly parametrized surface

I am studying Stokes' Theorem and its applications, and dealing with the following two problems: Consider the surface $S ={(x, y, z) \mid z=1-x^{2}-y^{2}>0}$. Find a smooth parametrisation $H$ of ...
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