Multiple integrals and vector calculus (PS4): Divergence and Stokes theorems

5. Verifying Stokes theorem

The vector $\mb{A}(\mb{r}) = [y, -x, z]$. Verify Stokes' Theorem for the hemispherical surface $\mb{r} = 1$, $z \geq 0$.

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6. A closed loop integral over a cylinder

Let $C$ be any closed loop on the surface of the cylinder $(x-3)^2 + y^2 = 2$. Find $\int \mb{A} \cdot \de{\mb{r}}$ for $\mb{A} = [y, -x, 0]$.

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9. Vector line integrals

If $\phi = 2xyz^2$, $\mb{F} = [xy, -z, x^2]$ and $C$ is the curve $x = t^2$, $y = 2t$, $z = t^3$ from $t = 0$ to $t = 1$, evaluate the following:

a) Vector line integral

\[ I_1 = \int_C \phi \, \de{\mb{r}} \]

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b) Vector line integral

\[ I_2 = \int_C \mb{F} \times \, \de{\mb{r}} \]

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