Collections TT13: Multivariable Calculus

3. Volume and surface integrals

a) Calculate the volume of the cylinder $x^2/a^2 + y^2/b^2 = 1$ between the planes $z = 0$ and $z = 1-x/a$.

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b) let $S$ be the surface $x^2 + y^2 + z^2 = 1$, with unit outward normal n and let v be the vector field \[ \mathbf{v} = [x^2 + yz, x^2 = z^2 + y, (x^3 y + 1)z ]. \]

Evluate \[ \iint_S \mathbf{v} \cdot \mathbf{n} \, \de{S}. \]

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4. Stokes' Theorem

a) i. State Stokes' Theorem for a smooth vector field $\mathbf{F}: \mathbb{R}^3 \to \mathbb{R}^3$, explaining the condition about orientations involved.

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ii. Let $\mb{a} \in\mathbb{R}^3$ be a fixed vector and consider the vector field $\mb{r} = [x, y, z]$. Prove that \[ \nabla \wedge (\mb{a} \wedge \mb{r}) = 2\mb{a}. \]

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iii. Hence, or otherwise, show that if $S$ is a smooth surface bounded by a simple closed curve, $C$, then \[ \int_C \mb{r} \wedge \de{\mb{s}} = 2 \iint_S \de\mb{S}. \]

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b) Verify Stokes' Theorem when $\mb{F}(\mb{r}) = [-y, x^2, 0]$, where $S$ is the surface of the hemisphere $x^2 + y^2 + z^2 = a^2$, with $z \geq 0$, with bounding curve, $C$, the circumference of the circle $x^2 + y^2 = a^2$.

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