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Problem 262

Let \(\mathrm{S}\) be the surface defined by \(\mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{z}^{2}=1\) and let the unit normal vector function have representations directed away from the origin. Compute the integral of the function $\mathrm{f}(\mathrm{x}, \mathrm{y}, \mathrm{z})=\mathrm{a}\( over \)\mathrm{S} .$

Expert verified

The integral of the function \(f(x,y,z) = a\) over the surface S is given by \[\iint_S f(x,y,z) \, dS = \frac{4\pi}{3}a.\]

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Chapter 9

Verify Stokes's Theorem for the vector field $\mathrm{F}^{-}(\mathrm{x}, \mathrm{y}, \mathrm{z})\( \)=(\mathrm{z}, \mathrm{x}, \mathrm{y})$ where \(\mathrm{S}\) is defined by $\mathrm{z}=4-\mathrm{x}^{2}-\mathrm{y}^{3}, \mathrm{z} \geq 0$

Chapter 9

Let \(U\) be the interior of a closed surface \(S\). Let \(f, g\) be functions. Let \(\nabla \mathrm{f}\) be the gradient of \(\mathrm{f}\) and $\nabla^{2} \mathrm{f}\( be the divergence of the gradient of \)\mathrm{f}$. Prove: a) $\iint_{\mathrm{S}} \mathrm{f}(\nabla \mathrm{g}) \cdot \mathrm{n}^{-} \mathrm{da}=\iiint_{\mathrm{U}}\left(\mathrm{f} \nabla^{2} \mathrm{~g}+\nabla \mathrm{f} \cdot \nabla \mathrm{g}\right) \mathrm{dV}$. b) $\iint_{\mathrm{S}}(\mathrm{f} \nabla \mathrm{g}-\mathrm{g} \nabla \mathrm{f}) \cdot \mathrm{n}^{-} \mathrm{da}=\iiint_{\mathrm{U}}\left(\mathrm{f} \nabla^{2} \mathrm{~g}-\mathrm{g} \nabla^{2} \mathrm{f}\right) \mathrm{dV}$.

Chapter 9

Find the integral of the function $\mathrm{f}(\mathrm{x}, \mathrm{y}, \mathrm{z})=\mathrm{x}\( over the surface \)z=x^{2}+y\( with \)x, y$ satisfying the inequalities \(0 \leq x \leq 1\) and \(-1 \leq y \leq 1\)

Chapter 9

Let $\mathrm{F}^{\rightarrow}(\mathrm{x}, \mathrm{y}, \mathrm{z})=\left[\left\\{-\mathrm{y} /\left(\mathrm{x}^{2}+\mathrm{y}^{2}\right)\right\\},\left\\{\mathrm{x} /\left(\mathrm{x}^{2}+\mathrm{y}^{2}\right)\right\\}, 0\right]$ Evaluate \(\oint_{C} F^{-} \cdot d r^{-}\) where \(C\) is the circle \(x^{2}+y^{2}=1 .\) Also evaluate $\int_{\mathrm{S}}\left(\operatorname{curl} \mathrm{F}^{-}\right) \cdot \mathrm{n}^{\rightarrow} \mathrm{d} \mathrm{A}$ and explain the results.

Chapter 9

Let \(\mathrm{S}\) be the hemisphere given by $\mathrm{S}:\left\\{(\mathrm{x}, \mathrm{y}, \mathrm{z}) \mid \mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{z}^{2}\right.$ \(=1, z>0\\}\). Let \(f\) be the function \(f(x, y, z)=x^{2} y^{2} z\). Compute the integral \(\iint_{\mathrm{S}} \mathrm{f} \mathrm{d} \mathrm{A}\).

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