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Problem 255

Consider the transformation $\mathrm{x}-\mathrm{u}+\mathrm{v} \quad \mathrm{y}=\mathrm{v}-\mathrm{u}^{2}$ Let \(D\) be the set in the \(u-v\) plane bounded by the lines $\mathrm{u}=0, \mathrm{v}=0\(, and \)\mathrm{u}+\mathrm{v}=2$ Find the area of \(\mathrm{D}^{*}\), the image of \(\mathrm{D}\), directly and by a change of variables.

Expert verified

The area of the image of D in the xy-plane is 4 square units, which can be found using the direct method by applying the transformation to the vertices and using the Shoelace formula, or by using a change of variables, computing the Jacobian determinant, and integrating with respect to u and v. In both methods, the resulting area is 4 square units.

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Chapter 9

Find the area of the upper hemisphere of the sphere given by the equation \(x^{2}+y^{2}+z^{2}=3^{2}\).

Chapter 9

Let \(\mathrm{S}\) be the surface \(\mathrm{S}=\left[\mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{z}^{2}=1\right\\}\) and let \(\mathrm{f}\) be the function $\mathrm{f}(\mathrm{x}, \mathrm{y}, \mathrm{z})=\mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{z}^{2}\(. Let \)\nabla \mathrm{f}^{-}$ represent the gradient of \(\mathrm{f}(\mathrm{x}, \mathrm{y}, \mathrm{z}) .\) Compute the integral $\iint_{\mathrm{S}} \nabla \mathrm{f} \cdot \mathrm{n}^{-} \mathrm{d} \mathrm{A}$.

Chapter 9

Compute the integral of the vector field $\mathrm{F}^{\rightarrow}(\mathrm{x}, \mathrm{y}, \mathrm{z})\( \)=\left(\mathrm{y},-\mathrm{x}, \mathrm{z}^{2}\right)$ over the paraboloid \(\mathrm{z}=\mathrm{x}^{2}+\mathrm{y}^{2}\) with \(0 \leq \mathrm{z} \leq 1\)

Chapter 9

Find the area of the torus whose parametrization is given by $\mathrm{x}=(\mathrm{R}-\cos \mathrm{v}) \cos \mathrm{u} \quad-\pi \leq \mathrm{u} \leq \pi$ \(\mathrm{y}=(\mathrm{R}-\cos \mathrm{v}) \sin \mathrm{u}\) \(-\pi \leq \mathrm{v} \leq \pi\) \(z=\sin v\) where \(\mathrm{R}>1\).

Chapter 9

Verify Stokes's Theorem for $\mathrm{F}^{-}(\mathrm{x}, \mathrm{y}, \mathrm{z})=\left(3 \mathrm{y},-\mathrm{xz}, \mathrm{yz}^{2}\right)$ where \(\mathrm{S}\) is the surface \(2 \mathrm{z}=\mathrm{x}^{2}+\mathrm{y}^{2}\) bounded by \(\mathrm{z}=2\) and \(\mathrm{C}\) is its boundary.

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