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\newcommand{\myCourseNumber}{Math 20E}
\newcommand{\myAssignmentDueDate}{11:00pm Tuesday, October 10, 2023}
\newcommand{\myAssignmentTitle}{Homework Assignment 1}
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\begin{questions}

\question Find an equation for the tangent plane to \enspace $\displaystyle f(x, y, z) = \frac{x y z}{x^2 + y^2 + z^2}$ \enspace at $(x_0, y_0, z_0) = (1,0,1)$.

\question Compute the matrix of partial derivatives of \enspace $f(x,y,z) = (x + y, x - y, x y)$.

\question Let $w = x^2 + y^2 + z^2$, $x = u v$, $y = u \cos(v)$, and $z = u \sin(v)$.  Use the chain rule to find $\frac{\partial w}{\partial u}$ when $(u,v) = (1, 0)$.

\question Let \enspace $g(u,v) = (e^u, u + \sin(v))$ \enspace and \enspace $f(x,y,z) = (x y, y z)$. \enspace Compute $\mathbf{D}(g \circ f)(0,1,0)$ using the chain rule.

\question Evaluate the iterated integral \enspace $\displaystyle \int_1^3 \int_1^2 \frac{x y}{\left(x^2 + y^2\right)^{\frac{3}{2}}} \, dx \, dy$.

\question Evaluate the double integral \enspace $\displaystyle \iint_R \left(x^2 y^2 + x\right) \, dy \, dx$, \enspace where $R = [0,2] \times [-1,0]$.

\question Compute the volume of the region over the rectangle $[0,1] \times [0,1]$ and under the graph $z = x y$.

\question Compute the volume of the solid bounded by the $x z$ plane, the $y z$ plane, the $x y$ plane, the planes $x = 1$ and $y = 1$, and the surface $z = x^2 + y^4$.

\question Evaluate the double integral \enspace $\displaystyle \iint_D x y \, dA$, \enspace where $D$ is the triangular region whose vertices are $(0,0)$, $(0,2)$, $(2,0)$.

\question Evaluate \enspace $\displaystyle \iint_D y \, dA$, \enspace where $D$ is the set of points $(x,y)$ such that $\displaystyle 0 \leq \frac{2 x}{\pi} \leq y \leq \sin(x)$.


\end{questions}

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