Chapter 9

Find the directional derivative(s) of \(f(x, y)=x+y^{2}\) at \((3,4)\) in the direction of a tangent vector to the graph of \(2 x^{2}+y^{2}=9\) at \((2,1)\)

Evaluate the given iterated integral by changing to polar coordinates. $$ \int_{-5}^{5} \int_{0}^{\sqrt{25-x^{2}}}(4 x+3 y) d y d x $$

Suppose \(\mathbf{F}\) is a conservative force field with potential function \(\phi\). In physics the function \(p=-\phi\) is called potential energy. Since \(\mathbf{F}=-\nabla p\), Newton's second law becomes $$ m \mathbf{r}^{\prime \prime}=-\nabla p \quad \text { or } \quad m \frac{d \mathbf{v}}{d t}+\nabla p=\mathbf{0} $$ By integrating \(m \frac{d v}{d t} \cdot \frac{d \mathbf{r}}{d t}+\nabla p \cdot \frac{d \mathbf{r}}{d t}=0\) with respect to \(t\), derive the law of conservation of mechanical energy: \(\frac{1}{2} m v^{2}+p=\) constant.

Verify the given identity. Assume continuity of all partial derivatives. \(\operatorname{div}(\mathbf{F} \times \mathbf{G})=\mathbf{G} \cdot \operatorname{curl} \mathbf{F}-\mathbf{F} \cdot \operatorname{curl} \mathbf{G}\)

Find the directional derivatives(s) of \(f(x, y)=x+y^{2}\) at \((3,4)\) in the direction of a tangent vector to the graph of \(2 x^{2}+y^{2}=9\) at \((2,1)\).

Find the indicated derivative. Assume that all vector functions are differentiable. $$ \frac{d}{d t}\left[\mathbf{r}_{1}(2 t)+\mathbf{r}_{2}\left(\frac{1}{t}\right)\right] $$

In Problems, evaluate the given iterated integral by changing to polar coordinates. $$ \int_{0}^{1} \int_{0}^{\sqrt{1-y^{2}}} \frac{1}{1+\sqrt{x^{2}+y^{2}}} d x d y $$

Let \(\mathbf{F}\) be a vector field. Find the flux of \(\mathbf{F}\) through the given surface. Assume the surface \(S\) is oriented upward. \(\mathbf{F}=-x^{3} y \mathbf{i}+y z^{3} \mathbf{j}+x y^{3} \mathbf{k} ; S\) that portion of the plane \(z=x+3\) in the first octant within the cylinder \(x^{2}+y^{2}=2 x\)

Find the volume of the solid bounded by the graphs of the given equations. $$ z=4-x^{2}-\frac{1}{4} y^{2}, z=0 $$

In Problems \(25-32\), verify the given identity. Assume continuity of all partial derivatives. $$ \operatorname{curl}(\operatorname{curl} \mathbf{F}+\operatorname{grad} f)=\operatorname{curl}(\operatorname{curl} \mathbf{F}) $$

Find the work done by the force \(\mathbf{F}(x, y)=2 x y \mathbf{i}+4 y^{2} \mathbf{j}\) acting along the piecewise-smooth curve consisting of the line segments from \((-2,2)\) to \((0,0)\) and from \((0,0)\) to \((2,3)\).

Show that the sum of the \(x-, y-\), and \(z\) -intercepts of every tangent plane to the graph of \(\sqrt{x}+\sqrt{y}+\sqrt{z}=\sqrt{a}\) \(a>0\), is the number \(a\).

Find the first partial derivatives of the given function. $$ G(p, q, r, s)=\left(p^{2} q^{3}\right)^{r^{4} s^{5}} $$

If \(f(x, y)=x^{2}+x y+y^{2}-x\), find all points where \(D_{\mathrm{u}} f(x, y)\) in the direction of \(\mathbf{u}=(1 / \sqrt{2})(\mathbf{i}+\mathbf{j})\) is zero.

Evaluate the given iterated integral by changing to polar coordinates. $$ \int_{0}^{1} \int_{0}^{\sqrt{1-y^{2}}} \frac{1}{1+\sqrt{x^{2}+y^{2}}} d x d y $$

Verify the given identity. Assume continuity of all partial derivatives. \(\operatorname{curl}(\operatorname{curl} \mathbf{F}+\operatorname{grad} f)=\operatorname{curl}(\operatorname{curl} \mathbf{F})\)

Find the indicated derivative. Assume that all vector functions are differentiable. $$ \frac{d}{d t}\left[t^{3} \mathbf{r}\left(t^{2}\right)\right] $$

Find the moment of inertia about the z-axis of the solid in the first octant that is bounded by the coordinate planes and the graph of \(x+y+z=1\) if the density is constant.

Let \(\mathbf{F}\) be a vector field. Find the flux of \(\mathbf{F}\) through the given surface. Assume the surface \(S\) is oriented upward. \(\mathbf{F}=\frac{1}{2} x^{2} \mathbf{i}+\frac{1}{2} y^{2} \mathbf{j}+z \mathbf{k} ; S\) that portion of the paraboloid \(z=4-x^{2}-y^{2}\) for \(0 \leq z \leq 4\)

Show that $$\nabla \cdot \nabla f=\frac{\partial^{2} f}{\partial x^{2}}+\frac{\partial^{2} f}{\partial y^{2}}+\frac{\partial^{2} f}{\partial z^{2}}$$ This is known as the Laplacian and is also written \(\nabla^{2} f\).

Find the work done by the force \(\mathbf{F}(x, y)=(x+2 y) \mathbf{i}+\) (6y \(-2 x\) )j acting counterclockwise once around the triangle with vertices \((1,1),(3,1)\), and \((3,2)\).

In Problems, evaluate the given integral. $$ \int_{-1}^{2}\left(t \mathrm{i}+3 t^{2} \mathrm{j}+4 t^{3} \mathbf{k}\right) d t $$

Verify that the given function satisfies Laplace's equation: $$\frac{\partial^{2} z}{\partial x^{2}}+\frac{\partial^{2} z}{\partial y^{2}}=0$$ $$ z=\ln \left(x^{2}+y^{2}\right) $$

Suppose \(\nabla f(a, b)=4 \mathbf{i}+3 \mathbf{j}\). Find a unit vector \(\mathbf{u}\) so that (a) \(D_{\mathrm{u}} f(a, b)=0\) (b) \(D_{\mathrm{a}} f(a, b)\) is a maximum, and (c) \(D_{\mathrm{a}} f(a, b)\) is a minimum.

Find the volume of the solid bounded by the graphs of the given equations. $$ z=4-y^{2}, x^{2}+y^{2}=2 x, z=0 $$

Find parametric equations for the normal line at the indicated point. In Problems 35 and 36, find symmetric equations for the normal line. $$ x^{2}+2 y^{2}+z^{2}=4 ;(1,-1,1) $$

Evaluate the given integral. $$ \int_{-1}^{2}\left(t \mathbf{i}+3 t^{2} \mathbf{j}+4 t^{3} \mathbf{k}\right) d t $$

Find the moment of inertia about the \(y\) -axis of the solid bounded by the graphsof \(z=y, z=4-y, z=1, z=0, x=2\) and \(x=0\) if the density at a point \(P\) is directly proportional to the distance from the \(y z\) -plane.

Let \(\mathbf{F}\) be a vector field. Find the flux of \(\mathbf{F}\) through the given surface. Assume the surface \(S\) is oriented upward. \(\mathbf{F}=e^{y} \mathbf{i}+e^{x} \mathbf{j}+18 y \mathbf{k} ; S\) that portion of the plane \(x+y+z=6\) in the first octant

Find the volume of the solid bounded by the graphs of the given equations. $$ z=1-x^{2}, z=1-y^{2}, x=0, y=0, z=0, \text { first octant } $$

Find the work done by the force \(\mathbf{F}(x, y, z)=y z \mathbf{i}+x z \mathbf{j}+x y \mathbf{k}\) acting along the curve given by \(\mathbf{r}(t)=t^{3} \mathbf{i}+t^{2} \mathbf{j}+t \mathbf{k}\) from \(t=1\) to \(t=3\)

In Problems, evaluate the given integral. $$ \int_{0}^{4}(\sqrt{2 t+1} \mathbf{i}-\sqrt{t} \mathbf{j}+\sin \pi t \mathbf{k}) d t $$

Find parametric equations for the normal line at the indicated point. In Problems 35 and 36, find symmetric equations for the normal line. $$ z=2 x^{2}-4 y^{2} ;(3,-2,2) $$

Verify that the given function satisfies Laplace's equation: $$ \frac{\partial^{2} z}{\partial x^{2}}+\frac{\partial^{2} z}{\partial y^{2}}=0 $$ $$ z=e^{x^{2}-y^{2}} \cos 2 x y $$

Evaluate the given integral. $$ \int_{0}^{4}(\sqrt{2 t+1} \mathbf{i}-\sqrt{t} \mathbf{j}+\sin \pi t \mathbf{k}) d t $$

The improper integral \(\int_{0}^{\infty} e^{-x^{2}} d x\) is important in the theory of probability, statistics, and other areas of applied mathematics. If \(I\) denotes the integral, then $$ I=\int_{0}^{\infty} e^{-x^{2}} d x \quad \text { and } \quad I=\int_{0}^{\infty} e^{-y^{2}} d y $$ and consequently $$ I^{2}=\left(\int_{0}^{\infty} e^{-x^{2}} d x\right)\left(\int_{0}^{\infty} e^{-y^{2}} d y\right)=\int_{0}^{\infty} \int_{0}^{\infty} e^{-\left(x^{2}+y^{2}\right)} d x d y $$ Discuss how to use polar coordinates to evaluate the last integral. Find the value of \(I\).

In Problems \(35-38\), convert the point given in cylindrical coardinates to rectangular cocrdinates. $$ \left(10, \frac{3 \pi}{4}, 5\right) $$

Evaluate the given iterated integral by reversing the order of integration. $$ \int_{0}^{1} \int_{x}^{1} x^{2} \sqrt{1+y^{4}} d y d x $$

Find symmetric equations for the normal line. $$ z=4 x^{2}+9 y^{2}+1 ;\left(\frac{1}{2}, \frac{1}{3}, 3\right) $$

In Problems, evaluate the given integral. $$ \int\left(t e^{t} \mathbf{i}-e^{-2 t} \mathbf{j}+t e^{t^{2}} \mathbf{k}\right) d t $$

Verify that the given function satisfies the wave equation: $$a^{2} \frac{\partial^{2} u}{\partial x^{2}}=\frac{\partial^{2} u}{\partial t^{2}}$$ $$ u=\cos \text { at } \sin x $$

(a) If \(f(x, y)=x^{3}-3 x^{2} y^{2}+y^{3}\), find the directional derivative of \(f\) at a point \((x, y)\) in the direction of \(\mathbf{u}=(1 / \sqrt{10})(3 \mathbf{i}+\mathbf{j})\). (b) If \(F(x, y)=D_{\mathrm{u}} f(x, y)\) in part (a), find \(D_{\mathrm{u}} F(x, y)\).

Convert the point given in cylindrical cocudinates to rectangular cocrdinates. $$ \left(10, \frac{3 \pi}{4}, 5\right) $$

Find the work done by a constant force \(\mathbf{F}(x, y)=a \mathbf{i}+b \mathbf{j}\) acting counterclockwise once around the circle \(x^{2}+y^{2}=9\).

Find parametric equations for the normal line at the indicated point. In Problems 35 and 36, find symmetric equations for the normal line. $$ z=4 x^{2}+9 y^{2}+1 ;\left(\frac{1}{2}, \frac{1}{3}, 3\right) $$

Evaluate the given integral. $$ \int\left(t e^{t} \mathbf{i}-e^{-2 t} \mathbf{j}+t e^{t^{2}} \mathbf{k}\right) d t $$

In Problems \(35-38\), convert the point given in cylindrical coardinates to rectangular cocrdinates. $$ \left(2, \frac{5 \pi}{6},-3\right) $$

Find the flux of \(\mathbf{F}=-y \mathbf{i}+x \mathbf{j}+6 z^{2} \mathbf{k}\) out of the closed surface \(S\) bounded by the paraboloids \(z=4-x^{2}-y^{2}\) and \(z=x^{2}+y^{2}\).

Evaluate the given iterated integral by reversing the order of integration. $$ \int_{0}^{1} \int_{2 y}^{2} e^{-y / x} d x d y $$

In an inverse square force field \(\mathbf{F}=c \mathbf{r} /\|\mathbf{r}\|^{3}\), where \(c\) is a constant and \(\mathbf{r}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k},^{*}\) find the work done in moving a particle along the line from \((1,1,1)\) to \((3,3,3)\).