Chapter 9

The temperature at a point \((x, y)\) on a rectangular metal plate is given by \(T(x, y)=100-2 x^{2}-y^{2}\). Find the path a heatseeking particle will take, starting at \((3,4)\), as it moves in the direction in which the temperature increases most rapidly.

Convert the point given in rectangular cocrdinates to cylindrical coondinates. $$ (-\sqrt{2}, \sqrt{6}, 2) $$

Assume that \(f\) and \(g\) have continuous second partial derivatives. Show that the given vector field is solenoidal. $$ \mathbf{F}=\nabla f \times \nabla g $$

Find the length of the curve traced by the given vector function on the indicated interval. $$ \mathbf{r}(t)=a \cos t \mathbf{i}+a \sin t \mathbf{j}+c t \mathbf{k} ; 0 \leq t \leq 2 \pi $$

In Problems \(39-42\), convert the point given in rectangular coardinates to cylindrical coordinates. $$ (1,2,7) $$

Find the center of mass of the lamina that has the given shape and density. $$ x=0, y=0,2 x+y=4 ; \rho(x, y)=x^{2} $$

In Problems, find the length of the curve traced by the given vector function on the indicated interval. $$ \mathbf{r}(t)=t \mathbf{i}+t \cos t \mathbf{j}+t \sin t \mathbf{k} ; 0 \leq t \leq \pi $$

Use the Chain Rule to find the indicated partial derivatives. $$ z=\frac{x-y}{x+y} ; x=\frac{u}{v}, y=\frac{v^{2}}{u} ; \frac{\partial z}{\partial u}, \frac{\partial z}{\partial v} $$

The temperature \(T\) at a point \((x, y, z)\) in space is inversely proportional to the square of the distance from \((x, y, z)\) to the origin. It is known that \(T(0,0,1)=500\). Find the rate of change of \(T\) at \((2,3,3)\) in the direction of \((3,1,1)\). In which direction from \((2,3,3)\) does the temperature \(T\) increase most rapidly? At \((2,3,3)\) what is the maximum rate of change of \(T ?\)

Convert the point given in rectangular cocrdinates to cylindrical coondinates. $$ (1,2,7) $$

Assume that \(f\) and \(g\) have continuous second partial derivatives. Show that the given vector field is solenoidal. $$ \mathbf{F}=\nabla f \times(f \nabla g) $$

Find the length of the curve traced by the given vector function on the indicated interval. $$ \mathbf{r}(t)=t \mathbf{i}+t \cos t \mathbf{j}+t \sin t \mathbf{k} ; 0 \leq t \leq \pi $$

In Problems \(43-46\), convert the given equation to cylindrical coardinates. $$ x^{2}+y^{2}+z^{2}=25 $$

Find the center of mass of the lamina that has the given shape and density. $$ y=x, x+y=6, y=0 ; \rho(x, y)=2 y $$

In Problems, find the length of the curve traced by the given vector function on the indicated interval. $$ \mathbf{r}(t)=e^{t} \cos 2 t \mathbf{i}+e^{t} \sin 2 t \mathbf{j}+e^{t} \mathbf{k} ; 0 \leq t \leq 3 \pi $$

Use the Chain Rule to find the indicated partial derivatives. $$ w=\left(u^{2}+v^{2}\right)^{3 / 2} ; u=e^{-t} \sin \theta, v=e^{-t} \cos \theta ; \frac{\partial w}{\partial t}, \frac{\partial w}{\partial \theta} $$

Find a function \(f\) such that $$ \nabla f=\left(3 x^{2}+y^{3}+y e^{x}\right) \mathbf{i}+\left(-2 y^{2}+3 x y^{2}+x e^{x}\right) \mathbf{j} $$

Conven the given equation to cylindrical codrdinates. $$ x^{2}+y^{2}+z^{2}=25 $$

Find the length of the curve traced by the given vector function on the indicated interval. $$ \mathbf{r}(t)=e^{t} \cos 2 t \mathbf{i}+e^{t} \sin 2 t \mathbf{j}+e^{t} \mathbf{k} ; 0 \leq t \leq 3 \pi $$

In Problems, find the length of the curve traced by the given vector function on the indicated interval. $$ \mathbf{r}(t)=3 t \mathbf{i}+\sqrt{3} t^{2} \mathbf{j}+\frac{2}{3} t^{3} \mathbf{k} ; 0 \leq t \leq 1 $$

Use the Chain Rule to find the indicated partial derivatives. $$ w=\tan ^{-1} \sqrt{u v} ; u=r^{2}-s^{2}, v=r^{2} s^{2} ; \frac{\partial w}{\partial r}, \frac{\partial w}{\partial s} $$

Let \(f_{x}, f_{y}, f_{x y}, f_{y x}\) be continuous and \(\mathbf{u}\) and \(\mathbf{v}\) be unit vectors. Show that \(D_{\mathrm{u}} D_{\mathrm{v}} f=D_{\mathrm{v}} D_{\mathrm{u}} f\)

Find the length of the curve traced by the given vector function on the indicated interval. $$ \mathbf{r}(t)=3 t \mathbf{i}+\sqrt{3} t^{2} \mathbf{j}+\frac{2}{3} t^{3} \mathbf{k} ; 0 \leq t \leq 1 $$

In Problems \(43-46\), convert the given equation to cylindrical coardinates. $$ x^{2}+y^{2}-z^{2}=1 $$

Find the center of mass of the lamina that has the given shape and density. $$ y=x^{2}, x=1, y=0 ; \rho(x, y)=x+y $$

Consider the vector field \(\mathbf{F}=x^{2} y z \mathbf{i}-x y^{2} z \mathbf{j}+(z+5 x) \mathbf{k}\) Explain why \(\mathbf{F}\) is not the curl of another vector field \(\mathbf{G}\).

Use the Chain Rule to find the indicated partial derivatives. $$ R=r s^{2} t^{4} ; r=u e^{v^{2}}, s=v e^{-u^{2}}, t=e^{u^{2} v^{2}} ; \frac{\partial R}{\partial u}, \frac{\partial R}{\partial v} $$

Conven the given equation to cylindrical codrdinates. $$ x^{2}+y^{2}-z^{2}=1 $$

Express the vector equation of a circle \(\mathbf{r}(t)=a \cos t \mathbf{i}+\) \(a \sin t \mathbf{j}\) as a function of arc length \(s\). Verify that \(\mathbf{r}^{\prime}(s)\) is a unit vector.

In Problems \(43-46\), convert the given equation to cylindrical coardinates. $$ x^{2}+z^{2}=16 $$

Find the center of mass of the lamina that has the given shape and density. $$ x=y^{2}, x=4 ; \rho(x, y)=y+5 $$

Use the Chain Rule to find the indicated partial derivatives. $$ Q=\ln (p q r) ; p=t^{2} \sin ^{-1} x, q=\frac{x}{t^{2}}, r=\tan ^{-1} \frac{x}{t} ; \frac{\partial Q}{\partial x}, \frac{\partial Q}{\partial t} $$

In Problems, assume that \(f\) and \(g\) are differentiable functions of two variables. Prove the given identity. $$ \nabla(f+g)=\nabla f+\nabla g $$

Conven the given equation to cylindrical codrdinates. $$ x^{2}+z^{2}=16 $$

Assume that \(f\) and \(g\) are differentiable functions of two variables. Prove the given identity. $$ \nabla(f+g)=\nabla f+\nabla g $$

Find the center of mass of the lamina that has the given shape and density. \(y=1-x^{2}, y=0 ;\) density at a point \(P\) directly proportional to the distance from the \(x\) -axis

Suppose \(\mathbf{r}\) is a differentiable vector function for which \(\|\mathbf{r}(t)\|=c\) for all \(t\). Show that the tangent vector \(\mathbf{r}^{\prime}(t)\) is perpendicular to the position vector \(\mathbf{r}(t)\) for all \(t\).

Use the Chain Rule to find the indicated partial derivatives. $$ \begin{aligned} &w=\sqrt{x^{2}+y^{2}} ; x=\ln (r s+t u) \\ &y=\frac{t}{u} \cosh r s ; \frac{\partial w}{\partial t}, \frac{\partial w}{\partial r}, \frac{\partial w}{\partial u} \end{aligned} $$

In Problems, assume that \(f\) and \(g\) are differentiable functions of two variables. Prove the given identity. $$ \nabla(f g)=f \nabla g+g \nabla f $$

Assume that \(f\) and \(g\) are differentiable functions of two variables. Prove the given identity. $$ \nabla(f g)=f \nabla g+g \nabla f $$

In Problems \(47-50\), convert the given equation to rectangular coardinates. $$ z=2 r \sin \theta $$

Find the center of mass of the lamina that has the given shape and density. \(y=1-x^{2}, y=0 ;\) density at a point \(P\) directly proportional to the distance from the \(x\) -axis

Use the Chain Rule to find the indicated partial derivatives. $$ \begin{aligned} &s=p^{2}+q^{2}-r^{2}+4 t ; p=\phi e^{3 \theta}, q=\cos (\phi+\theta), r=\phi \theta^{2}, \\ &t=2 \phi+8 \theta ; \frac{\partial s}{\partial \phi^{\prime}}, \frac{\partial s}{\partial \theta} \end{aligned} $$

In Problems, assume that \(f\) and \(g\) are differentiable functions of two variables. Prove the given identity. $$ \nabla\left(\frac{f}{g}\right)=\frac{g \nabla f-f \nabla g}{g^{2}} $$

Assume that \(f\) and \(g\) are differentiable functions of two variables. Prove the given identity. $$ \nabla\left(\frac{f}{g}\right)=\frac{g \nabla f-f \nabla g}{g^{2}} $$

In Problems \(47-50\), convert the given equation to rectangular coardinates. $$ r=5 \sec \theta $$

Find the center of mass of the lamina that has the given shape and density. $$ y=e^{x}, x=0, x=1, y=0 ; \rho(x, y)=y^{3} $$

Use (8) to find the indicated derivative. $$ z=\ln \left(u^{2}+v^{2}\right) ; u=t^{2}, v=t^{-2} ; \frac{d z}{d t} $$

If \(\mathbf{F}(x, y, z)=f_{1}(x, y, z) \mathbf{i}+f_{2}(x, y, z) \mathbf{j}+f_{3}(x, y, z) \mathbf{k}\) and \(\nabla=\mathbf{i} \frac{\partial}{\partial x}+\mathbf{j} \frac{\partial}{\partial y}+\mathbf{k} \frac{\partial}{\partial z}\) show that \(\nabla \times \mathbf{F}=\left(\frac{\partial f_{3}}{\partial y}-\frac{\partial f_{2}}{\partial z}\right) \mathbf{i}+\left(\frac{\partial f_{1}}{\partial z}-\frac{\partial f_{3}}{\partial x}\right) \mathbf{j}+\left(\frac{\partial f_{2}}{\partial x}-\frac{\partial f_{1}}{\partial y}\right) \mathbf{k} .\)

Convert the given equation to rectangular cocrdinates. $$ r=5 \sec \theta $$