Chapter 9

In Problems, find the gradient of the given function at the indicated point. $$ F(x, y, z)=\ln \left(x^{2}+y^{2}+z^{2}\right) ;(-4,3,5) $$

In Problems, graph the curve traced by the given vector function. \(\mathbf{r}(t)=t \mathbf{i}+t^{3} \mathbf{j}+t \mathbf{k}\)

Find the Jacobian of the transformation \(T\) from the \(u v\)-plane to the \(x y\)-plane. $$ x=e^{3 u} \sin v, y=e^{3 u} \cos v $$

Use the divergence theorem to find the outward flux \(\iint_{S}(\mathbf{F} \cdot \mathbf{n}) d S\) of the given vector field \(\mathbf{F}\). \(\mathbf{F}=\left(x^{2}+\sin y\right) \mathbf{i}+z^{2} \mathbf{j}+x y^{3} \mathbf{k} ; D\) the region bounded by $$ y=x^{2}, z=9-y, z=0 $$

Use Stokes' theorem to evaluate \(\oint_{C} \mathbf{F} \cdot d \mathbf{r}\). Assume \(C\) is oriented counterclockwise as viewed from above. \(\mathbf{F}=(x+2 z) \mathbf{i}+(3 x+y) \mathbf{j}+(2 y-z) \mathbf{k} ; C\) the curve of intersection of the plane \(x+2 y+z=4\) with the coordinate planes

Show that the given line integral is independent of the path. Evaluate in two ways: (a) Find a potential function \(\phi\) and then use Theorem 9.9.1, and (b) Use any convenient path between the endpoints of the path. $$ \int_{(0,0)}^{(2,8)}\left(y^{3}+3 x^{2} y\right) d x+\left(x^{3}+3 y^{2} x+1\right) d y $$

Find the curl and the divergence of the given vector field. $$ \mathbf{F}(x, y, z)=10 y z \mathbf{i}+2 x^{2} z \mathbf{j}+6 x^{3} \mathbf{k} $$

Find the gradient of the given function at the indicated point. $$ F(x, y, z)=\ln \left(x^{2}+y^{2}+z^{2}\right) ;(-4,3,5) $$

Describe the level surfaces but do not graph. $$ F(x, y, z)=x^{2}+y^{2}+z^{2} $$

\(\mathbf{r}(t)\) is the position vector of a moving particle. Find the tangential and normal components of the acceleration at any \(t\). $$ \mathbf{r}(t)=3 \cos t \mathbf{i}+2 \sin t \mathbf{j}+t \mathbf{k} $$

Graph the curve traced by the given vector function. $$ \mathbf{r}(t)=t \mathbf{i}+t^{3} \mathbf{j}+t \mathbf{k} $$

In Problems \(7-10\), find the Jacobian of the transformation \(T\) from the \(u v\) -plane to the \(x y\) -plane. $$ u=\frac{y}{x^{2}}, v=\frac{y^{2}}{x} $$

Use the divergence theorem to find the outward flux \(\iint_{S}(\mathbf{F} \cdot \mathbf{n}) d S\) of the given vector field \(\mathbf{F}\). $$ \begin{aligned} &\mathbf{F}=(x \mathbf{i}+y \mathbf{j}+z \mathbf{k}) /\left(x^{2}+y^{2}+z^{2}\right) ; D \text { the region bounded by }\\\ &\text { the concentric spheres } x^{2}+y^{2}+z^{2}=a^{2}, x^{2}+y^{2}+z^{2}=b^{2}\\\ &b>a \end{aligned} $$

Use Stokes' theorem to evaluate \(\oint_{C} \mathbf{F} \cdot d \mathbf{r}\). Assume \(C\) is oriented counterclockwise as viewed from above. $$ \begin{aligned} &\mathbf{F}=y^{3} \mathbf{i}-x^{3} \mathbf{j}+z^{3} \mathbf{k} ; C \text { the trace of the cylinder } x^{2}+y^{2}=1\\\ &\text { in the plane } x+y+z=1 \end{aligned} $$

Use Green's theorem to evaluate the given line integral. $$ \begin{aligned} &\oint_{C} 2 x y d x+3 x y^{2} d y \text { , where } C \text { is the triangle with vertices }\\\ &(1,2),(2,2),(2,4) \end{aligned} $$

Find the surface area of the portions of the sphere \(x^{2}+y^{2}+z^{2}=a^{2}\) that are within the cylinder \(x^{2}+y^{2}=a y\)

In Problems \(7-16, \mathbf{r}(t)\) is the position vector of a moving particle. Find the tangential and normal components of the acceleration at any \(t\). $$ \mathbf{r}(t)=t^{2} \mathbf{i}+\left(t^{2}-1\right) \mathbf{j}+2 t^{2} \mathbf{k} $$

Sketch the region of integration for the given iterated integral. $$ \int_{0}^{2} \int_{1}^{2 x+1} f(x, y) d y d x $$

In Problems, show that the given line integral is independent of the path. Evaluate in two ways: (a) Find a potential function \(\phi\) and then use Theorem \(9.9 .1\), and (b) Use any convenient path between the endpoints of the path. $$ \int_{10,0)}^{(2,8)}\left(y^{3}+3 x^{2} y\right) d x+\left(x^{3}+3 y^{2} x+1\right) d y $$

In Problems \(7-16\), find the curl and the divergence of the given vector field. $$ \mathbf{F}(x, y, z)=4 x y \mathbf{i}+\left(2 x^{2}+2 y z\right) \mathbf{j}+\left(3 z^{2}+y^{2}\right) \mathbf{k} $$

Describe the level surfaces but do not graph. $$ F(x, y, z)=x^{2}+3 y^{2}+6 z^{2} $$

Suppose \(\mathbf{r}(t)=t^{2} \mathbf{i}+\left(t^{3}-2 t\right) \mathbf{j}+\left(t^{2}-5 t\right) \mathbf{k}\) is the position vector of a moving particle. At what points does the particle pass through the \(x y\) -plane? What are its velocity and acceleration at these points?

In Problems, use Definition \(9.5 .1\) to find \(D_{\mathrm{a}} f(x, y)\) given that \(\mathbf{u}\) makes the indicated angle with the positive \(x\) -axis. $$ f(x, y)=x^{2}+y^{2} ; \theta=30^{\circ} $$

In Problems, graph the curve traced by the given vector function. \(\mathbf{r}(t)=e^{t} \cos t \mathbf{i}+e^{t} \sin t \mathbf{j}+e^{t} \mathbf{k}\)

Find the Jacobian of the transformation \(T\) from the \(u v\)-plane to the \(x y\)-plane. $$ u=\frac{y}{x^{2}}, v=\frac{y^{2}}{x} $$

Find the volume of the solid bounded by the graphs of the given equations. $$ r \quad 1+\cos \theta, z \quad y, z \quad 0, \text { first octant } $$

Show that the given line integral is independent of the path. Evaluate in two ways: (a) Find a potential function \(\phi\) and then use Theorem 9.9.1, and (b) Use any convenient path between the endpoints of the path. $$ \int_{(0,0)}^{(2,8)}\left(y^{3}+3 x^{2} y\right) d x+\left(x^{3}+3 y^{2} x+1\right) d y $$

Find the curl and the divergence of the given vector field. $$ \mathbf{F}(x, y, z)=4 x y \mathbf{i}+\left(2 x^{2}+2 y z\right) \mathbf{j}+\left(3 z^{2}+y^{2}\right) \mathbf{k} $$

Sketch the level curve or surface passing through the indicated point. Sketch the gradient at the point. $$ F(x, y, z)=y+z ;(3,1,1) $$

\(\mathbf{r}(t)\) is the position vector of a moving particle. Find the tangential and normal components of the acceleration at any \(t\). $$ \mathbf{r}(t)=t^{2} \mathbf{i}+\left(t^{2}-1\right) \mathbf{j}+2 t^{2} \mathbf{k} $$

Suppose \(r(t)=t^{2} \mathbf{i}+\left(t^{3}-2 t\right) \mathbf{j}+\left(t^{2}-5 t\right) \mathbf{k}\) is the position vector of a moving particle. At what points does the particle pass through the \(x y\)-plane? What are its velocity and acceleration at these points?

Graph the curve traced by the given vector function. $$ \mathbf{r}(t)=e^{t} \cos t \mathbf{i}+e^{t} \sin t \mathbf{j}+e^{t} \mathbf{k} $$

In Problems \(7-10\), find the Jacobian of the transformation \(T\) from the \(u v\) -plane to the \(x y\) -plane. $$ u=\frac{2 x}{x^{2}+y^{2}}, v=\frac{-2 y}{x^{2}+y^{2}} $$

Use the divergence theorem to find the outward flux \(\iint_{S}(\mathbf{F} \cdot \mathbf{n}) d S\) of the given vector field \(\mathbf{F}\). $$ \begin{aligned} &\mathbf{F}=2 y z \mathbf{i}+x^{3} \mathbf{j}+x y^{2} \mathbf{k} ; D \text { the region bounded by the ellipsoid }\\\ &x^{2} / a^{2}+y^{2} / b^{2}+z^{2} / c^{2}=1 \end{aligned} $$

Use Green's theorem to evaluate the given line integral. $$ \begin{aligned} &\oint_{C} e^{2 x} \sin 2 y d x+e^{2 x} \cos 2 y d y, \text { where } C \text { is the ellipse }\\\ &9(x-1)^{2}+4(y-3)^{2} \quad 36 \end{aligned} $$

In Problems \(7-16, \mathbf{r}(t)\) is the position vector of a moving particle. Find the tangential and normal components of the acceleration at any \(t\). $$ \mathbf{r}(t)=t^{2} \mathbf{i}-t^{3} \mathbf{j}+t^{4} \mathbf{k} $$

Sketch the region of integration for the given iterated integral. $$ \int_{1}^{4} \int_{-\sqrt{y}}^{\sqrt{y}} f(x, y) d x d y $$

In Problems \(7-16\), find the curl and the divergence of the given vector field. $$ \mathbf{F}(x, y, z)=(x-y)^{3} \mathbf{i}+e^{-y z} \mathbf{j}+x y e^{2 y} \mathbf{k} $$

Sketch the level curve or surface passing through the indicated point. Sketch the gradient at the point. $$ F(x, y, z)=x^{2}+y^{2}-z ;(1,1,3) $$

Describe the level surfaces but do not graph. $$ F(x, y, z)=4 y-2 z+1 $$

Suppose a parlicle movesin space so that \(\mathbf{a}(t)=\mathbf{0}\) for all time \(t\). Describe its path.

In Problems, graph the curve traced by the given vector function. \(\mathbf{r}(t)=\left\langle t \cos t, t \sin t, t^{2}\right\rangle\)

Fill in the blank or answer true/false. Where appropriate, assume continuity of \(P, O\), and their first partial derivatives. If the work \(\int_{C} F \cdot d r\) depends on the curve \(C\), then \(F\) is nonconservative.___

Find the Jacobian of the transformation \(T\) from the \(u v\)-plane to the \(x y\)-plane. $$ u=\frac{2 x}{x^{2}+y^{2}}, v=\frac{-2 y}{x^{2}+y^{2}} $$

Find the surface area of the portions of the cone \(z^{2}=\frac{1}{4}\left(x^{2}+y^{2}\right)\) that are within the cylinder \((x-1)^{2}+y^{2}=1\).

Show that the given line integral is independent of the path. Evaluate in two ways: (a) Find a potential function \(\phi\) and then use Theorem 9.9.1, and (b) Use any convenient path between the endpoints of the path. $$ \int_{(-2,0)}^{(1,0)}\left(2 x-y \sin x y-5 y^{4}\right) d x-\left(20 x y^{3}+x \sin x y\right) d y $$

Find the curl and the divergence of the given vector field. $$ \mathbf{F}(x, y, z)=(x-y)^{3} \mathbf{i}+e^{-y z} \mathbf{j}+x y e^{2 y} \mathbf{k} $$

\(\mathbf{r}(t)\) is the position vector of a moving particle. Find the tangential and normal components of the acceleration at any \(t\). $$ \mathbf{r}(t)=t^{2} \mathbf{i}-t^{3} \mathbf{j}+t^{4} \mathbf{k} $$

Suppose a particle movesin space so that \(\mathrm{a}(t)=\mathbf{0}\) for all time \(t\). Describe its path.

Graph the curve traced by the given vector function. $$ r(t)=\left\langle t \cos t, t \sin t, t^{2}\right\rangle $$