Chapter 9

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

Given that \(\mathbf{r}(t)=\frac{\sin \angle t}{t} \mathbf{i}+(t-2)^{5} \mathbf{j}+t \ln t \mathbf{k}\), find \(\lim _{t \rightarrow 0^{+}} \mathbf{r}(t)\).

Evaluate the given integral by means of the indicated change of variables. $$ \begin{aligned} &\iint_{R}\left(x^{2}+y^{2}\right)^{-3} d A \text { , where } R \text { is the region bounded by the circles }\\\ &x^{2}+y^{2}=2 x, x^{2}+y^{2}=4 x, x^{2}+y^{2}=2 y, x^{2}+y^{2}=6 y\\\ &u=\frac{2 x}{x^{2}+y^{2}}, v=\frac{2 y}{x^{2}+y^{2}}\left[\text { Hint: } \text { Form } u^{2}+v^{2}\right. \text { .] } \end{aligned} $$

Suppose there is a continuous distribution of charge throughout a closed and bounded region \(D\) enclosed by a surface \(S\). Then the natural extension of Gauss' law is given by $$ \iint_{S}(\mathbf{E} \cdot \mathbf{n}) d S=\iiint_{D} 4 \pi \rho d V $$ where \(\rho(x, y, z)\) is the charge density or charge per unit volume. (a) Proceed as in the derivation of the continuity equation (16) to show that \(\operatorname{div} \mathbf{E}=4 \pi \rho\). (b) Given that \(\mathbf{E}\) is an irrotational vector field, show that the potential \(\phi\) for \(\mathbf{E}\) satisfies Poisson's equation \(\nabla^{2} \phi=4 \pi \rho\). In Problems \(17-21\), assume that \(S\) forms the boundary of a closed and bounded region \(D\).

In Problems 13-16, use Stokes' theorem to evaluate \(\iint_{S}\) (curl \(\left.\mathbf{F}\right) \cdot \mathbf{n} d S\). Assume that the surface \(S\) is oriented upward. \(\mathbf{F}=2 x y^{2} z \mathbf{i}+2 x^{2} y z \mathbf{j}+\left(x^{2} y^{2}-6 x\right) \mathbf{k} ; S\) that portion of the plane \(z=y\) that lies inside the cylinder \(x^{2}+y^{2}=1\)

In Problems, find the center of mass of the lamina that has the given shape and density. \(r \quad 2+2 \cos \theta, y \quad 0\), first and second quadrants; \(\rho(r, \theta) \quad k\) (constant)

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 \(t\). $$ \mathbf{r}(t)=t \mathbf{i}+(2 t-1) \mathbf{j}+(4 t+2) \mathbf{k} $$

In Problems, determine whether the given vector field is a conservative field. If so, find a potential function \(\phi\) for \(\mathbf{F}\). $$ \mathbf{F}(x, y)=2 e^{2 y} \mathbf{i}+x e^{2 y} \mathbf{j} $$

Evaluate \(\int_{C}-y^{2} d x+x y d y\), where \(C\) is given by \(x=2 t\), \(y=t^{3}, 0 \leq t \leq 2\).

In Problems \(7-16\), find the curl and the divergence of the given vector field. $$ \mathbf{F}(x, y, z)=x^{2} \sin y z \mathbf{i}+z \cos x z^{3} \mathbf{j}+y e^{5 x y} \mathbf{k} $$

Find an equation of the tangent plane to the graph of the given equation at the indicated point. $$ 5 x^{2}-y^{2}+4 z^{2}=8 ;(2,4,1) $$

Find the first partial derivatives of the given function. $$ z=\tan \left(x^{3} y^{2}\right) $$

In Problems, find the directional derivative of the given function at the given point in the indicated direction. $$ f(x, y)=x^{2} \tan y ;(2, \pi / 3) \text { , in the direction of the negative } x \text { -axis } $$

Given that \(\lim _{t \rightarrow a} r_{1}(t)=\mathbf{i}-2 \mathbf{j}+\mathbf{k}\) and \(\lim _{t \rightarrow a} \mathbf{r}_{2}(t)=2 \mathbf{i}+\) \(5 \mathbf{j}+7 \mathbf{k}\), find: (a) \(\lim _{t \rightarrow a}\left[-4 \mathbf{r}_{1}(t)+3 \mathbf{r}_{2}(t)\right]\) (b) \(\lim _{t \rightarrow a} \mathbf{r}_{1}(t) \cdot \mathbf{r}_{2}(t) .\)

Sketch the region \(D\) whose volume \(V\) is given by the iterated integral. $$ 4 \int_{0}^{3} \int_{0}^{\sqrt{9-y^{2}}} \int_{4}^{\sqrt{25-x^{2}-y^{2}}} d z d x d y $$

Use Stokes' theorem to evaluate \(\iint_{S}(\operatorname{curl} \mathbf{F}) \cdot \mathbf{n} d S\). Assume that the surface \(S\) is oriented upward. \(\mathbf{F}=2 x y^{2} z \mathbf{i}+2 x^{2} y z \mathbf{j}+\left(x^{2} y^{2}-6 x\right) \mathbf{k} ; S\) that portion of the plane \(z=y\) that lies inside the cylinder \(x^{2}+y^{2}=1\)

Find the center of mass of the lamina that has the given shape and density. \(r \quad 2+2 \cos \theta, y \quad 0\), first and second quadrants; \(\rho(r, \theta) \quad k\) (constant)

Determine whether the given vector field is a conservative field. If so, find a potential function \(\phi\) for \(\mathbf{F}\). $$ \mathbf{F}(x, y)=2 e^{2 y} \mathbf{i}+x e^{2 y} \mathbf{j} $$

Find the curl and the divergence of the given vector field. $$ \mathbf{F}(x, y, z)=x^{2} \sin y z \mathbf{i}+z \cos x z^{3} \mathbf{j}+y e^{5 x y} \mathbf{k} $$

Find the directional derivative of the given function at the given point in the indicated direction. $$ f(x, y)=x^{2} \tan y ;\left(\frac{1}{2}, \pi / 3\right) \text {, in the direction of the negative } x \text {-axis } $$

\(\mathbf{r}(t)\) is the position vector of a moving particle. Find the tangential and normal components of the acceleration at any \(t\). $$ \mathbf{r}(t)=t \mathbf{i}+(2 t-1) \mathbf{j}+(4 t+2) \mathbf{k} $$

Evaluate the given integral by means of the indicated change of variables. \(\iint_{R}\left(x^{2}+y^{2}\right) d A\), where \(R\) is the region in the first quadrant bounded by the graphs of \(x^{2}-y^{2}=a, x^{2}-y^{2}=b, 2 x y=c\) \(2 x y=d, 0

Sketch the region \(D\) whose volume \(V\) is given by the iterated integral. $$ \int_{-1}^{1} \int_{-\sqrt{1-x^{2}}}^{\sqrt{1-x^{2}}} \int_{0}^{5} d z d y d x $$

$$ \begin{aligned} &\text { Find the curvature of an elliptical helix that is described by }\\\ &\mathbf{r}(t)=a \cos t \mathbf{i}+b \sin t \mathbf{j}+c t \mathbf{k}, a>0, b>0, c>0 \end{aligned} $$

Evaluate the double integral over the region \(R\) that is bounded by the graphs of the given equations. Choose the most convenient order of integration. $$ \iint_{R} 2 x y d A ; y=x^{3}, y=8, x=0 $$

Evaluate \(\int_{C} 2 x^{3} y d x+(3 x+y) d y\), where \(C\) is given by \(x=y^{2}\) from \((1,-1)\) to \((1,1)\).

Let a be a constant vector and \(\mathbf{r}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}\). Verify the given identity. $$ \operatorname{div} \mathbf{r}=3 $$

Find an equation of the tangent plane to the graph of the given equation at the indicated point. $$ x^{2}-y^{2}-3 z^{2}=5 ;(6,2,3) $$

Find the first partial derivatives of the given function. $$ z=\frac{4 \sqrt{x}}{3 y^{2}+1} $$

In Problems, find \(\mathbf{r}^{\prime}(t)\) and \(\mathbf{r}^{\prime \prime}(t)\) for the given vector function. \(\mathbf{r}(t)=\ln t \mathbf{i}+\mathbf{j}, t>0\)

In Problems, find the directional derivative of the given function at the given point in the indicated direction. $$ F(x, y, z)=x^{2} y^{2}(2 z+1)^{2} ;(1,-1,1),\langle 0,3,3\rangle $$

Fill in the blank or answer true/false. Where appropriate, assume continuity of \(P, O\), and their first partial derivatives. If \(\Gamma\) is a conservative force field, then the sum of the potential and kinetic energies of an object is constant.___

Assume that \(S\) forms the boundary of a closed and bounded region \(D\). If a is a constant vector, show that \(\iint_{S}(\mathbf{a} \cdot \mathbf{n}) d S=0\).

Use Stokes' theorem to evaluate $$ \oint_{C} z^{2} e^{x^{2}} d x+x y^{2} d y+\tan ^{-1} y d z $$ where \(C\) is the circle \(x^{2}+y^{2}=9\), by finding a surface \(S\) with \(C\) as its boundary and such that the orientation of \(C\) is counterclockwise as viewed from above.

Let \(\mathbf{a}\) be a constant voctor and \(\mathbf{r}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}\). Verify the given identity. \(\operatorname{div} r=3\)

Find the directional derivative of the given function at the given point in the indicated direction. $$ F(x, y, z)=x^{2} y^{2}(2 z+1)^{2} ;(1,-1,1),\langle 0,3,3\rangle $$

Find the curvature of an elliptical helix that is described by \(\mathbf{r}(t)=a \cos t \mathbf{i}+b \sin t \mathbf{j}+c t \mathbf{k}, a>0, b>0, c>0 .\)

Find \(\mathbf{r}^{\prime}(t)\) and \(\mathbf{r}^{\prime \prime}(t)\) for the given vector function. $$ \mathbf{r}(t)=\ln t \mathbf{i}+\mathbf{j}, t>0 $$

Evaluate the given integral by means of the indicated change of variables. \(\begin{aligned} &\iint_{R}\left(x^{2}+y^{2}\right) \sin x y d A, \text { where } R \text { is the region bounded by the }\\\ &\text { graphs of } x^{2}-y^{2}=1, x^{2}-y^{2}=9, x y=2, x y=-2\\\ &u=x^{2}-y^{2}, v=x y \end{aligned}\)

Consider the surface integral \(\iint_{S}\) (curl \(\left.\mathbf{F}\right) \cdot \mathbf{n} d S\), where \(\mathbf{F}=x y z \mathbf{k}\) and \(S\) is that portion of the paraboloid \(z=1-x^{2}-y^{2}\) for \(z \geq 0\) oriented upward. (a) Evaluate the surface integral by the method of Section 9.13; that is, do not use Stokes' theorem. (b) Evaluate the surface integral by finding a simpler surface that is oriented upward and has the same boundary as the paraboloid. (c) Use Stokes' theorem to verify the result in part (b).

Sketch the region \(D\) whose volume \(V\) is given by the iterated integral. $$ \int_{0}^{2} \int_{0}^{\sqrt{4-x^{2}}} \int_{x^{2}+y^{2}}^{4} d z d y d x $$

Evaluate the surface integral \(\iint_{S} G(x, y, z) d S\). \(G(x, y, z)=x+y+z ; S\) the cone \(z=\sqrt{x^{2}+y^{2}}\) between \(z=1\) and \(z=4\)

Evaluate the double integral over the region \(R\) that is bounded by the graphs of the given equations. Choose the most convenient order of integration. $$ \iint_{R} \frac{x}{\sqrt{y}} d A ; y=x^{2}+1, y=3-x^{2} $$

Evaluate \(\int_{C} 4 x d x+2 y d y\), where \(C\) is given by \(x=y^{3}+1\) from \((0,-1)\) to \((9,2)\).

Let a be a constant vector and \(\mathbf{r}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}\). Verify the given identity. $$ \operatorname{curl} \mathbf{r}=\mathbf{0} $$

Find an equation of the tangent plane to the graph of the given equation at the indicated point. $$ x y+y z+z x=7 ;(1,-3,-5) $$

Find the first partial derivatives of the given function. $$ z=4 x^{3}-5 x^{2}+8 x $$

In Problems, find \(\mathbf{r}^{\prime}(t)\) and \(\mathbf{r}^{\prime \prime}(t)\) for the given vector function. \(\mathbf{r}(t)=\langle t \cos t-\sin t, t+\cos t\rangle\)

In Problems, find the directional derivative of the given function at the given point in the indicated direction. $$ F(x, y, z)=\frac{x^{2}-y^{2}}{z^{2}} ;(2,4,-1), \mathbf{i}-2 \mathbf{j}+\mathbf{k} $$

Fill in the blank or answer true/false. Where appropriate, assume continuity of \(P, O\), and their first partial derivatives. If \(\int_{C} P d x+Q\) dy isindependent of the path, then \(F=P i+Q \mathbf{j}\) is the gradient of some function \(\phi\).___