Chapter 9

In Problems, find the vector function that describes the curve \(C\) of intersection between the given surfaces. Sketch the curve \(C\). Use the indicated parameter. $$ x^{2}+y^{2}=9, z=9-x^{2} ; x=3 \cos t $$

Fill in the blank or answer true/false. Where appropriate, assume continuity of \(P, O\), and their first partial derivatives. Assuming continuity of all partial derivatives, \(\nabla \times \nabla f=0\).___

Use the divergence theorem to find the outward flux \(\iint_{S}(\mathbf{F} \cdot \mathbf{n}) d S\) of the given vector field \(\mathbf{F}\). \(\mathbf{F}=3 x^{2} y^{2} \mathbf{i}+y \mathbf{j}-6 z x y^{2} \mathbf{k} ; D\) the region bounded by the paraboloid \(z=x^{2}+y^{2}\) and the plane \(z=2 y\)

Use Stokes' theorem to evaluate \(\iint_{S}(\operatorname{curl} \mathbf{F}) \cdot \mathbf{n} d S\). Assume that the surface \(S\) is oriented upward. \(\mathbf{F}=6 y z \mathbf{i}+5 x \mathbf{j}+y z e^{x^{2}} \mathbf{k} ; S\) that portion of the paraboloid \(z=\frac{1}{4} x^{2}+y^{2}\) for \(0 \leq z \leq 4\)

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

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

Find the directional derivative of the given function at the given point in the indicated direction. $$ f(x, y)=\tan ^{-1} \frac{y}{x} ;(2,-2), \mathbf{i}-3 \mathbf{j} $$

\(\mathbf{r}(t)\) is the position vector of a moving particle. Find the tangential and normal components of the acceleration at any \(t\). $$ \mathbf{r}(t)=5 \cos t \mathbf{i}+5 \sin t \mathbf{j} $$

Find the vector function that describes the curve \(C\) of intersection between the given surfaces. Sketch the curve \(C\). Use the indicated parameter. $$ x^{2}+y^{2}=9, z=9-x^{2} ; x=3 \cos t $$

Evaluate the given integral by means of the indicated change of variables. $$ \begin{aligned} &\iint_{R} \frac{\cos \frac{1}{2}(x-y)}{3 x+y} d A, \text { where } R \text { is the region bounded by the }\\\ &\text { graphs of } y=x, y=x-\pi, y=-3 x+3, y=-3 x+6 \text { ; }\\\ &u=x-y, v=3 x+y \end{aligned} $$

Use the divergence theorem to find the outward flux \(\iint_{S}(\mathbf{F} \cdot \mathbf{n}) d S\) of the given vector field \(\mathbf{F}\). $$ \begin{aligned} &\mathbf{F}=x y^{2} \mathbf{i}+x^{2} y \mathbf{j}+6 \sin x \mathbf{k} ; D \text { the region bounded by the }\\\ &\text { cone } z=\sqrt{x^{2}+y^{2}} \text { and the planes } z=2, z=4 \end{aligned} $$

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. $$ \begin{aligned} &\mathbf{F}=y \mathbf{i}+(y-x) \mathbf{j}+z^{2} \mathbf{k} ; S \text { that portion of the sphere }\\\ &x^{2}+y^{2}+(z-4)^{2}=25 \text { for } z \geq 0 \end{aligned} $$

Use Green's theorem to evaluate the given line integral. \(\oint_{C} x y^{2} d x+3 \cos y d y\), where \(C\) is the boundary of the region in the first quadrant determined by the graphs of \(y \quad x^{2}, y \quad x^{3}\)

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 \(t\). $$ \mathbf{r}(t)=\cosh t \mathbf{i}+\sinh t \mathbf{j} $$

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. $$ \iint_{R}(x+1) d A ; y=x, x+y=4, x=0 $$

In Problems, determine whether the given vector field is a conservative field. If so, find a potential function \(\phi\) for \(\mathbf{F}\). $$ \mathbf{F}(x, y)=\left(x^{2}+y^{2}+1\right)^{-2}(x \mathbf{i}+y \mathbf{j}) $$

Evaluate \(\int_{C} y d x+x d y\) on the given curve \(C\) between \((0,0)\) and \((1,1)\). \(C\) consists of the line segments from \((0,0)\) to \((1,0)\) and from \((1,0)\) to \((1,1)\).

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

Find the points on the given surface at which the gradient is parallel to the indicated vector. $$ x^{3}+y^{2}+z=15 ; 27 \mathbf{i}+8 \mathbf{j}+\mathbf{k} $$

Find the first partial derivatives of the given function. $$ z=-x^{3}+6 x^{2} y^{3}+5 y^{2} $$

A small projectile is launched from ground level with an initial speed of \(98 \mathrm{~m} / \mathrm{s}\). Find the possible angles of elevation so that its range is \(490 \mathrm{~m}\).

In Problems, find the directional derivative of the given function at the given point in the indicated direction. $$ f(x, y)=\frac{x y}{x+y} ;(2,-1), 6 \mathbf{i}+8 \mathbf{j} $$

In Problems, find the vector function that describes the curve \(C\) of intersection between the given surfaces. Sketch the curve \(C\). Use the indicated parameter. $$ z=x^{2}+y^{2}, z=1 ; x=\sin t $$

Fill in the blank or answer true/false. Where appropriate, assume continuity of \(P, O\), and their first partial derivatives. The surface integral of the nomal component of the curl of a conservative vector field \(F\) over a surface \(S\) is equal to zero.___

Use Stokes' theorem to evaluate \(\iint_{S}(\operatorname{curl} \mathbf{F}) \cdot \mathbf{n} d S\). Assume that the surface \(S\) is oriented upward. \(\mathbf{F}=y \mathbf{i}+(y-x) \mathbf{j}+z^{2} \mathbf{k} ; S\) that portion of the sphere \(x^{2}+y^{2}+(z-4)^{2}=25\) for \(z \geq 0\)

Determine whether the given vector field is a conservative field. If so, find a potential function \(\phi\) for \(\mathbf{F}\). $$ \mathbf{F}(x, y)=\left(x^{2}+y^{2}+1\right)^{-2}(x \mathbf{i}+y \mathbf{j}) $$

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

Find the directional derivative of the given function at the given point in the indicated direction. $$ f(x, y)=\frac{x y}{x+y} ;(2,-1), 6 \mathbf{i}+8 \mathbf{j} $$

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

Find the vector function that describes the curve \(C\) of intersection between the given surfaces. Sketch the curve \(C\). Use the indicated parameter. $$ z=x^{2}+y^{2}, z=1 ; x=\sin t $$

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. \(\mathbf{F}=3 x^{2} \mathbf{i}+8 x^{3} y \mathbf{j}+3 x^{2} y \mathbf{k} ; S\) that portion of the plane \(z=x\) that lies inside the rectangular cylinder defined by the planes \(x=0, y=0, x=2, y=2\)

In Problems \(15-20\), sketch the region \(D\) whose volume \(V\) is given by the iterated integral. $$ \int_{0}^{4} \int_{0}^{3} \int_{0}^{2-2 / 3} d x d z d y $$

Evaluate the given integral on any piecewise-smooth simple closed curve \(C\). $$ \oint_{C} a y d x+b x d y $$

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 \(t\). $$ \mathbf{r}(t)=e^{-t}(\mathbf{i}+\mathbf{j}+\mathbf{k}) $$

Evaluate the surface integral \(\iint_{S} G(x, y, z) d S\). \(G(x, y, z)=x ; S\) the portion of the cylinder \(z=2-x^{2}\) in the first octant bounded by \(x=0, y=0, y=4, z=0\).

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. $$ \iint_{R \atop f C}(2 x+4 y+1) d A ; y=x^{2}, y=x^{3} $$

In Problems, determine whether the given vector field is a conservative field. If so, find a potential function \(\phi\) for \(\mathbf{F}\). $$ \mathbf{F}(x, y)=\left(x^{3}+y\right) \mathbf{i}+\left(x+y^{3}\right) \mathbf{j} $$

Evaluate \(\int_{C}\left(6 x^{2}+2 y^{2}\right) d x+4 x y d y\), where \(C\) is given by \(x=\sqrt{t}, y=t, 4 \leq t \leq 9\).

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

Find an equation of the tangent plane to the graph of the given equation at the indicated point. $$ x^{2}+y^{2}+z^{2}=9 ;(-2,2,1) $$

Find the first partial derivatives of the given function. $$ z=5 x^{4} y^{3}-x^{2} y^{6}+6 x^{5}-4 y $$

In Problems, find the directional derivative of the given function at the given point in the indicated direction. $$ f(x, y)=(x y+1)^{2} ;(3,2) \text { , in the direction of }(5,3) $$

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

Fill in the blank or answer true/false. Where appropriate, assume continuity of \(P, O\), and their first partial derivatives. Work done by a force \(F\) along a curve \(C\) is due entirely to the tangential component of \(\mathrm{F}\).___

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

Use Stokes' theorem to evaluate \(\iint_{S}(\operatorname{curl} \mathbf{F}) \cdot \mathbf{n} d S\). Assume that the surface \(S\) is oriented upward. \(\mathbf{F}=3 x^{2} \mathbf{i}+8 x^{3} y \mathbf{j}+3 x^{2} y \mathbf{k} ; S\) that portion of the plane \(z=x\) that lies inside the rectangular cylinder defined by the planes \(x=0, y=0, x=2, y=2\)

Find the center of mass of the lamina that has the given shape and density. Outside \(r \quad 2\) and inside \(r \quad 2+2 \cos \theta, y \quad 0\), first quadrant; density at a point \(P\) inversely proportional to the distance from the pole

Determine whether the given vector field is a conservative field. If so, find a potential function \(\phi\) for \(\mathbf{F}\). $$ \mathbf{F}(x, y)=\left(x^{3}+y\right) \mathbf{i}+\left(x+y^{3}\right) \mathbf{j} $$

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

Find the directional derivative of the given function at the given point in the indicated direction. $$ f(x, y)=(x y+1)^{2} ;(3,2) \text {, in the direction of }(5,3) $$