Chapter 14

Use a line integral to compute the area of the given region. The region bounded by \(y=x^{2}\) and \(y=4\)

Compute the work done by the force field \(\mathbf{F}\) along the curve \(C.\) \(\mathbf{F}(x, y)=\langle 2 x, 2 y\rangle, C\) is the line segment from (3,1) to (5,4)

Determine whether the given vector field is conservative and/or incompressible. $$\left\langle e^{y}, x e^{y}+z^{2}, 2 y z-1\right\rangle$$

Find the flux of \(\mathbf{F}\) over \(\partial Q\). \(Q\) is bounded by \(x+2 y+3 z=12\) and the coordinate planes, \(\mathbf{F}=\left\langle x^{2} y, 3 x, 4 y-x^{2}\right\rangle\)

Determine whether or not the vector field is conservative. If it is, find a potential function. $$\langle y, 1\rangle$$

Evaluate \(\int_{C} \mathbf{F} \cdot d \mathbf{r}.\) \(\mathbf{F}(x, y)=\left(4 x y-2 x, 2 x^{2}-x\right), C\) is the portion of the parabola \(y=x^{2}\) from \((-2,4)\) to \((2,4)\)

Use a line integral to compute the area of the given region. The region bounded by \(y=x^{2}\) and \(y=2 x\)

Compute the work done by the force field \(\mathbf{F}\) along the curve \(C.\) \(\mathbf{F}(x, y)=\\{2 y,-2 x\rangle, C\) is the line segment from (4,2) to (0,4)

Let \(\mathbf{F}(x, y)=\langle M(x, y), N(x, y)\rangle\) be a vector field whose components \(M\) and \(N\) have continuous first partial derivatives in all of \(\mathbb{R}^{2} .\) Show that \(\nabla \cdot \mathbf{F}=0\) if and only if \(\int_{C} \mathbf{F} \cdot \mathbf{n} d s=0\) for all simple closed curves \(C\). (Hint: Use a vector form of Green's Theorem.)

Set up a double integral and evaluate the surface integral \(\iint g(x, y, z) d S\) \(\iint_{S} x z d S, S\) is the portion of the plane \(z=2 x+3 y\) above the rectangle \(1 \leq x \leq 2,1 \leq y \leq 3\)

Determine whether or not the vector field is conservative. If it is, find a potential function. $$(x-2 x y) \mathbf{i}+\left(y^{2}-x^{2}\right) \mathbf{j}$$

Label each expression as a scalar quantity, a vector quantity or undefined, if \(f\) is a scalar function and \(\mathbf{F}\) is a vector field. a. \(\nabla \cdot(\nabla f)\) b. \(\nabla \times(\nabla \cdot \mathbf{F})\) c. \(\nabla(\nabla \times \mathbf{F})\) d. \(\nabla(\nabla \cdot \mathbf{F})\) e. \(\nabla \times(\nabla f)\).

Find the flux of \(\mathbf{F}\) over \(\partial Q\). \(Q\) is bounded by \(z=1-x^{2}, z=-3, y=-2\) and \(y=2\) \(\mathbf{F}=\left\langle x^{2}, y^{3}, x^{3} y^{2}\right\rangle\)

Evaluate \(\int_{C} \mathbf{F} \cdot d \mathbf{r}.\) \(\mathbf{F}(x, y)=\left(y^{2} e^{x y^{2}}-y, 2 x y e^{x y^{2}}-x-1\right), C\) is the line segment from \(( 2,3 )\) to \((3,0)\)

Use Green's Theorem to show that the center of mass of the region bounded by the positive curve \(C\) with constant density is given by \(\bar{x}=\frac{1}{2 A} \oint_{C} x^{2} d y\) and \(\bar{y}=-\frac{1}{2 A} \oint_{C} y^{2} d x,\) where \(A\) is the area of the region.

Compute the work done by the force field \(\mathbf{F}\) along the curve \(C.\) \(\mathbf{F}(x, y)=\langle 2 x, 2 y\rangle, C\) is the quarter-circle from (4,0) to (0,4)

Set up a double integral and evaluate the surface integral \(\iint_{S} g(x, y, z) d S\) \(\iint_{S}\left(z-y^{2}\right) d S, S\) is the portion of the paraboloid \(z=x^{2}+y^{2}\) below \(z=4\)

Determine whether or not the vector field is conservative. If it is, find a potential function. $$\left(x^{2}-y\right) \mathbf{i}+(x-y) \mathbf{j}$$

Label each expression as a scalar quantity, a vector quantity or undefined, if \(f\) is a scalar function and \(\mathbf{F}\) is a vector field. a. \(\nabla(\nabla f)\) b. \(\nabla \cdot(\nabla \cdot \mathbf{F})\) c. \(\nabla \cdot(\nabla \times \mathbf{F})\) d. \(\nabla \times(\nabla \mathbf{F})\) e. \(\nabla \times(\nabla \times(\nabla \times \mathbf{F}))\).

Find the flux of \(\mathbf{F}\) over \(\partial Q\). \(Q\) is bounded by \(z=1-x^{2}, z=0, y=0\) and \(x+y=4\) \(\mathbf{F}=\left\langle y^{3}, x^{2}-z, z^{2}\right\rangle\)

Evaluate \(\int_{C} \mathbf{F} \cdot d \mathbf{r}.\) \(\mathbf{F}(x, y)=\left\langle 2 y e^{2 x}+y^{3}, e^{2 x}+3 x y^{2}\right\rangle, C\) is the line segment from \((4,3)\) to \((1,-3)\)

Compute the work done by the force field \(\mathbf{F}\) along the curve \(C.\) \(\mathbf{F}(x, y)=\langle 2 y,-2 x\rangle, C\) is the upper half-circle from (-3,0) to (3,0)

Set up a double integral and evaluate the surface integral \(\iint_{S} g(x, y, z) d S\) \begin{aligned} &\iint_{S}\left(x^{2}+y^{2}+z^{2}\right)^{3 / 2} d S, S \quad \text { is the lower hemisphere }\\\ &z=-\sqrt{9-x^{2}-y^{2}} \end{aligned}

Determine whether or not the vector field is conservative. If it is, find a potential function. $$\langle y \sin x y, x \sin x y\rangle$$

If \(\mathbf{r}=\langle x, y, z\rangle,\) prove that \(\nabla \times \mathbf{r}=\mathbf{0}\) and \(\nabla \cdot \mathbf{r}=3\).

Coulomb's law for an electrostatic field applied to a point charge \(q\) at the origin gives us \(\mathbf{E}(\mathbf{r})=q \frac{\mathbf{r}}{r^{3}},\) where \(r=\|\mathbf{r}\|\) Let \(Q\) be bounded by the sphere \(x^{2}+y^{2}+z^{2}=a^{2}\) for some constant \(a>0 .\) Show that the flux of E over \(\partial Q\) equals \(4 \pi q\) Discuss the fact that the flux does not depend on the value of \(a\)

Evaluate \(\int_{C} \mathbf{F} \cdot d \mathbf{r}.\) \(\mathbf{F}(x, y)=\left\langle\frac{1}{y}-e^{2 x}, 2 y-\frac{x}{y^{2}}\right\rangle, \)C\( \text { is the circle } \)(x-5)^{2}+(y+6)^{2}=16,$ \text { oriented counterclockwise}

Compute the work done by the force field \(\mathbf{F}\) along the curve \(C.\) \(\mathbf{F}(x, y)=\langle 2, x\rangle, C\) is the portion of \(y=x^{2}\) from (0,0) to (1,1)

Set up a double integral and evaluate the surface integral \(\iint_{S} g(x, y, z) d S\) \iint_{S} \sqrt{x^{2}+y^{2}+z^{2}} d S, S \text { is the sphere } x^{2}+y^{2}+z^{2}=9

Determine whether or not the vector field is conservative. If it is, find a potential function. $$(y \cos x, \sin x-y)$$

If \(\mathbf{r}=\langle x, y, z\rangle\) and \(r=\|\mathbf{r}\|,\) prove that \(\nabla \cdot(r \mathbf{r})=4 r\).

Evaluate \(\int_{C} \mathbf{F} \cdot d \mathbf{r}.\) \(\mathbf{F}(x, y)=\langle 3 y-\sqrt{y / x}, 3 x-\sqrt{x / y}\rangle, \)C\( \text { is the ellipse } 4(x-4)^{2}+9(y-4)^{2}=36, \text { oriented counterclockwise }\)

Compute the work done by the force field \(\mathbf{F}\) along the curve \(C.\) \(\mathbf{F}(x, y)=\langle 0, x y\rangle, C\) is the portion of \(y=x^{3}\) from (0,0) to (1,1)

Set up a double integral and evaluate the surface integral \(\iint_{S} g(x, y, z) d S\) \(\iint_{S}\left(x^{2}+y^{2}-z\right) d S, S\) is the portion of the paraboloid \(z=4-x^{2}-y^{2}\) between \(z=1\) and \(z=2\)

Determine whether or not the vector field is conservative. If it is, find a potential function. $$(4 x-z, 3 y+z, y-x)$$

Prove Green's first identity in three dimensions (see exercise 43 in section 14.5 for Green's first identity in two dimensions): \(\iiint_{Q} f \nabla^{2} g d V=\iint_{\partial Q} f(\nabla g) \cdot \mathbf{n} d S-\iiint_{Q}(\nabla f \cdot \nabla g) d V\) (Hint: Use the Divergence Theorem applied to \(\mathbf{F}=f \nabla g\).)

Use Green's Theorem to prove the change of variables formula $$\iint_{R} d A=\iint_{S}\left|\frac{\partial(x, y)}{\partial(u, v)}\right| d u d v$$ where \(x=x(u, v)\) and \(y=y(u, v)\) are functions with continuous partial derivatives.

Compute the work done by the force field \(\mathbf{F}\) along the curve \(C.\) \(\mathbf{F}(x, y)=\langle 3 x, 2\rangle, C\) is the line segment from (0,0) to (0,1) followed by the line segment to (4,1)

Determine whether or not the vector field is conservative. If it is, find a potential function. $$\left(z^{2}+2 x y, x^{2}-z, 2 x z-1\right\rangle$$

Prove Green's second identity in three dimensions (see exercise 44 in section 14.5 for Green's second identity in two dimensions): $$ \iiint_{Q}\left(f \nabla^{2} g-g \nabla^{2} f\right) d V=\iint_{\partial Q}(f \nabla g-g \nabla f) \cdot \mathbf{n} d S $$ (Hint: Use Green's first identity from exercise \(31 .\) )

For \(\mathbf{F}=\frac{1}{x^{2}+y^{2}}\langle-y, x\rangle\) and \(C\) any circle of radius \(r>0\) not containing the origin, show that \(\oint_{C} \mathbf{F} \cdot d \mathbf{r}=0\)

Compute the work done by the force field \(\mathbf{F}\) along the curve \(C.\) \(\mathbf{F}(x, y)=\langle y, x\rangle, C\) is the square from (0,0) to (1,0) to (1,1) to (0,1) to (0,0)

Set up a double integral and evaluate the surface integral \(\iint_{S} g(x, y, z) d S\) \(\iint_{S} z^{2} d S, S\) is the portion of the cone \(z^{2}=x^{2}+y^{2}\) between \(z=-4\) and \(z=4\)

Determine whether or not the vector field is conservative. If it is, find a potential function. $$\left\langle y^{2} z^{2}-1,2 x y z^{2}, 4 z^{3}\right\rangle$$

Exercises \(33-36\) use Gauss' Law \(\nabla \cdot \mathbf{E}=\frac{\rho}{\epsilon_{0}}\) for an electric field E, charge density \(\rho\) and permittivity \(\epsilon_{0}\) If \(S\) is a closed surface, show that the total charge \(q\) enclosed by \(S\) satisfies \(q=\epsilon_{0} \iint_{S} \mathbf{E} \cdot \mathbf{n} d S\)

Compute the work done by the force field \(\mathbf{F}\) along the curve \(C.\) \(\mathbf{F}(x, y, z)=(y, 0, z), C\) is the triangle from (0,0,0) to (2,1,2) to (2,1,0) to (0,0,0)

Set up a double integral and evaluate the surface integral \(\iint_{S} g(x, y, z) d S\) \(\iint_{S} z^{2} d S, S\) is the portion of the cone \(z=\sqrt{x^{2}+y^{2}}\) above the rectangle \(0 \leq x \leq 2,-1 \leq y \leq 2\)

Determine whether or not the vector field is conservative. If it is, find a potential function. $$\left\langle z^{2}+2 x y, x^{2}+1,2 x z-3\right\rangle$$

Compute the work done by the force field \(\mathbf{F}\) along the curve \(C.\) \(\mathbf{F}(x, y, z)=\langle z, y, 0\rangle, C\) is the line segment from (1,0,2) to (2,4,2)

Set up a double integral and evaluate the surface integral \(\iint_{S} g(x, y, z) d S\) \(\iint_{S} x d S, S\) is the portion of \(x^{2}+y^{2}-z^{2}=1\) between \(z=0\) and \(z=1\)