Chapter 14

The work done to increase the temperature of a gas from \(T_{1}\) to \(T_{2}\) and increase its pressure from \(P_{1}\) to \(P_{2}\) is given by \(\int_{C}\left(\frac{R T}{P} d P-R d T\right) .\) Here, \(R\) is a constant, \(T\) is temperature, \(P\) is pressure and \(C\) is the path of \((P, T)\) values as the changes occur. Compare the work done along the following two paths. (a) \(C_{1}\) consists of the line segment from \(\left(P_{1}, T_{1}\right)\) to \(\left(P_{1}, T_{2}\right),\) followed by the line segment to \(\left(P_{2}, T_{2}\right) ;\) (b) \(C_{2}\) consists of the line segment from \(\left(P_{1}, T_{1}\right)\) to \(\left(P_{2}, T_{1}\right),\) followed by the line segment to \(\left(P_{2}, T_{2}\right).\)

Use the formulas \(m=\int_{C} \rho d s, \bar{x}=\frac{1}{m} \int_{C} x \rho d s\) \(\bar{y}=\frac{1}{m} \int_{c} y \rho d s, I=\int_{C} w^{2} \rho d s.\) Compute the mass \(m\) of the helical spring \(x=\cos 2 t\) \(y=\sin 2 t, z=t, 0 \leq t \leq \pi,\) with density \(\rho=z^{2}.\)

Find the mass and center of mass of the region. The portion of the paraboloid \(z=x^{2}+y^{2}\) inside the cylinder \(x^{2}+y^{2}=4, \rho(x, y, z)=z\)

Use the notation \(r=\langle x, y\rangle\) and \(r=\|\mathbf{r}\|=\sqrt{x^{2}+y^{2}}\) Show that \(\frac{\mathbf{r}}{r^{n}}=\frac{\langle x, y\rangle}{\left(x^{2}+y^{2}\right)^{n / 2}}\) is conservative, for any integer \(n\)

If \(f\) is a scalar function, \(\mathbf{r}=\langle x, y\rangle\) and \(r=\|\mathbf{r}\|,\) show that $$\nabla^{2} f(r)=f^{\prime \prime}(r)+\frac{1}{r} f^{\prime}(r)$$

Use the formulas \(m=\int_{C} \rho d s, \bar{x}=\frac{1}{m} \int_{C} x \rho d s\) \(\bar{y}=\frac{1}{m} \int_{c} y \rho d s, I=\int_{C} w^{2} \rho d s.\) Compute the mass \(m\) of the ellipse \(x=4 \cos t, y=4 \sin t\) \(z=4 \cos t, 0 \leq t \leq 2 \pi,\) with density \(\rho=4.\)

A two-dimensional force acts radially away from the origin with magnitude \(3 .\) Write the force as a vector field.

Compute the Laplacian \(\Delta f\) for \(f(x, y, z)=\sqrt{x^{2}+y^{2}+z^{2}}\).

A two-dimensional force acts radially toward the origin with magnitude equal to the square of the distance from the origin. Write the force as a vector field.

Compute the Laplacian \(\Delta f\) for \(f(x, y, z)=\frac{1}{x^{2}+y^{2}+z^{2}}\).

The circulation of a fluid with velocity field \(\mathbf{v}\) around the closed path \(C\) is defined by \(\Gamma=\int_{C} \mathbf{v} \cdot d \mathbf{r} .\) For inviscid flow, \(\frac{d}{d t} \Gamma=\int_{C} \mathbf{v} \cdot d \mathbf{v} .\) Show that in this case \(\frac{d}{d t} \Gamma=0 .\) This is known as Kelvin's Circulation Theorem and explains why small whirlpools in a stream stay coherent and move for periods of time.

Use the formulas of exercises 53 and 54 to evaluate the surface integral. \(\iint_{S} z d S,\) where \(S\) is the portion of \(x^{2}+y^{2}=1\) with \(x \geq 0\) and \(z\) between \(z=1\) and \(z=2\)

A three-dimensional force acts radially toward the origin with magnitude equal to the square of the distance from the origin. Write the force as a vector field.

Suppose that \(\mathbf{F}(x, y)=\left\langle x^{2}, y^{2}-4 x\right\rangle\) represents the velocity field of a fluid in motion. For a small box centered at \((x, y)\) determine whether the flow into the box is greater than, less than or equal to the flow out of the box. (a) \((x, y)=(0,0)\) and (b) \((x, y)=(1,0)\).

Find the surface area extending from the given curve in the \(x y\) -plane to the given surface. Above the quarter-circle of radius 2 centered at the origin from (2,0,0) to (0,2,0) up to the surface \(z=x^{2}+y^{2}.\)

Use the formulas of exercises 53 and 54 to evaluate the surface integral. \(\iint y z \, d S,\) where \(S\) is the portion of \(x^{2}+y^{2}=1\) with \(x \geq 0\) and \(z\) between \(z=1\) and \(z=4-y\)

A three-dimensional force acts radially away from the z-axis (parallel to the \(x y\) -plane) with magnitude equal to the cube of the distance from the \(z\) -axis. Write the force as a vector field.

Find the surface area extending from the given curve in the \(x y\) -plane to the given surface. Above the portion of \(y=x^{2}\) from (0,0,0) to (2,4,0) up to the surface \(z=x^{2}+y^{2}.\)

Use the formulas of exercises 53 and 54 to evaluate the surface integral. \(\iint_{S}\left(y^{2}+z^{2}\right) d S,\) where \(S\) is the portion of the paraboloid \(x=9-y^{2}-z^{2}\) in front of the \(y z\) -plane

Derive the electrostatic field for positive charges \(q\) at (-1,0) and (1,0) and negative charge \(-q\) at (0,0)

Give an example of a vector field \(\mathbf{F}\) such that \(\nabla \cdot \mathbf{F}\) is a positive function of \(y\) only.

Find the surface area extending from the given curve in the \(x y\) -plane to the given surface. Above the line segment from (2,0,0) to (-2,0,0) up to the surface \(z=4-x^{2}-y^{2}.\)

Use the formulas of exercises 53 and 54 to evaluate the surface integral. \(\iint_{S}\left(y^{2}+z^{2}\right) d S,\) where \(S\) is the hemisphere \(x=\sqrt{4-y^{2}-z^{2}}\)

Give an example of a vector field \(\mathbf{F}\) such that \(\nabla \times \mathbf{F}\) is a function of \(x\) only.

Find the surface area extending from the given curve in the \(x y\) -plane to the given surface. Above the line segment from (1,1,0) to (-1,1,0) up to the surface \(z=\sqrt{x^{2}+y^{2}}.\)

Use the formulas of exercises 53 and 54 to evaluate the surface integral. \(\iint_{S} x^{2} d S,\) where \(S\) is the portion of the paraboloid \(y=x^{2}+z^{2}\) to the left of the plane \(y=1\)

If \(T(x, y, z)\) gives the temperature at position \((x, y, z)\) in space, the velocity field for heat flow is given by \(\mathbf{F}=-k \nabla T\) for a constant \(k>0 .\) This is known as Fourier's law. Use this vector field to determine whether heat flows from hot to cold or vice versa. Would anything change if the law were \(\mathbf{F}=k \nabla T ?\)

Gauss' law states that \(\nabla \cdot \mathbf{E}=\frac{\rho}{\epsilon_{0}} .\) Here, \(\mathbf{E}\) is an electrostatic field, \(\rho\) is the charge density and \(\epsilon_{0}\) is the permittivity. If E has a potential function \(-\phi,\) derive Poisson's equation \(\nabla^{2} \phi=-\frac{\rho}{\epsilon_{0}}\).

Find the surface area extending from the given curve in the \(x y\) -plane to the given surface. Above the unit square \(x \in[0,1], y \in[0,1]\) up to the plane \(z=4-x-y.\)

Use the formulas of exercises 53 and 54 to evaluate the surface integral. \(\iint_{S}\left(x^{2}+z^{2}\right) d S,\) where \(S\) is the hemisphere \(y=\sqrt{4-x^{2}-z^{2}}\)

For two-dimensional fluid flow, if \(\mathbf{v}=\left\langle v_{x}(x, y), v_{y}(x, y)\right\rangle\) is the velocity field, then \(v\) has a stream function \(g\) if \(\frac{\partial g}{\partial x}=-v_{y}\) and \(\frac{\partial g}{\partial y}=v_{x} .\) Show that if \(v\) has a stream function and the components \(v_{x}\) and \(v_{y}\) have continuous partial derivatives, then \(\nabla \cdot \mathbf{v}=0\).

Find the surface area extending from the given curve in the \(x y\) -plane to the given surface. Above the ellipse \(x^{2}+4 y^{2}=4\) up to the plane \(z=4-x.\)

For \(v=\left(2 x y,-y^{2}+x\right),\) show that \(\nabla \cdot v=0\) and find a stream function \(g\).

For \(\mathbf{v}=\left\langle x e^{x y}-1,2-y e^{x y}\right\rangle,\) show that \(\nabla \cdot \mathbf{v}=0\) and find a stream function \(g\).

Explain the following result geometrically. The flux integral of \(\mathbf{F}(x, y, z)=\langle x, y, z\rangle\) across the cone \(z=\sqrt{x^{2}+y^{2}}\) is 0

Sketch the function \(f(x)=\frac{1}{1+x^{2}}\) and use it to sketch the vector field \(\mathbf{F}=\left\langle 0, \frac{1}{1+x^{2}}, 0\right\rangle .\) If this represents the velocity field of a fluid and a paddle wheel is placed in the fluid at various points near the origin, explain why the paddle wheel would start spinning. Compute \(\nabla \times \mathbf{F}\) and label the fluid flow as rotational or irrotational. How does this compare to the motion of the paddle wheel?

In geometric terms, determine whether the flux integral of \(\quad \mathbf{F}(x, y, z)=\langle x, y, z\rangle\) across the hemisphere \(z=\sqrt{1-x^{2}-y^{2}}\) is 0

Sketch the vector field \(\mathbf{F}=\left\langle\frac{1}{1+x^{2}}, 0,0\right\rangle .\) If this represents the velocity field of a fluid and a paddle wheel is placed in the fluid at various points near the origin, explain why the paddle wheel would not start spinning. Compute \(\nabla \times \mathbf{F}\) and label the fluid flow as rotational or irrotational. How does this compare to the motion of the paddle wheel?

For the cone \(z=c \sqrt{x^{2}+y^{2}}\) (where \(c>0\) ), show that in spherical coordinates \(\tan \phi=\frac{1}{c} .\) Then show that parametric equations are \(x=\frac{u \cos v}{\sqrt{c^{2}+1}}, y=\frac{u \sin v}{\sqrt{c^{2}+1}}\) and \(z=\frac{c u}{\sqrt{c^{2}+1}}\)

Show that if \(\mathbf{G}=\nabla \times \mathbf{H},\) for some vector field \(\mathbf{H}\) with continuous partial derivatives, then \(\nabla \cdot \mathbf{G}=0\).

Show the converse of exercise \(65 ;\) that is, if \(\nabla \cdot \mathbf{G}=0\) then \(\mathbf{G}=\nabla \times \mathbf{H}\) for some vector field \(\mathbf{H}\). [ Hint: Let \(\mathbf{H}(x, y, z)=\left\langle 0, \int_{0}^{x} G_{3}(u, y, z) d u,-\int_{0}^{x} G_{2}(u, y, z) d u\right\rangle]\)

Find the flux of \(\langle x, y, z\rangle\) across the portion of \(z=c \sqrt{x^{2}+y^{2}}\) below \(z=1 .\) Explain in physical terms why this answer makes sense.

If \(C\) has parametric equations \(x=x(t), y=y(t), z=z(t)\) \(a \leq t \leq b,\) for differentiable functions \(x, y\) and \(z\) show that \(\int_{C} \mathbf{F} \cdot \mathbf{T} d s=\int_{a}^{b}\left[F_{1}(x, y, z) x^{\prime}(t)+F_{2}(x, y, z) y^{\prime}(t)+\right.\) \(\left.F_{3}(x, y, z) z^{\prime}(t)\right] d t,\) which is the work line integral \(\int_{C} \mathbf{F} \cdot d \mathbf{r}.\)

Find the flux of \(\langle x, y, 0\rangle\) across the portion of \(z=c \sqrt{x^{2}+y^{2}}\) below \(z=1 .\) Explain in physical terms why this answer makes sense.

If the two-dimensional vector \(\mathbf{n}\) is normal (perpendicular to the tangent) to the curve \(C\) at each point and \(\mathbf{F}(x, y)=\left\langle F_{1}(x, y), F_{2}(x, y)\right\rangle,\) show that \(\int_{C} \mathbf{F} \cdot \mathbf{n} d s=\) \(\int_{C} F_{1} d y-F_{2} d x.\)

If \(T(x, y)\) is the temperature function, the line integral \(\int_{C}(-k \nabla T) \cdot\) nds gives the rate of heat loss across C. For \(T(x, y)=60 e^{y / 50}\) and \(C\) the rectangle with sides \(x=-20, x=20, y=-5\) and \(y=5,\) compute the rate of heat loss. Explain in terms of the temperature function why the integral is 0 along two sides of \(C.\)