Use the Ratio Test to determine whether each series converges or diverges or state that it provides no information. $\sum_{k=1}^{\infty} \frac{(2 k) !}{5^{k} 3^{k-1}}$

What is the (a) specific weight, and (b) density of air at an absolute pressure of 600 kPa and a temperature of 50$^{\circ}C$?

(a) Show that $\sum_{k=1}^{\infty} \frac{\ln k}{k^{p}}$ diverges for $0<p \leq 1$. (b) Show that $\sum_{k=1}^{\infty} \frac{(\ln k)^{r}}{k^{p}}$, where r is a positive real number and $0<p \leq 1$, diverges.

Use the method of separation of variables to solve each of the following boundary-value problems. $\frac{\partial u}{\partial t}=\frac{\partial u}{\partial y} ; \quad u(t, 0)=e^{-3 t}+e^{2 t}$

Find the lengths of the curves. $x=1+t^{3}, y=1-t^{2},(-1 \leq t \leq 2)$

A dentist uses a spherical mirror to examine a tooth. The tooth is 1.00 cm in front of the mirror, and the image is formed 10.0 cm behind the mirror. Determine the mirror’s radius of curvature.

The vertical displacement of a nonuniform membrane satisfies $\frac{\partial^{2} u}{\partial t^{2}}=c^{2}\left(\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}\right),$ where c depends on x and y. Suppose that u = 0 on the boundary of an irregularly shaped membrane. (a) Show that the time variable can be separated by assuming that $u(x, y, t)=\phi(x, y) h(t).$ Show that $\phi(x, y)$ satisfies the eigenvalue problem $\nabla^{2} \phi+\lambda \sigma(x, y) \phi=0$ with $\phi=0$ on the boundary. What is $\sigma(x, y) ?$

Find all relative extrema and saddle points of the function. Use the Second Partials Test where applicable. $f(x, y)=-4\left(x^{2}+y^{2}+81\right)^{1 / 4}$

The maximum wave force on a 1/36 model seawall was found to be 80 N. For a corresponding wave in the full-scale wall, what full-scale force would you expect? Assume freshwater is used in the model study. Assume T=10$^{\circ}C$ for both model and prototype water.

Consider a three-dimensional ket space. If a certain set of orthonormal kets—say, $|1\rangle,|2\rangle,$ and $|3\rangle$—are used as the base kets, the operators A and B are represented by

$$A \doteq\left(\begin{array}{ccc}{a} & {0} & {0} \\ {0} & {-a} & {0} \\ {0} & {0} & {-a}\end{array}\right), \quad B \doteq\left(\begin{array}{ccc}{b} & {0} & {0} \\ {0} & {0} & {-i b} \\ {0} & {i b} & {0}\end{array}\right)$$

with a and b both real. (a) Obviously A exhibits a degenerate spectrum. Does B also exhibit a degenerate spectrum? (b) Show that A and B commute. (c) Find a new set of orthonormal kets that are simultaneous eigenkets of both A and B. Specify the eigenvalues of A and B for each of the three eigenkets. Does your specification of eigenvalues completely characterize each eigenket?

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Review. (a) A homeowner has a solar water heater installed on the roof of his house. The heater is a flat, closed box with excellent thermal insulation. Its interior is painted black, and its front face is made of insulating glass. Its emissivity for visible light is 0.900, and its emissivity for infrared light is 0.700. Light from the noontime Sun is incident perpendicular to the glass with an intensity of 1 000 W/m2, and no water enters or leaves the box. Find the steady-state temperature of the box’s interior. (b) What If? The homeowner builds an identical box with no water tubes. It lies flat on the ground in front of the house. He uses it as a cold frame, where he plants seeds in early spring. Assuming the same noontime Sun is at an elevation angle of 50.0$^ { \circ }$, find the steady-state temperature of the interior of the box when its ventilation slots are tightly closed.

Sutherland’s equation (select all that apply): a. relates temperature and viscosity, b. must be calculated using Kelvin, c. requires use of a single universal constant for all gases, d. requires use of a different constant for each gas.

Explain why two parallel wires carrying currents in opposite directions repel each other.