Find the Laplacian of the following scalar functions: (a) $V=4 x y^{2} z^{3}$ (b) $V=x y+y z+z x$ (c) $V=3 /\left(x^{2}+y^{2}\right)$ (d) $V=5 e^{-r} \cos \phi$ (e) $V=10 e^{-R} \sin \theta$

Find the charateristic vibration frequences (normal frequences) and the characteristic modes of vibration for systems of masses and springs as in Figure 12.1 and Examples 3, 4, and 6 for the following array:

$$3k,~3m,~2k,~4m,~2k.$$

Determine the dimensions of Nul A and Col A for the matrices shown.

$$A=\left[ \begin{array}{rrrrrr}{1} & {3} & {-4} & {2} & {-1} & {6} \\ {0} & {0} & {1} & {-3} & {7} & {0} \\ {0} & {0} & {0} & {1} & {4} & {-3} \\ {0} & {0} & {0} & {0} & {0} & {0}\end{array}\right]$$

Use the given information about f(x) and its derivatives to determine whether Trapezoidal Rule would be exact, underestimate or overestimate the integral (or there’s not enough information to tell). $f^{\prime \prime}(x)>0, f^{\prime}(x)<0$

What are the differences and similarities between the quantization of angular momentum in the Bohr model and the Schrodinger theory?

Find the inverse of each function. Then graph the function and its inverse. $f(x)=5 x+10$, $f^{-1}(x)=$ _____

How are blood concentrations of aldosterone and cortisol regulated?

Verify Stokes’s theorem for the vector field $\mathbf{A}=\hat{\mathbf{R}} \cos \theta+\hat{\phi} \sin \theta$ by evaluating it on the hemisphere of unit radius.

How can the Uds. commands be used to make a sentence with the words? comprar los boletos / cinco dólares

Express each statement in “if... then” form. (More than one correct wording in “if... then” form may be possible.) Then write the statement’s converse, inverse, and contrapositive. Being a citizen is a necessary condition for voting.

How does the energy level of a hybrid orbital compare with the energy levels of the orbitals from which it was formed?

Find the matrix product. Note that each product can be found mentally, without the use of a calculator or pencil-and-paper calculations.

$$\left[\begin{array}{ll}{3} & {0} \\ {0} & {3}\end{array}\right]\left[\begin{array}{ll}{1} & {3} \\ {5} & {7}\end{array}\right]$$

If sediment input to a beach from rivers is greater than the output carried by the longshore current, will a beach grow or shrink? Explain your answer.

You place a long straight wire on top of your compass, and the wire is a height of 5 mm above the compass needle (Figure 17.81). If the conventional current in the wire is I = 0.2 A and runs left to right as shown, calculate the approximate angle the needle deflects away from north and draw the position of the compass needle.