Using the figure as a guide, transform Cartesian velocity components $(u, v, w)$ into cylindrical velocity components $\left(u_r, u_\theta, u_z\right)$. (Hint: Since the $z$-component of velocity remains the same in such a transformation, we need only to consider the $x y$-plane.)

Find the Fourier transform of $f(x)=e^{-x^{2} /\left(2 \sigma^{2}\right)}$. Hint: Complete the square in the x terms in the exponent and make the change of variable $y=x+\sigma^{2} i \alpha$. Use tables or computer to evaluate the definite integral.

For the following problem, use the composite argument properties to transform the left side of the equation to a single function of a composite argument. Then solve the equation algebraically to get

The general solution for x or $\theta$

$$\cos 3 x \cos x+\sin 3 x \sin x=-1$$

Use the integral definition to calculate the Laplace transform of the function

$$f(t)=\left\{\begin{array}{ll}{t-1} & {\text { if } 0 \leq t<2} \\ {1} & {\text { if } t \geq 2}\end{array}\right.$$

.

Prove that if F($\omega$) is the Fourier transform of f(t),

$$\mathcal{F}\left[f(t) \sin \omega_{0} t\right]=\frac{j}{2}\left[F\left(\omega+\omega_{0}\right)-F\left(\omega-\omega_{0}\right)\right]$$

Solve the following initial value problems by the Laplace transform. (If necessary, use partial fraction expansion. Show all details.)

$y^{\prime \prime}-2 y^{\prime}-3 y=0, \quad y(1)=-3$, $y^{\prime}(1)=-17$

Use the Laplace transform to solve the given initial-value problem. Use the table of Laplace transforms in Appendix III as needed. $y^{\prime \prime}+9 y=\cos 3 t, \quad y(0)=2, \quad y^{\prime}(0)=5$

Sketch or graph the given function (which is assumed to be zero outside the given interval). Represent it using unit step functions. Find its transform. Show the details of your work.

$$\sinh t(0<t<2)$$

Find the Laplace transform of $f(t).$

$$f(t)=\left\{\begin{array}{ll}{1-\cos (2 t)} & {\text { for } 0 \leq t<3 \pi} \\ {0} & {\text { for } t \geq 3 \pi}\end{array}\right.$$

Let g(t)=x(t)+αx(it), where

$$x(t)=βe^{-t}$$

u(t) and the Laplace transform of g(t) is G(s)=δ/s²-1, -1<Re{s}<t. Determine the values of the constants α and β.

Use the Laplace transform to solve the given differential equation subject to the indicated initial conditions. $y^{\prime \prime}+y=\delta(t-\pi / 2)+\delta(t-3 \pi / 2), \quad y(0)=0, y^{\prime}(0)=0$

Using the Laplace transform, solve the following problem. (Show the details.) $y^{\prime \prime}+3 y^{\prime}+2 y=4 t$ if $0<t<1$ and 8 if $t>1 ; \quad y(0)=0, \quad y^{\prime}(0)=0$

Suppose that you have a double integral over a region R in the xy-plane and you want to transform that integral into an equivalent double integral over a region S in the uv-plane. Describe the procedure you would use.

Find the Laplace transform of $f(t).$

$$f(t)=\left\{\begin{array}{ll}{t^{2}} & {\text { for } 0 \leq t<2} \\ {1-t-4 t^{2}} & {\text { for } t \geq 2}\end{array}\right.$$