Familiarize yourself with parametric representations of important surfaces by deriving a representation, by finding the parameter curves (curves u=const and v=const) of the surface and a normal vector N=ru*rv of the surface. Show the details of your work. Helicoid r(u, v)=[u cos v, u sin v, v]. Explain the name.

Familiarize yourself with parametric representations of important surfaces by deriving a representation , by finding the parameter curves ( curves u=const and v=const) of the surface and a normal vector N=ru*rv of the surface. Show the details of your work. Cone r(u,v)=[u cos v, u sin v, cu]

Familiarize yourself with parametric representations of important surfaces by deriving a representation, by finding the parameter curves (curves u=const and v=const) of the surface and a normal vector N=ru*rv of the surface. Show the details of your work. Ellipsoid r(u, v)=[a cos v cos u, b cos v sin u, c sin v]

Familiarize yourself with parametric representations of practically important surfaces by deriving a representation $z=f(x, y) \text { or } g(x, y, z)=0 \text {. }$, by finding the parameter curves (curves $u=$ const and $v=$ const) on the surface and a normal vector $\mathbf{N}=\mathbf{r}_u \times \mathbf{r}_v$ of the surface. (Show the details of your work.) Hyperbolic paraboloid $\mathbf{r}(u, v)=\left[a u \cosh v, \quad b u \sinh v, u^2\right]$.

Parametric representation of curves was used in the proof of Theorem 1 (Conformality of mapping by analytic functions) and will be needed in integration, etc. Familiarize yourself with this basic concept by representing the following curve parametrically.

$$x^2+9 y^2=9$$

Parametric representation of curves was used in the proof of Theorem 1 (Conformality of mapping by analytic functions) and will be needed in integration, etc. Familiarize yourself with this basic concept by representing the following curve parametrically.

$$y=k x^2$$

Find the value of ∫_C δw/δn ds taken counterclockwise over the boundary C of the region R. w=x²y+xy², R:x²+y²≤1, x≥0, y≥0

Check, and if independent, integrate from (0, 0, 0) to (a, b, c).

$$2e^{x^2}(xcos2ydx-sin2ydy)$$

Stocks A and B have the following historical returns:

$$\begin{array}{ccc} \textbf{Year} & \textbf{Stock A's Returns}, \mathbf{r}_{\mathbf{A}} & \textbf{Stock B's Returns}, \mathbf{r}_{\mathbf{B}} \\ \hline 2011 & (24.25 \%) & 5.50 \% \\ 2012 & 18.50 & 26.73 \\ 2013 & 38.67 & 48.25 \\ 2014 & 14.33 & (4.50) \\ 2015 & 39.13 & 43.86 \end{array}$$

Calculate the standard deviation of returns for each stock and for the portfolio. Use Equation

$$\text { Estimated } \sigma=\sqrt{\frac{\sum_{\mathrm{t}=1}^{\mathrm{N}}\left(\overline{\mathbf{r}}_{\mathrm{t}}-\overline{\mathbf{r}}_{\text {Avg }}\right)^2}{\mathrm{~N}-1}}$$

Stocks A and B have the following historical returns:" $$ \begin{matrix} \text{Year} & \text{Stock A's Returns, }{r_A} & \text{Stock B's Returns, }{r_B}\\ \text{2011} & \text{(18.00\\%)} & \text{(14.50\\%)}\\ \text{2012} & \text{33.00} & \text{21.80}\\ \text{2013} & \text{15.00} & \text{30.50}\\ \text{2014} & \text{(0.50)} & \text{(7.60)}\\ \text{2015} & \text{27.00} & \text{26.30}\\ \end{matrix} $$ a. Calculate the average rate of return for each stock during the period 2011 through 2015. b. Assume that someone held a portfolio consisting of 50% of Stock A and 50% of Stock B. What would the realized rate of return on the portfolio have been each year? What would the average return on the portfolio have been during this period? c. Calculate the standard deviation of returns for each stock and for the portfolio. d. Calculate the coefficient of variation for each stock and for the portfolio. e. Assuming you are a risk-averse investor, would you prefer to hold Stock A, Stock B, or the portfolio? Why?

Solve the semi-infinite plate problem if the bottom edge of width $\pi$ is held at $T=\cos x$ and the other sides are at $0^{\circ}$.

Evaluate ∫_C F(r)*dr counterclockwise around the boundary C of the region R by Green’s theorem, where F=[6y², 2x-2y^4], R the square with vertices ±(2,2), ±(2,-2)

Describe the region of integration and evaluate.

$$∫_0^3∫_{-y}^y (x^2+y^2)dxdy$$

Show that the form under the integral sign is exact in the plane and evaluate the integral. Show the details of your work. ∫_(π/2,π)^(π,0) (½cos½x cos 2y dx-2 sin ½ x sin 2y dy)