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. xy-plane r(u, v)=(u, v)(thus ui+vj)

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 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]

Find a parametric representation for the surface. The part of the ellipsoid x^2+y^2+3z^2=1 that lies to the left of the xz-plane

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.) Elliptic paraboloid $\mathbf{r}(u, v)=\left[a u \cos v, \quad b u \sin v, u^2\right]$.

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.

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

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. Elliptic cylinder r(u, v)=[a cos v, b sin v, u]

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.

$$y=k x^2$$

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 $\mathbf{r}(u, v)=\left[a u \cosh v, \quad b u \sinh v, u^2\right]$.

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 $0^{\circ}$.

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.) Hyperboloid $\mathbf{r}(u, v)=[a \sinh u \cos v, \quad b \sinh u \sin v, \quad c \cosh u]$.

Find $u_{x x}^{\prime \prime}$ when $u$ and $v$ are defined as functions of $x$ and $y$ by the equations $x y+u v=1$ and $x u+y v=0$.