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]

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.) Cone $\mathbf{r}(u, v)=[u \cos v, \quad u \sin v, \quad c u]$.

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. Hyperbolic paraboloid r(u, v)=[au cosh v, bu sinh v, u²]

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]

Using $\iint_R \nabla^2 w d x d y=\oint_C \frac{\partial w}{\partial n} d s$, evaluate $\oint_C \frac{\partial w}{\partial n} d s$ counterclockwise over the boundary curve $C$ of the region $R$. (Show the details of your work.) $w=\cosh x, \quad R$ the triangle with vertices $(0,0),(4,2),(0,2)$

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)

Evaluate ∫_C F(r)*dr counterclockwise around the boundary C of the region R by Green’s theorem, where F=[y,-x], C the circle x²+y²=1/4

Find the volume of the given region in space. The region beneath z=4x²+9y² and above the rectangle with vertices (0, 0), (3, 0), (3, 2), (0, 2) in the xy-plane.

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

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

A 20-lb weight stretches a spring 6 in. The weight is released from rest 6 in. below the equilibrium position. Find the position of the weight at $t=\pi / 12, \pi / 8, \pi / 6, \pi / 4,9 \pi / 32$ seconds.

A mass weighing 20 pounds stretches one spring 6 inches and another spring 2 inches. The springs are attached parallel to each other to a common rigid support and then to a metal plate of negligible mass. The mass is attached to the center of the plate in the double-spring arrangement. Determine the effective spring constant of this system. Find the equation of motion if the mass is initially released from the equilibrium position with a downward velocity of 2 ft/s.

A mass weighing 20 pounds stretches a spring 6 inches. The mass is initially released from rest from a point 6 inches below the equilibrium position. (a) Find the position of the mass at the times $t = \pi/12, \pi/8, \pi/6, \pi/4$, and $9\pi /32 \text{ s}$. (b) What is the velocity of the mass when $t = 3\pi/16 \text{ s}$? In which direction is the mass heading at this instant? (c) At what times does the mass pass through the equilibrium position?