Find the deflection u(x, y, t) of the square membrane of side π and c²=1 for initial velocity 0 and initial deflection 0.1 xy(π-x)(π-y)

Find the Fourier transform of $f(x)$ . Show the details.

$$f(x)=\left\{\begin{aligned} e^{2 i x} & \text { if }-1<x<1 \\ 0 & \text { otherwise }\end{aligned}\right.$$

Evaluate the surface integral ∫∫_S F*ndA by the divergence theorem. Show the details. F=[e^x, e^y, e^z], S the surface of the cube |x|≤1, |y|≤1, |z|≤1

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.

Solve the equation $u_x + 2xy^2u_y = 0$.

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 $x u+y v=0$.

Suppose the system

$$\left\{\begin{array}{l}{x u + y v-u v=0} \\ {y u-x v+u v=0}\end{array}\right.$$

can be solved for u and v in terms of x and y, so that u = u(x, y) and v = v(x, y). Use implicit differentiation to find the partial derivatives $\frac{\partial u}{\partial x}$ and $\frac{\partial v}{\partial x}$.

Find the type, transform to normal form, and solve. Show your work in detail.

$$u_{xx}-6u_{xy}+9u_{yy}=0$$

Find u(x, t) for the string of length :=1 and c²=1 when the initial velocity is zero and the inital deflection with small k(say, 0.01) is as follows. Sketch or graph u(x, t). kx²(1-x)

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

Find the Fourier series of f(x) as given over one period and sketch f(x) and partial sums. f(x)=|x| (-π<x<π)

Find a parametric representation. Circle in the plane z=1 with center (3, 2) and passing through the origin.

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