APM346 Self check: Calculus II

This is not a home assignment!

Differential calculus
1. Calculate $$\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}+\frac{\partial^2u}{\partial z^2}$$ as $u=r^{-1}e^{i\omega r}$ with $r=\sqrt{x^2+y^2+z^2}>0$.
2. Calculate $$\frac{\partial^3 e^{xyz}}{\partial x \partial y\partial z}.$$
3. Let $u(t)=U(x,y,z,t)$ with, $x=x(t)$, $y=y(t), z=z(t)$. Find $\frac{d u}{dt}$.
4. Find extremums of $$e^{-\frac{1}{2}(x^2+y^2)}xy$$ and classify them.
5. Find extremums of $$e^{-\frac{1}{2}(x^2+y^2)}xy$$ under constrain $x^2+y^2=R^2$; consider different cases.
6. Find $\frac{\partial y}{\partial x}$ as $x^3+y^3-3xy=0$.
7. Find $\frac{\partial y}{\partial t}$ as $x^3+y^3-3xyt=0$.
Integral calculus
1. Calculate $$\iiint_\Omega xyz \,dxdydz$$ with $\Omega = \{x>0,y>0,z>0, x+y+z <1\}$.
2. In ellipse $D=\{ x^2 /a^2+y^2 /b^2 \le 1\}$ with $a=\cosh (k)$, $b=\sinh (k)$. Introduce elliptical coordinates $x=\cosh (u)\cos(v)$, $y=\sinh(u)\sin(v)$ with $(u,v)\in D'=\{ 0<u<k, 0<v<2\pi\}$ and rewrite $\int_D f(x,y)\,dxdy$ in these coordinates.
3. Calculate $$\iint_D \frac{dx dy}{\sqrt{x^2+y^2}}$$ over $D=\{x^2+y^2<1\}$ using polar coordinates;
4. Calculate $$\iint_D \frac{dx dy dz}{\sqrt{x^2+y^2+z^2}}$$ over $\Omega=\{x^2+y^2+z^2<1\}$ using spherical coordinates;
5. Check Green formula (calculate directly both parts) as $D=\{x^2+y^2\le 1\}$ and $\mathbf{A}=y\mathbf{i}-x\mathbf{j}$.
6. Check Gauss formula (calculate directly both parts) as $\Omega=\{x^2+y^2+z^2\le 1\}$ and $\mathbf{A}=x\mathbf{i}+y\mathbf{j}+z\mathbf{k}$.
7. Check Stokes formula (calculate directly both parts) as $\Sigma = \{x^2+y^2+z^2=1, z>0\}$ and normal makes an accute angle with $\mathbf{k}$, $\mathbf{A}=\mathbf{i}-x\mathbf{j}$.
8. Check formula $$\Delta \mathbf{A}=-\Delta \mathbf{A} +\nabla \cdot (\nabla\cdot \mathbf{A})$$ as $\mathbf{A}=x^2 \mathbf{i}- y^2\mathbf{j}+2xy \mathbf{k}$ (calculate directly both parts).