Vector calculus: final 2

Grading

These are exercises for Vector calculus: course.

1. Give a parametric formula for a curve similar to an ascending spiral.
2. Write the equation of the plane which passes through the origin and is perpendicular to the line $x=0,y=t,z=2t$.
3. Find either the highest or the lowest point (whichever applicable) of the surface given by the equation $z=f(x,y)=2x^2+y^2-2x-y+1$.
4. Measure the volume of the solid lying above the rectangle $R$ in the $xy$-plane consisting of all points $(x,y,0)$ with $0 \leq x \leq 2,0 \leq y \leq 2$ and bounded from above by the surface given by $z=3y^2+4xy$.
5. Go with the vector field $$F[x,y]=\frac{1}{\sqrt{x^2+y^2}}(y,-x), (x,y)\neq (0,0). $$ Choose a few points on the plane and pencil in the field vector $F[x,y]$ with tail at $(x,y)$. Pencil in a few typical trajectories of the field.
6. Measure the net flow across the curve $y=x^2-1,0 \leq x \leq 1$, of a fluid whose velocity is given by a vector field $F[x,y]=(y,2x)$.
7. Here is a plot of a few level curves of a function $f[x,y]$ with a minimizer at $(1,0)$ and a maximizer at $(-1,0)$. Sketch a few trajectories of the vector field $grad\,f[x,y]$.
8. Let $$F[x,y]=(\sin (xy)+xy\cos (xy),x²\cos (xy)). $$ Show that $F[x,y]$ is a gradient field.
9. Suppose $$grad\,f[x,y]=(2xy+y^3,x^2+3y^2x).$$ Find $f[x,y]$.
10. Suppose $$f[x,y]=\sin x \cdot \ln (\cos y+1). $$ Find the flow of the vector field $F[x,y]=grad\,f[x,y]$ along any curve from $(\frac{\pi}{2},0)$ to $(\frac{\pi}{2},\frac{\pi}{2})$.
11. Let $A$ be the area of that part of the surface $z=x^2$ that lies above the unit circle. Write $A$ as a double integral. Do not evaluate.
12. Use an appropriate change of variables to find the area of the first-quadrant region bounded by the curves $xy=1,xy=2$ and $xy^2=2,xy^2=4$.
13. Find the volume conversion factor $V_{(x,y,z)}[u,v,w]$ of the transformation $x[u,v,w]=e^{uv},y[u,v,w]=uv^2w,z[u,v,w]=vw$.
14. Use Gauss's formula to evaluate the flow of the vector field $$F[x,y,z]=(xy,x^2+\frac{1}{2}y^2,yz)$$ across the surface of the solid body consisting of all points $(x,y,z)$ with $-1 \leq x \leq 1,|y| \leq 2-x^2,0 \leq z \leq 3$.