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Vector calculus: midterm 2
The test is for Vector calculus: course.
1. True or false?
- a. Every closed set is bounded.
- b. The empty set is path-connected.
- c. If the partial derivatives exist then the function is differentiable.
- d. Every continuous $1$-dimensional vector field is conservative.
- e. If $C$ is a vertical segment in $R^2$ and $F(x,y)=(\sin(xy),0)$ for all $x,y$, then
$$\displaystyle\int_{C}F(X) \cdot dX=0.$$
2. Give an example of a function of two variables such that:
- a. its image is the circle $x^2+y^2=1$,
- b. its image is the sphere $x^2+y^2+z^2=1$,
- c. its graph is the sphere $x^2+y^2+z^2=1$,
- d. its domain is the disk $x^2+y^2 \leq 1$,
- e. it is equal to its best affine approximation.
3. Find the best affine approximation of the function $$f(x)=x_1^2+x_2^2+x_3^2,$$ where $x=(x_1,x_2,x_3)$, at the point $a=(0,-1,1)$.
4. Find the global minima of the function $$f(x,y,z)=x^2z^2+y^2+y.$$
5. Suppose $f:{\bf R}^2 \rightarrow {\bf R}^2$ is a differentiable vector function such that the matrix of $f'(0,0)$ is the identity matrix. Find the total derivative of the vector function $h(x,y)=f(xy-1,x+y-2)$ at $(1,1)$.
6. Find the tangent plane to the parametric surface $f(u,v)=(u-v,v^2,u)$ at the point $(0,0,0)$.
7. Evaluate the integral $$\displaystyle\int\int\int_{B}xydV,$$ where $B$ is bounded by the planes $x=0,y=0,y=1-x,z=0,z=1$.
8. (a) Present the general formula of change of variables $T$ in an integral. (b) For the case of affine $T$, provide the Riemann sum for the integral and an illustration for the theorem.
9. Compute $$\displaystyle\int_{C}xydx+y^2dy,$$ where $C$ is the half of the unit circle from $(0,-1$) to $(0,1)$.
10. (a) State Stokes' Theorem for "simple regions". (b) Use part (a) to compute the area of a circle.
11. Suppose $$F(x,y)=(y\cos (xy),x\cos (xy)).$$ Find a potential function of $F$.
12. Suppose $C$ is a closed curve that makes one loop around the origin. Find the work of the force $$F(x,y)=\left(\frac{1}{x^2+y^2} \right)(-y,x)$$ along $C$. Justify.
