Math 240,

the course of Dr. Mihailovs


Midterm 2

July 23, 1998

  1. Let ${\bf F}=\begin{pmatrix} xy\\ yz\\ zx \end{pmatrix}$. Find ${\bf \nabla\negmedspace\cdot F}$ and ${\bf \nabla\negmedspace\times F}$.

  2. Compute $\int_D 6x^2y dx dy$ where D is a unit square $0\leq x\leq 1, 0\leq y\leq 1$.

  3. Find the rate of increase per unit distance of the function f=xy2z at $\begin{pmatrix} 1\\ 1\\ 1 \end{pmatrix}$ in the direction of $\begin{pmatrix} 2\\ 1\\ 2 \end{pmatrix}$.

  4. Let ${\bf C}$ be the half-circle $x^2+y^2=1,\thickspace x\!\geq 0$ oriented counter-clockwise. Compute $\int_{\bf C} ydx-xdy$.

  5. Let ${\bf C}$ be the boundary of unit square $0\leq x\leq 1, 0\leq y\leq 1$, oriented counter-clockwise. Use Green's Theorem to evaluate $\int_{\bf C} 2ydx + 3xdy$.

  6. Let $A=\begin{pmatrix} 2 & 3\\ 4 & 5 \end{pmatrix}$. Write down A-1. Let $B=\begin{pmatrix} 1 & -1 & 2\\ -2 & 2 & 1\\ -3 & 1 & -4 \end{pmatrix}$. Write down the Laplace expansion of $\det B$ by the first column. Do not evaluate it.

  7. Let D be the quater of the unit disc $x^2+y^2\leq 1$ which lies in the first quadrant, $x\geq 0, y\geq 0$. Use polar coordinates to evaluate $\int_D xdxdy$.

  8. V is the upper half of the unit ball $x^2+y^2+z^2\leq 1, \thickspace z\geq 0$. Use spherical polars to evaluate $\int_V\sqrt{x^2+y^2+z^2} dxdydz$.

  9. Compute the volume of the solid body bounded by z=2-x2-y2 and the plane z=0.

  10. Let $T_A: \mathbb {R} ^4\rightarrow \mathbb {R} ^3,\quad {\bf x}\mapsto A{\bf x}$ where $A=\begin{pmatrix} 1 & 2 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0 \end{pmatrix}$. Find a basis for the range of TA and a basis for the null space of TA.

  11. S is the surface $z=1-x^2-y^2, \thickspace z\geq 0$. D is the unit disc $x^2+y^2\leq 1$. dA is the surface element on S. Write $\int_S(5-4z) dA$ as an integral over D. Do not evaluate.

  12. S is the surface of the unit cube $0\leq x\leq 1, \thickspace 0\leq y\leq 1, \thickspace 0\leq z\leq 1$. ${\bf n}$ is the outward drawn unit normal on S. Use the divergence Theorem to evaluate $\int_S {\bf F\cdot n} dA$ where ${\bf F}=\begin{pmatrix} 2x\\ 3y\\ -4z\end {pmatrix}$.

  13. Write out a careful statement of Stokes' Theorem (use a diagram).

  14. Let ${\bf D}$ be the unit square $0\leq x\leq 1, \thickspace 0\leq y\leq 1$ oriented by taking the boundary clockwise. Evaluate $\int_{\bf D}x dy\wedge dx$.

  15. Suppose that A is a $2\times 2$ matrix and that the following sequence of basic operations reduces A to I:

    \begin{displaymath} A\xrightarrow[\frac{1}{2} \row 2]{\row 2}A_1\xrightarrow[\row 2-\row 1] {\row 2}I\end{displaymath}

    Find the elementary matrices which produce these operations. Find A-1.

  16. Let Q=x12-4x1x2+x22. Determine whether the origin is a $\max$, $\min$, saddle point, or none of these.

  17. Let ${\bf C}$ be a path in $ \mathbb {R} ^3$ running from the origin to the point $\begin{pmatrix} 1\\ 2\\ -1 \end{pmatrix}$. Determine whether $I=\int_{\bf C} xdx+ydy+zdz$ depends on the path or is the same for all paths from ${\bf 0}$ to $\begin{pmatrix} 1\\ 2\\ -1 \end{pmatrix}$.

  18. Let $A=\begin{pmatrix} 1 & -1\\ 1 & 1 \end{pmatrix}$. Write down the general real solution to $\Dot{x}=Ax$.

  19. Evaluate $\int_S 2x^2+3y^2+4z^2\thickspace dA$ where S is the unit sphere x2+y2+z2=1 and dA is the surface element.

    Copyright © Aleksandrs Mihailovs 7/23/1998