Good luck!

1. Let $F:\mathbb{R}^3 \to \mathbb{R}^3$ be a vector field. What is the curl of $F$?

   $\left(\frac{\partial F_2}{\partial z} - \frac{\partial F_3}{\partial y},\frac{\partial F_3}{\partial x} - \frac{\partial F_1}{\partial z},\frac{\partial F_1}{\partial y} - \frac{\partial F_2}{\partial x}\right)$
   $\left(\frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z},\frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x},\frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y}\right)$
   

2. What is the Gamma function?

   $\Gamma(z) := \int_0^\infty t^{z-1}e^{-t}\,dt$
   $\Gamma(z) := \int_0^\infty t^{z}e^{-t-1}\,dt$
   

3. What is $x \land y$?

   $\max(x,y)$
   $\min(x,y)$
   

4. What is Fatou's Lemma?

   For any sequence of measurable functions $\{f_n\}_n$, we have $$\liminf_{n \to \infty} \int_X f_n\,d\mu \leq \int_X \liminf_{n \to \infty} f_n\,d\mu.$$
   For any sequence of non-negative measurable functions $\{f_n\}_n$, we have $$\liminf_{n \to \infty} \int_X f_n\,d\mu \leq \int_X \liminf_{n \to \infty} f_n\,d\mu.$$
   For any sequence of measurable functions $\{f_n\}_n$, we have $$\liminf_{n \to \infty} \int_X f_n\,d\mu \geq \int_X \liminf_{n \to \infty} f_n\,d\mu.$$
   For any sequence of non-negative measurable functions $\{f_n\}_n$, we have $$\liminf_{n \to \infty} \int_X f_n\,d\mu \geq \int_X \liminf_{n \to \infty} f_n\,d\mu.$$
   

5. What is $\int_{-\infty}^\infty e^{-x^2}\,dx$?

   $\sqrt{\pi}$
   $\sqrt{2\pi}$
   

6. What is $(f/g)'$?

   $\frac{f'g-fg'}{g^2}$
   $\frac{fg'-f'g}{g^2}$
   

7. What is $\Gamma(n)$ for $n$ a positive integer?

   $(n-1)!$
   $(n+1)!$
   

8. Let $u,v \in \mathbb{R}^3$ be vectors forming an angle $\theta$. Which is true?

   $|u \times v| = |u||v|\sin\theta$
   $|u \times v| = |u||v|\cos\theta$
   

9. What is Pascal's Identity?

   $\binom{n}{k} + \binom{n+1}{k} = \binom{n+1}{k+1}$
   $\binom{n}{k} + \binom{n}{k+1} = \binom{n+1}{k+1}$
   

10. What is $(x,y)$ rotated about the origin by $90^\circ$ counter-clockwise?

    $(-y,x)$
    $(y,-x)$
    

11. Which of the following is a correct form of spherical coordinates?

    $\begin{cases}x = r\sin\theta\cos\phi, \\ y = r\sin\theta\sin\phi, \\ z = r\cos\theta \end{cases}$
    $\begin{cases}x = r\cos\theta\sin\phi, \\ y = r\cos\theta\cos\phi, \\ z = r\sin\theta \end{cases}$
    

12. What is the Holder conjugate of $p$?

    $(p-1)/p$
    $p/(p-1)$
    

13. $x,y \in \mathbb{R}^N$, $E \subseteq \mathbb{R}^N$. Which is equivalent to the statement that $x \in E+y$?

    $x+y \in E$
    $x-y \in E$
    

14. What is the Taylor expansion of $\log(1-x)$ at $x=0$?

    $x+\frac{x^2}{2}+\frac{x^3}{3}+\ldots$
    $-x-\frac{x^2}{2}-\frac{x^3}{3}-\ldots$
    $-x+\frac{x^2}{2}-\frac{x^3}{3}+\ldots$
    $x-\frac{x^2}{2}+\frac{x^3}{3}-\ldots$
    

15. Let $f:\mathbb{R} \to \mathbb{R}$ be convex. Let $X$ be a random variable with $\mathbb{E}|X|<\infty$. What does Jensen's inequality state?

    $\mathbb{E} f(X) \leq f(\mathbb{E}X)$
    $\mathbb{E} f(X) \geq f(\mathbb{E}X)$
    

16. What are the dimensions of this matrix? $$\begin{bmatrix}1 & 2 & 3 \\ 4 & 5 & 6\end{bmatrix}$$

    $2 \times 3$
    $3 \times 2$
    

17. Let $(X,\tau)$ be a topological space, let $E \subseteq X$. Which of the following implications necessarily holds?

    $E$ closed $\implies$ $E$ sequentially closed
    $E$ closed $\impliedby$ $E$ sequentially closed
    Both
    Neither
    

18. Let $(X,\tau)$ be a topological space, let $K \subseteq X$. Which of the following implications necessarily holds?

    $K$ compact $\implies$ $K$ sequentially compact
    $K$ compact $\impliedby$ $K$ sequentially compact
    Both
    Neither
    

19. A family has two children with binary gender identities chosen uniformly at random. Given that there exists a boy among those children, what is the probability that the other child is a girl?

    $1/3$
    $1/2$
    $2/3$
    

20. Let $f:\mathbb{R} \to \mathbb{R}$ be smooth. Which of the following necessarily holds for some $\xi_x$ between $0$ and $x$?

    $f(x) = f(0) + f'(0)x + f''(0)x^2/2 + f''(\xi_x)x^2/2$
    $f(x) = f(0) + f'(0)x + f''(0)x^2/2 + f'''(\xi_x)x^3/6$
    

21. What is $\operatorname{cosh}(x)$?

    $\frac{e^x-e^{-x}}{2}$
    $\frac{e^x+e^{-x}}{2}$
    

22. Let $\{A_n\}_{n=1}^\infty$ be a sequence of sets. What is $\liminf_{n \to \infty} A_n$?

    $\displaystyle\bigcup_{n=1}^\infty\bigcap_{k=n}^\infty A_k$
    $\displaystyle\bigcap_{n=1}^\infty\bigcup_{k=n}^\infty A_k$
    

23. Let $f = (u,v)$ with $u,v:\mathbb{R}^2 \to \mathbb{R}$. What are the Cauchy-Riemann equations for $f$ (i.e. the equations that guarnatee that $f:\mathbb{C} \to \mathbb{C}$ is holomorphic, with the standard identification $\mathbb{R}^2 \cong \mathbb{C}$)?

    $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$
    $\frac{\partial u}{\partial x} = -\frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y} = \frac{\partial v}{\partial x}$
    

24. Assume that $f:\mathbb{R} \to \mathbb{R}$ is smooth and that $g:\mathbb{R} \to \mathbb{R}$ is a smooth injection. Select the correct identity.

    $\displaystyle\int_{g(E)} f(x)g'(x)\,dx = \int_{E} f(g(y))\,dy$
    $\displaystyle\int_{g(E)} f(x)\,dx = \int_{E} f(g(y))g'(y)\,dy$
    

25. Let $p \geq 1$. Which of the following is true?

    $(x+y)^p \leq C(x^p+y^p)$ for all $x,y \geq 0$, for some $C > 0$ depending only on $p$
    $(x+y)^p \geq C(x^p+y^p)$ for all $x,y \geq 0$, for some $C > 0$ depending only on $p$
    Both
    Neither
    

26. How is the vector $\overrightarrow{AB}$ defined?

    $A-B$
    $B-A$
    

27. What is the first isomorphism theorem?

    $G / \operatorname{ker}\phi \cong \operatorname{im}\phi$
    $G / \operatorname{im}\phi \cong \operatorname{ker}\phi$
    

28. $U \subseteq \mathbb{R}^2$ is open, bounded, simply connected with $\partial U$ oriented counter-clockwise. What does Green's Theorem state?

    $\displaystyle\int_U \frac{\partial F_1}{\partial y} - \frac{\partial F_2}{\partial x}\,dA = \int_{\partial U} F \cdot dr$
    $\displaystyle\int_U \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y}\,dA = \int_{\partial U} F \cdot dr$
    

29. Select the correct identity.

    $\sin(x+\frac{\pi}{2}) = \cos x$
    $\cos(x+\frac{\pi}{2}) = \sin x$
    

30. The integral $\int_0^1 \frac{1}{x^{0.999}}\,dx$

    converges
    diverges
    

31. $A \in \mathbb{R}^{n \times n}$, $B \in \mathbb{C}^{n \times n}$ satisfy $A^T=A^{-1}$ and $B^* = B^{-1}$. Select the correct classification.

    $A$ is orthogonal, $B$ is unitary
    $A$ is unitary, $B$ is orthogonal
    

32. Let $(X,\tau)$ be a topological space.

    $\tau$ is closed under finite intersections, but not necessarily countable intersections.
    $\tau$ is closed under countable intersections, but not necessarily arbitrary intersections.
    $\tau$ is closed under arbitrary intersections.
    

33. Let $\mathcal{F}$ be a sigma algebra.

    $\mathcal{F}$ is closed under finite intersections, but not necessarily countable intersections.
    $\mathcal{F}$ is closed under countable intersections, but not necessarily arbitrary intersections.
    $\mathcal{F}$ is closed under arbitrary intersections.
    

34. What is the Baire Category Theorem?

    In a complete metric space, the countable intersection of open dense sets has non-empty interior
    In a complete metric space, the countable intersection of open dense sets is dense
    

35. What is the Baire Category Theorem?

    If a complete metric space is the countable union of closed sets, then one of those closed sets has non-empty interior
    If a complete metric space is the countable union of closed sets, then one of those closed sets is dense
    

36. A simple polygon with lattice vertices and area $A$ has $B$ lattice points on its boundary and $I$ lattice points in its interior. What does Pick's Theorem state?

    $A = B + I/2 + 1$
    $A = B - I/2 + 1$
    $A = B + I/2 - 1$
    $A = I + B/2 + 1$
    $A = I - B/2 + 1$
    $A = I + B/2 - 1$
    

37. Let $A$ be an $n \times n$ matrix with real entries. Let $B = \begin{bmatrix}v_1 & v_2 & \cdots & v_n\end{bmatrix}$ be another $n \times n$ matrix whose columns $\{v_1,\cdots,v_n\}$ form a basis of $\mathbb{R}^n$. The matrix representation of $A$ with respect to this basis is

    $BAB^{-1}$
    $B^{-1}AB$