--- date: 2022-12-25 04:01 modification date: Sunday 25th December 2022 04:01:30 title: "Grad Math Facts" flashcard: "Math::Grad Math Facts" --- # Undergrad ## Misc algebra - [x] What is Descartes' rule of signs? ✅ 2022-12-25 - [ ] The number of positive real roots of $f$ is equivalent $\mod 2$ to the number of sign changes in the coefficients of $f(x)$. The number of *negative* real roots is the same process but for $f(-x)$. E.g. $f(x)=x^5+4 x^4-3 x^2+x-6$ has 3 changes of sign, so an odd number of real roots. - [x] Factor $x^n-y^n$. ✅ 2022-12-25 - [ ] $$x^n-y^n=(x-y)\left(x^{n-1}+x^{n-2} y+\ldots+x y^{n-2}+y^{n-1}\right) = (x-y)\sum_{k=0}^{n-1}x^{n-k-1}y^k$$ How to prove: long division to obtain ${x^n-y^n\over x-y} = x^n + y\qty{x^{n-1}-y^{n-1}\over x-y}$ and induct. How to remember: $x^n-1 = (x-1)(x^{n-1} + x^{n-2} + \cdot + x + 1)$ and set $x=b/a$. - [x] Factor $x^n + y^n$ for $n$ an odd prime. ✅ 2022-12-25 - [ ] $$x^n+y^n=(x+y)\left(x^{n-1}-x^{n-2} y+\cdots-x y^{n-2}+y^{n-1}\right)$$ ## Logs - [x] Write $\log$ and $\exp$ as group homomorphisms; i.e. what are the domains and codomains? ✅ 2022-12-25 - [ ] $$\GG_m \underset{\exp}{\overset{\log}{\mapstofrom}} \GG_a$$ - [x] True or false: $\log(a-b) = \log(a) / \log(b)$. ✅ 2022-12-25 - [ ] False: take $b=1$ and $a$ to be anything, this would force dividing by zero in general. - [x] True or false: $\log(a+b) = \log(a) + \log(b)$ ✅ 2022-12-25 - [ ] False: $\log_2(4+4) \log_2(8) = 3 \neq 4 = 2 + 2 = \log_2(4) + \log_2(4)$. - [x] True or false: $\log(a+b) = \log(a)\cdot \log(b)$. ✅ 2022-12-25 - [ ] False: $\log_2(4+4) = \log_2(8) = 3 \neq 4 = 2 \cdot 2 = \log_2(4) \cdot \log_2(4)$. ## Trig - [x] $\sin(\pi/6)$ ✅ 2023-01-25 - [ ] $1/2$ - [x] $\cos(\pi/6)$ ✅ 2023-01-25 - [ ] $\sqrt 3 /2$. - [x] $\sin(\pi/3)$ ✅ 2023-01-25 - [ ] $\sqrt 3/2$ - [x] $\cos(\pi/3)$ ✅ 2023-01-25 - [ ] $1/2$. - [x] $\arctan(0)$ ✅ 2023-01-25 - [ ] $0$ - [x] $\arctan(1)$ ✅ 2023-01-25 - [ ] $\pi/4$. - [x] $\arctan(\infty)$ ✅ 2023-01-25 - [ ] $\pi/2$ - [x] $\arctan\qty{\sqrt 3\over 3}$ ✅ 2023-01-25 - [ ] $\pi/6$ - [x] $\arctan\qty{\sqrt 3}$ ✅ 2023-01-25 - [ ] $\pi/3$ - [x] $\sin(a+b) = \cdots$ ✅ 2023-01-09 - [ ] $\sin(a)\cos(b) + \cos(a)\sin(b)$. - [x] $\cos(a+b) = \cdots$ ✅ 2023-01-09 - [ ] $\cos(a)\cos(b) - \sin(a)\sin(b)$. - [x] $\tan(a+b) = \cdots$ ✅ 2023-01-09 - [ ] ${\tan(a)\tan(b) \over 1-\tan(a)\tan(b)}$. - [x] Double angle formulas converting $f(2t)$ into $\tan$ for $f=\sin,\cos$. ✅ 2023-01-09 - [ ] $$\begin{align}\sin(2t) &= {2\tan(t) \over 1+\tan^2(t)} \\ \cos(2t) &= {1-\tan^2(2t) \over 1+\tan^2(t)}\end{align}.$$ - [x] Standard parameterization of a circle in $\CC$ ✅ 2023-01-09 - [ ] $$F(t) = {1-x^2\over 1+x^2} + i{2x\over 1+x^2}$$ where $x = \tan(t)$ and $t\in (-\pi/2, \pi/2)$. ## Hyperbolic Trig - [x] Exponential expansions of $\sin(z)$ ✅ 2023-01-09 - $$\begin{align*} \sin(\theta) &= \frac{e^{i\theta} - e^{-i\theta}}{2i} = \frac{z - z\inv}{2i},\qquad z\da e^{i\theta}\\ d\theta &= \frac{dz}{iz} \end{align*}$$ - [x] Exponential definitions of $\sin,\cos, \sinh, \cosh$. ✅ 2023-01-09 - [ ] $$\begin{align} \cos(z) &= {e^{iz} + e^{-iz}\over 2} \qquad \cosh(z) = {e^{z} - e^{-z}\over 2} \\ \sin(z) &= {e^{iz} + e^{-iz}\over 2i} \qquad \sinh(z) = {e^{z} - e^{-z}\over 2} \end{align}.$$ - [x] Relating hyperbolic functions to usual ones: $\cosh(iz) = \cdots$ ✅ 2023-01-09 - [ ] $$\cosh(iz) = \cos(z), \qquad \cos(iz) = \cosh(z).$$ - [x] Relating hyperbolic functions to usual ones: $\sin(z) = f(\sinh(\cdots)$ ✅ 2023-01-09 - [ ] $$\sinh(iz) = i\sin(z),\qquad \sin(iz) = i\sinh(z).$$ - [x] Angle addition formulas: $\cosh(x+iy) = \cdots$ ✅ 2023-01-09 - $$\cosh (x+i y) =\cosh (x) \cos (y)+i \sinh (x) \sin (y) .$$ - [x] Angle addition formulas: $\sinh(x+iy) = \cdots$ ✅ 2023-01-09 - $$\sinh (x+i y) =\sinh (x) \cos (y)+i \cosh (x) \sin (y).$$ ## Series - [x] Series expansion for $\arctan(x)$. ✅ 2023-01-25 - [ ] $$z - {1\over 3}z^3 + {1\over 5}z^5 + \cdots$$ - [x] Series expansion for $\log$ ✅ 2023-01-09 - [ ] $$-\log(1-z) = \sum_{n\geq 1} {z^n\over n}$$ - [x] Series expansion for $\cosh(z)$ ✅ 2022-12-30 - [ ] $$\cosh x=1+\frac{x^{2}}{2 !}+\frac{x^{4}}{4 !}+\frac{x^{6}}{6 !}+\cdots=\sum_{n=0}^{\infty} \frac{x^{2 n}}{(2 n) !}.$$ - [x] Series expansion for $\sinh(z)$ ✅ 2022-12-30 - [ ] $$\sinh x=x+\frac{x^{3}}{3 !}+\frac{x^{5}}{5 !}+\frac{x^{7}}{7 !}+\cdots=\sum_{n=0}^{\infty} \frac{x^{2 n+1}}{(2 n+1) !}.$$ - [x] Series expansion for $\sech(z) = {1\over \cosh(z)}$ ✅ 2022-12-30 - $$\operatorname{sech} x=1-\frac{x^{2}}{2}+\frac{5 x^{4}}{24}-\frac{61 x^{6}}{720}+\cdots.$$ - [x] Series expansion for $\csch(z) = {1\over \sinh(z)}$ ✅ 2022-12-30 - $$\operatorname{csch} x=x^{-1}-\frac{x}{6}+\frac{7 x^{3}}{360}-\frac{31 x^{5}}{15120}+\cdots.$$ - [x] Inverting series: ${1\over \sin(z) } = \cdots$ ✅ 2022-12-30 - $${1\over \sin(z)} = \frac{1}{z}+\frac{1}{3 !} z+\frac{7}{360} z^{3}+\mathrm{O}\left(z^{5}\right).$$ - [x] Inverting series: for $A(z) = \sum a_k z^k$ and $1/A(z) = \sum b_k z^k$, the formula for the $b_k$ in terms of $a_k$. ✅ 2023-01-25 - $b_0 = a_0\inv$ - $b_1 = -a_0\inv(a_1 b_0 )$ - $b_2 = -a_0\inv(a_1b_1 + a_2b_0)$ - $b_3 = -a_0\inv(a_1b_2 + a_2b_1 + a_3 b_0)$ - $\cdots$ - $$b_n = -a_0\inv\sum_{k=1}^n a_n b_{n-k} = -a_0\inv\qty{a_1b_{n-1} + a_2b_{n-2} + \cdots + a_n b_0}$$ i.e. count up the $a$s and down the $b$s, starting at $b_{n-1}$ since $b_n$ is unknown. ## Set theoretic calculations - [x] $f^{-1} \left( \Union_i U_i \right) =_? \Union_i f^{-1}(U_i)$? ✅ 2022-12-30 - [ ] True. $$\begin{aligned} f^{-1}\left[\bigcup_{i \in I} Y_i\right] & =\left\{x \in X \mid f(x) \in \bigcup_{i \in I} Y_i\right\} \\ & =\left\{x \in \lambda \mid \quad \exists i \in I \text { such that } f(x) \in Y_i\right\} \\ & =\bigcup_{i \in I}\left\{x \in X \mid f(x) \in Y_i\right\} \\ & =\bigcup_{i \in I} f^{-1}\left[Y_i\right] \end{aligned}$$ - [x] $$f^{-1} \left( \bigcap U_i \right) =_? \bigcap_i f^{-1} U_i$$ ✅ 2022-12-30 - [ ] Yes - [x] True or false: $X =_? (f^{-1} \circ f)(X)$ ✅ 2022-12-30 - [ ] Only when $f$ is injective, otherwise just $\subseteq$. - [x] True or false: $X =_? (f\circ f^{-1})(X)$ ✅ 2022-12-30 - [ ] Only when $f$ is surjective, otherwise just $\subseteq$. - [x] True or false: $f^{-1}(A) \setminus f^{-1}(B) =_? f^{-1}(A\setminus B)$ ✅ 2022-12-30 - [ ] Yes. For forward direction, only an inclusion. - [x] What inverses exist for injective (resp. surjective) functions? ✅ 2022-12-30 - Injections: left inverses, i.e. $f(x) = f(y) \implies x=y$ - Surjections: right inverses - [ ] What is the continuum hypothesis? - [ ] What is $\aleph_0$? - [ ] What are the axioms for an equivalence relation? ## Calc I - [x] What is the limit definition of the exponential function? ✅ 2023-01-24 - [ ] $e^x = \lim_{n \to \infty} \qty{1 + {x\over n}}^n$ - [ ] What is the MVT? - [ ] $\del_x \csc(x) = \cdots$ - [ ] $\del_x \sec(x) =\cdots$ - [ ] $\del_x \tan(x) = \cdots$ - [ ] $\int \sec(x) \dx = \cdots$ - [ ] $\int \tan(x) \dx = \cdots$. - [ ] $\del_x \arcsin(x) = \cdots$ - [ ] $\del_x \arccos(x) = \cdots$ - [ ] $\del_x \arctan(x) = \cdots$ - [ ] $\del_x f\inv(x) = \cdots$ - [ ] $\int \log(x) \dx = \cdots$ ## Calc II - [x] $\sin ^2 \theta=\cdots$ ✅ 2022-12-30 - [ ] $\sin ^2 \theta=\frac{1-\cos (2 \theta)}{2}$ - [x] $\cos ^2 \theta=\cdots$ ✅ 2022-12-30 - [ ] $\cos ^2 \theta=\frac{1+\cos (2 \theta)}{2}$ - [x] What is the weird trig substitution that bizarrely works sometimes? ✅ 2022-12-30 - [ ] $\theta = 2\tan^{-1}(x)$. - [x] What are the allowed indeterminate forms for L'Hopital's rule? ✅ 2022-12-30 - ![Grad Math Facts-2](attachments/2022-12-30.png) ### Series Tests - [x] What is the divergence test? ✅ 2022-12-30 - [ ] $\sum c_n < \infty \implies c_n\convergesto{n\to\infty}0$. - [x] What is the $p\dash$test? ✅ 2022-12-30 - [ ] $\sum_{n\geq 0} 1/n^p < \infty \iff p >1$ and diverges otherwise. - [x] What is the comparison test? ✅ 2022-12-30 - [ ] If $a_n \leq b_n$ for $n \gg 0$, then $\sum b_n < \infty \implies \sum a_n < \infty$. - [ ] If $a_n\geq b_n$ for $n\gg 0$, then $\sum b_n \not<\infty \implies \sum a_n\not< \infty$. - [x] What is the limit comparison test? ✅ 2022-12-30 - [ ] Let $L\da \lim_{n\to\infty}{a_n\over b_n}$, then - [ ] $L\neq 0:\, \sum a_n < \infty \iff \sum b_n < \infty$. - [ ] $L=0:\, \sum b_n < \infty \implies \sum a_n < \infty$ - [ ] $L= \infty:\, \sum b_n \not < \infty \implies \sum a_n \not < \infty$. - [x] What is the integral test? ✅ 2022-12-30 - [ ] If $c_n = f(n)$, then $\sum_{n\geq 0} c_n < \infty \iff \int_{N}^\infty f(x)\dx < \infty$ for any $N$. - [x] What is the alternating series test? ✅ 2022-12-30 - [ ] If $c_{n+1}\leq c_n$ for all $n$ and $c_n\to 0$ then $\sum (-1)^n c_n < \infty$. - [x] What is the ratio test? ✅ 2022-12-30 - [ ] ![Grad Math Facts-3](attachments/2022-12-30-1.png) - [x] What is the root test? ✅ 2022-12-30 - [ ] ![Grad Math Facts-3](attachments/2022-12-30-2.png) - [ ] What is the volume of the region between $f(x), g(x) = x=a, x=b$ revolved around the $x\dash$axis? ## Calc III - [ ] What is the total derivative $D_f$ of a function $f$? - [ ] What is the Hessian $H_f$ of a function $f$? - [ ] What is $\iint_S f(x,y,z) \,\mathrm{dS}$? - [ ] What is $\int_\gamma F \,\mathrm{ds}$? - [ ] What is $\int_\gamma f \,\mathrm{ds}$? - [ ] How do you set up Lagrange multipliers? - [x] How is $\nabla \cross \vector F$ computed? ✅ 2022-12-30 - [ ] $$\nabla\cross F = \det \begin{bmatrix}\hat i & \hat j & \hat k \\ \del_x &\del_y &\del_z \\ F_1 & F_2 & F_3 \end{bmatrix}$$ - [x] How is $\nabla \cdot \vector F$ computed? ✅ 2022-12-30 - [ ] $$\nabla \cdot \vector F \da \nabla \cdot \tv{F_1, \cdots, F_n} = \sum \del_{x_i} F_i$$ - [x] What are the parametric and symmetric equations of lines in $\RR^3$? ✅ 2022-12-30 - [ ] Parameteric: $r(t) = t\vector p_0 + (1-t)\vector p_1$. - [ ] Symmetric: solve for $t$ in each component and set equal to get$$\frac{x-x_0}{n_0}=\frac{y-y_0}{n_1}=\frac{z-z_0}{n_2}$$ - [x] What is the vector/scalar equation of a plane? ✅ 2022-12-30 - [ ] For $\vector n$ a normal and $\vector p_0$ on the plane, $\inp{\vector x - \vector {p}_0}{\vector n} = 0$ - [x] What are the changes of variables for cartesian to spherical coordinates? ✅ 2022-12-30 - [ ] $$\begin{aligned} & x=r \sin \theta \cos \varphi \\ & y=r \sin \theta \sin \varphi \\ & z=r \cos \theta \end{aligned}$$ - [x] What is the differential for a change of rectangular to spherical coordinates? ✅ 2022-12-30 - [ ] $\dx\dy\dz \leadsto \rho^2 \sin(\phi)\, \drho \dphi 0\dtheta$ - [x] What is Green's theorem? ✅ 2022-12-30 - [ ] Circulation form: for a vector field $\vector F(x, y) = \tv{F_1(x, y), F_2(x, y) }$ on $\RR^2$, $$\begin{align*}\oint_{\bd R} \vector F(x,y)\cdot d\vector{r} &= \oint_{\bd R} \vector F(x,y)\cdot \vector T(x,y) \ds \\ &= \oint_{\bd R}F_1(x,y)\dx + F_2(x,y)\dy \\ &= \iint_R \qty{\del_x F_2(x,y) - \del_y F_1(x,y)} \dA \\ &=\iint_R (\nabla\cross F(x,y) )\dA \end{align*}$$ - [ ] Flux form: $$\oint_{\bd R} \vector F(x,y)\cdot \vector N(x,y)\ds = \iint_R \del_x F_1(x,y) + \del_y F_2(x,y)\dA = \iint_R (\nabla \cdot \vector F(x,y)) \dA$$ - [x] What is Stokes' theorem? ✅ 2022-12-30 - [ ] $$\int_{\bd S} \vector F \cdot d\vector r = \iint_S (\nabla\cross \vector F) \cdot d\vector{S} \da \iint_S (\nabla\cross \vector F) \cdot \vector N \, dS$$ - [x] What is Faraday's law? ✅ 2022-12-30 - [ ] $$\nabla \cross \vector E = -\dd{}{t} \vector B$$ - [x] What is the fundamental theorem of line integrals? ✅ 2022-12-30 - [ ] $$\int_C f\cdot d\vector r = F(p_1) - F(p_2) \iff \nabla F = f$$ - [x] What is the divergence theorem? ✅ 2022-12-30 - [ ] $$\iiint_M \nabla\cdot \vector F(x,y,z)\dV = \iint_{\bd M}\vector F(x,y,z) \cdot d\vector S \da \iint_{S} \vector F \cdot \vector N \, dS$$ - [x] What are the div-grad-curl theorems? ✅ 2022-12-30 - [ ] $\nabla \cdot(\nabla \cross \mathbf F) = 0$. - [ ] $\nabla\cross(\nabla \mathbf F) = 0$. - [ ] How does one find the area using Green's theorem? - [ ] What is Gauss' theorem? - [x] How do you solve a linear second order ODE? ✅ 2022-12-30 - [ ] ![Grad Math Facts-3](attachments/2022-12-30-3.png) ## Misc - [ ] What is Bayes' theorem? - [ ] What is Stirling's approximation? - [ ] What is the QR decomposition of a matrix? - [ ] What is the SVD? - [ ] How is the SVD related to the pseudoinverse? - [ ] What is the formula for $\proj_{\vector u}(\vector v)$? - [ ] Describe the separation axioms. - [ ] What is Urysohn's theorem? - [ ] What is the rank-nullity theorem? - [ ] What is the spectral theorem? - [ ] What is the surface area of a cone? - [ ] What is the volume of a cone? - [ ] What are the properties of a metric? - [ ] What is the Euler totient theorem? - [ ] What is the Moore-Penrose pseudoinverse? - [ ] Give several formulas for the totient. - [ ] What is a limit point? - [x] Define $\log(z)$ for $z\in \CC$. ✅ 2022-12-30 - [ ] $\log(z) = \log\left(re^{i\theta}\right) = \log(r) + i\theta = \log(|z|)+i\arg(z)$ - [x] What are the implications for fields, UFDs, PIDs, etc? ✅ 2022-12-30 - [ ] $$\begin{aligned}\text { field } & \Longrightarrow \text { Euclidean Domain } \Longrightarrow \text { PID } \\ & \Longrightarrow \text { UFD } \Longrightarrow \text { integral domain }\end{aligned}$$ - [ ] ED not a field: $\CC[x]$. - [ ] PID not an ED: $\ZZ\adjoin{ 1 + \sqrt{-19} \over 2}$. - [ ] UFD not a PID: $\ZZ[x]$; take $\gens{x, 2}$. - [ ] ID not a UFD: $\ZZ\adjoin{\sqrt{-5}}$.