Complete Mathematical Formulas Reference

Parentheses → Exponents → Multiplication/Division → Addition/Subtraction
\( (a + b) \times c = ac + bc \)

\( \text{Percentage} = \frac{\text{Part}}{\text{Whole}} \times 100\% \)
\( \text{Percentage Change} = \frac{\text{New - Old}}{\text{Old}} \times 100\% \)
\( \text{Compound Percentage: } A = P(1 + \frac{r}{100})^n \)

\( \frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd} \)
\( \frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd} \)
\( \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} \)

\( a^m \times a^n = a^{m+n} \)
\( \frac{a^m}{a^n} = a^{m-n} \)
\( (a^m)^n = a^{mn} \)
\( a^{-n} = \frac{1}{a^n} \)
\( a^{1/n} = \sqrt[n]{a} \)
\( a^0 = 1 \ (a \neq 0) \)

\( \sqrt[n]{ab} = \sqrt[n]{a} \times \sqrt[n]{b} \)
\( \sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}} \)
\( \sqrt[m]{\sqrt[n]{a}} = \sqrt[mn]{a} \)
\( a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m \)

\( \log_b(xy) = \log_b x + \log_b y \)
\( \log_b(\frac{x}{y}) = \log_b x - \log_b y \)
\( \log_b(x^n) = n \log_b x \)
\( \log_b a = \frac{\log_c a}{\log_c b} \) (Change of base)
\( b^{\log_b x} = x \)

\( a_n = a_1 + (n-1)d \)
\( S_n = \frac{n}{2}[2a_1 + (n-1)d] = \frac{n}{2}(a_1 + a_n) \)

\( a_n = a_1 r^{n-1} \)
\( S_n = \frac{a_1(1 - r^n)}{1 - r} \ (r \neq 1) \)
\( S_\infty = \frac{a_1}{1 - r} \ (|r| < 1) \)

\( \sum_{k=1}^n k = \frac{n(n+1)}{2} \)
\( \sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6} \)
\( \sum_{k=1}^n k^3 = \left[\frac{n(n+1)}{2}\right]^2 \)

\( ax^2 + bx + c = 0 \)
\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
Discriminant: \( \Delta = b^2 - 4ac \)

For \( ax^2 + bx + c = 0 \) with roots α, β:
\( \alpha + \beta = -\frac{b}{a} \)
\( \alpha\beta = \frac{c}{a} \)

\( ax^2 + bx + c = a\left(x + \frac{b}{2a}\right)^2 + \left(c - \frac{b^2}{4a}\right) \)

\( a^2 - b^2 = (a-b)(a+b) \)
\( a^2 + 2ab + b^2 = (a+b)^2 \)
\( a^2 - 2ab + b^2 = (a-b)^2 \)
\( a^3 + b^3 = (a+b)(a^2 - ab + b^2) \)
\( a^3 - b^3 = (a-b)(a^2 + ab + b^2) \)

\( a^4 + b^4 = (a^2 + \sqrt{2}ab + b^2)(a^2 - \sqrt{2}ab + b^2) \)
\( a^n - b^n = (a-b)(a^{n-1} + a^{n-2}b + \dots + b^{n-1}) \)
\( a^n + b^n = (a+b)(a^{n-1} - a^{n-2}b + \dots - ab^{n-2} + b^{n-1}) \) (n odd)

\( (a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \)
\( \binom{n}{k} = \frac{n!}{k!(n-k)!} \)

For \( P(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_0 \):
Sum of roots: \( -\frac{a_{n-1}}{a_n} \)
Product of roots: \( (-1)^n \frac{a_0}{a_n} \)

For dividing by \( (x-c) \):
Bring down leading coefficient, multiply by c, add to next coefficient

For \( ax + by = e \), \( cx + dy = f \):
\( x = \frac{\begin{vmatrix} e & b \\ f & d \end{vmatrix}}{\begin{vmatrix} a & b \\ c & d \end{vmatrix}} \),
\( y = \frac{\begin{vmatrix} a & e \\ c & f \end{vmatrix}}{\begin{vmatrix} a & b \\ c & d \end{vmatrix}} \)

\( a^2 + b^2 = c^2 \)
Generalized: \( \cos C = \frac{a^2 + b^2 - c^2}{2ab} \)

Triangle: \( A = \frac{1}{2}bh = \frac{1}{2}ab\sin C \)
Rectangle: \( A = lw \)
Circle: \( A = \pi r^2 \)
Trapezoid: \( A = \frac{1}{2}(a+b)h \)
Parallelogram: \( A = bh \)

Circle: \( C = 2\pi r = \pi d \)
Rectangle: \( P = 2(l + w) \)
Triangle: \( P = a + b + c \)

Cube: \( V = s^3 \)
Rectangular Prism: \( V = lwh \)
Sphere: \( V = \frac{4}{3}\pi r^3 \)
Cylinder: \( V = \pi r^2 h \)
Cone: \( V = \frac{1}{3}\pi r^2 h \)
Pyramid: \( V = \frac{1}{3}Bh \)

Sphere: \( SA = 4\pi r^2 \)
Cylinder: \( SA = 2\pi r(h + r) \)
Cone: \( SA = \pi r(r + \sqrt{h^2 + r^2}) \)
Cube: \( SA = 6s^2 \)

2D: \( d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \)
3D: \( d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2} \)
Point to line: \( d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \)

Slope-intercept: \( y = mx + b \)
Point-slope: \( y - y_1 = m(x - x_1) \)
Two-point: \( \frac{y - y_1}{y_2 - y_1} = \frac{x - x_1}{x_2 - x_1} \)
General: \( Ax + By + C = 0 \)

Standard: \( (x-h)^2 + (y-k)^2 = r^2 \)
General: \( x^2 + y^2 + Dx + Ey + F = 0 \)
Parametric: \( x = h + r\cos\theta, y = k + r\sin\theta \)

Law of Sines: \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \)
Law of Cosines: \( c^2 = a^2 + b^2 - 2ab\cos C \)

\( A = \frac{1}{2}ab\sin C \)
Heron's: \( A = \sqrt{s(s-a)(s-b)(s-c)} \), \( s = \frac{a+b+c}{2} \)

\( \frac{d}{dx}(c) = 0 \)
\( \frac{d}{dx}(x^n) = nx^{n-1} \)
\( \frac{d}{dx}(e^x) = e^x \)
\( \frac{d}{dx}(a^x) = a^x \ln a \)
\( \frac{d}{dx}(\ln x) = \frac{1}{x} \)
\( \frac{d}{dx}(\log_a x) = \frac{1}{x \ln a} \)

\( \frac{d}{dx}(\sin x) = \cos x \)
\( \frac{d}{dx}(\cos x) = -\sin x \)
\( \frac{d}{dx}(\tan x) = \sec^2 x \)
\( \frac{d}{dx}(\cot x) = -\csc^2 x \)
\( \frac{d}{dx}(\sec x) = \sec x \tan x \)
\( \frac{d}{dx}(\csc x) = -\csc x \cot x \)

\( \frac{d}{dx}(\sin^{-1} x) = \frac{1}{\sqrt{1-x^2}} \)
\( \frac{d}{dx}(\cos^{-1} x) = -\frac{1}{\sqrt{1-x^2}} \)
\( \frac{d}{dx}(\tan^{-1} x) = \frac{1}{1+x^2} \)

Product: \( (fg)' = f'g + fg' \)
Quotient: \( \left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2} \)
Chain: \( (f(g(x)))' = f'(g(x)) \cdot g'(x) \)
Implicit: \( \frac{dy}{dx} = -\frac{F_x}{F_y} \)

\( \int x^n dx = \frac{x^{n+1}}{n+1} + C \ (n \neq -1) \)
\( \int \frac{1}{x} dx = \ln|x| + C \)
\( \int e^x dx = e^x + C \)
\( \int a^x dx = \frac{a^x}{\ln a} + C \)

\( \int \sin x dx = -\cos x + C \)
\( \int \cos x dx = \sin x + C \)
\( \int \sec^2 x dx = \tan x + C \)
\( \int \csc^2 x dx = -\cot x + C \)

Substitution: \( \int f(g(x))g'(x)dx = \int f(u)du \)
Parts: \( \int u dv = uv - \int v du \)
Partial Fractions: \( \int \frac{P(x)}{Q(x)}dx \)
Trig Sub: \( x = a\sin\theta, a\tan\theta, a\sec\theta \)

\( \int_a^b f(x)dx = F(b) - F(a) \)
\( \int_a^b f(x)dx = -\int_b^a f(x)dx \)
\( \int_a^b [f(x) \pm g(x)]dx = \int_a^b f(x)dx \pm \int_a^b g(x)dx \)

\( f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(x-a)^n \)
Maclaurin (a=0): \( f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!}x^n \)

\( e^x = \sum_{n=0}^\infty \frac{x^n}{n!} \)
\( \sin x = \sum_{n=0}^\infty \frac{(-1)^n x^{2n+1}}{(2n+1)!} \)
\( \cos x = \sum_{n=0}^\infty \frac{(-1)^n x^{2n}}{(2n)!} \)
\( \ln(1+x) = \sum_{n=1}^\infty \frac{(-1)^{n+1} x^n}{n} \)

\( f_x = \frac{\partial f}{\partial x} \)
\( f_{xy} = \frac{\partial^2 f}{\partial y \partial x} \)
Chain: \( \frac{dz}{dt} = \frac{\partial z}{\partial x}\frac{dx}{dt} + \frac{\partial z}{\partial y}\frac{dy}{dt} \)

\( \nabla f = \left\langle \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right\rangle \)
\( D_{\vec{u}} f = \nabla f \cdot \vec{u} \)

Double: \( \iint_R f(x,y) dA \)
Triple: \( \iiint_E f(x,y,z) dV \)
Polar: \( \iint f(r,\theta) r dr d\theta \)

Mean: \( \mu = \frac{\sum x_i}{N} \) (population)
Sample Mean: \( \bar{x} = \frac{\sum x_i}{n} \)
Median: middle value when sorted
Mode: most frequent value

Variance: \( \sigma^2 = \frac{\sum (x_i - \mu)^2}{N} \)
Sample Variance: \( s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1} \)
Standard Deviation: \( \sigma = \sqrt{\sigma^2} \)
Range: \( \max - \min \)

Q1: 25th percentile
Q2: 50th percentile (median)
Q3: 75th percentile
IQR: \( Q3 - Q1 \)

Binomial: \( P(X=k) = \binom{n}{k} p^k (1-p)^{n-k} \)
Poisson: \( P(X=k) = \frac{\lambda^k e^{-\lambda}}{k!} \)
Geometric: \( P(X=k) = (1-p)^{k-1}p \)

Normal: \( f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}} \)
Exponential: \( f(x) = \lambda e^{-\lambda x} \)
Uniform: \( f(x) = \frac{1}{b-a} \)

Mean CI: \( \bar{x} \pm z_{\alpha/2} \frac{\sigma}{\sqrt{n}} \)
Proportion CI: \( \hat{p} \pm z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \)
t-interval: \( \bar{x} \pm t_{\alpha/2} \frac{s}{\sqrt{n}} \)

z-test: \( z = \frac{\bar{x} - \mu}{\sigma/\sqrt{n}} \)
t-test: \( t = \frac{\bar{x} - \mu}{s/\sqrt{n}} \)
Chi-square: \( \chi^2 = \sum \frac{(O-E)^2}{E} \)

\( y = \beta_0 + \beta_1 x + \epsilon \)
\( \hat{\beta}_1 = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \)
\( \hat{\beta}_0 = \bar{y} - \hat{\beta}_1 \bar{x} \)

Pearson's r: \( r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}} \)
R²: coefficient of determination

\( \sin^2\theta + \cos^2\theta = 1 \)
\( 1 + \tan^2\theta = \sec^2\theta \)
\( 1 + \cot^2\theta = \csc^2\theta \)

\( \sin(A \pm B) = \sin A \cos B \pm \cos A \sin B \)
\( \cos(A \pm B) = \cos A \cos B \mp \sin A \sin B \)
\( \tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B} \)

\( \sin 2\theta = 2\sin\theta\cos\theta \)
\( \cos 2\theta = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta \)
\( \tan 2\theta = \frac{2\tan\theta}{1 - \tan^2\theta} \)
\( \sin\frac{\theta}{2} = \pm\sqrt{\frac{1 - \cos\theta}{2}} \)
\( \cos\frac{\theta}{2} = \pm\sqrt{\frac{1 + \cos\theta}{2}} \)

\( \sin A \cos B = \frac{1}{2}[\sin(A+B) + \sin(A-B)] \)
\( \cos A \cos B = \frac{1}{2}[\cos(A+B) + \cos(A-B)] \)
\( \sin A \sin B = \frac{1}{2}[\cos(A-B) - \cos(A+B)] \)

\( \sin A + \sin B = 2\sin\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right) \)
\( \sin A - \sin B = 2\cos\left(\frac{A+B}{2}\right)\sin\left(\frac{A-B}{2}\right) \)
\( \cos A + \cos B = 2\cos\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right) \)
\( \cos A - \cos B = -2\sin\left(\frac{A+B}{2}\right)\sin\left(\frac{A-B}{2}\right) \)

\( \sin\theta = \sin\alpha \Rightarrow \theta = n\pi + (-1)^n\alpha \)
\( \cos\theta = \cos\alpha \Rightarrow \theta = 2n\pi \pm \alpha \)
\( \tan\theta = \tan\alpha \Rightarrow \theta = n\pi + \alpha \)

\( \sin^{-1}x \in [-\frac{\pi}{2}, \frac{\pi}{2}] \)
\( \cos^{-1}x \in [0, \pi] \)
\( \tan^{-1}x \in (-\frac{\pi}{2}, \frac{\pi}{2}) \)

\( \sin^{-1}x + \cos^{-1}x = \frac{\pi}{2} \)
\( \tan^{-1}x + \cot^{-1}x = \frac{\pi}{2} \)
\( \sec^{-1}x + \csc^{-1}x = \frac{\pi}{2} \)

Divisible by 2: last digit even
Divisible by 3: sum of digits divisible by 3
Divisible by 4: last two digits divisible by 4
Divisible by 5: ends with 0 or 5
Divisible by 6: divisible by 2 and 3
Divisible by 9: sum of digits divisible by 9

Prime Counting: \( \pi(x) \sim \frac{x}{\ln x} \)
Prime Factorization: \( n = p_1^{a_1} p_2^{a_2} \dots p_k^{a_k} \)
Number of divisors: \( d(n) = (a_1+1)(a_2+1)\dots(a_k+1) \)

\( a \equiv b \pmod{m} \iff m \mid (a-b) \)
\( (a+b) \mod m = ((a \mod m) + (b \mod m)) \mod m \)
\( (ab) \mod m = ((a \mod m)(b \mod m)) \mod m \)

Fermat: \( a^{p-1} \equiv 1 \pmod{p} \) (p prime, a not divisible by p)
Euler: \( a^{\phi(n)} \equiv 1 \pmod{n} \) (gcd(a,n)=1)
\( \phi(n) = n \prod_{p|n}(1 - \frac{1}{p}) \)

\( \gcd(a,b) = \gcd(b, a \mod b) \)
Extended: \( ax + by = \gcd(a,b) \)
\( \text{lcm}(a,b) = \frac{ab}{\gcd(a,b)} \)

\( (AB)_{ij} = \sum_{k=1}^n A_{ik}B_{kj} \)
\( A(BC) = (AB)C \)
\( A(B+C) = AB + AC \)

2×2: \( \begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc \)
3×3: Sarrus' rule or cofactor expansion
\( \det(AB) = \det(A)\det(B) \)
\( \det(A^{-1}) = \frac{1}{\det(A)} \)

2×2: \( A^{-1} = \frac{1}{ad-bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \)
\( AA^{-1} = A^{-1}A = I \)
\( (AB)^{-1} = B^{-1}A^{-1} \)

Dot product: \( \vec{a} \cdot \vec{b} = \sum a_i b_i = |a||b|\cos\theta \)
Cross product (3D): \( \vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix} \)
Magnitude: \( |\vec{a}| = \sqrt{\sum a_i^2} \)

Vectors linearly independent if \( c_1\vec{v}_1 + \dots + c_n\vec{v}_n = 0 \) implies all \( c_i = 0 \)
Rank: dimension of column space
Nullity: dimension of null space

\( \det(A - \lambda I) = 0 \)
For 2×2: \( \lambda^2 - \text{tr}(A)\lambda + \det(A) = 0 \)
\( A\vec{v} = \lambda\vec{v} \)

\( A = PDP^{-1} \) where D diagonal, P eigenvectors
\( A^n = PD^nP^{-1} \)

\( \frac{dy}{dx} = f(x)g(y) \)
Solution: \( \int \frac{dy}{g(y)} = \int f(x)dx + C \)

\( \frac{dy}{dx} + P(x)y = Q(x) \)
Integrating factor: \( \mu(x) = e^{\int P(x)dx} \)
Solution: \( y = \frac{1}{\mu(x)}\int \mu(x)Q(x)dx + \frac{C}{\mu(x)} \)

\( M(x,y)dx + N(x,y)dy = 0 \) exact if \( \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} \)
Solution: \( \int M dx + \int (N - \frac{\partial}{\partial y}\int M dx)dy = C \)

\( ay'' + by' + cy = 0 \)
Characteristic: \( ar^2 + br + c = 0 \)
Real distinct roots: \( y = C_1e^{r_1x} + C_2e^{r_2x} \)
Real repeated: \( y = (C_1 + C_2x)e^{rx} \)
Complex: \( y = e^{\alpha x}(C_1\cos\beta x + C_2\sin\beta x) \)

\( ay'' + by' + cy = f(x) \)
General solution: \( y = y_h + y_p \)
Method of undetermined coefficients
Variation of parameters

\( ax^2y'' + bxy' + cy = 0 \)
Assume \( y = x^r \), solve \( ar(r-1) + br + c = 0 \)

\( y' + P(x)y = Q(x)y^n \)
Substitute \( v = y^{1-n} \)

\( P(A \cup B) = P(A) + P(B) - P(A \cap B) \)
\( P(A^c) = 1 - P(A) \)
\( P(A|B) = \frac{P(A \cap B)}{P(B)} \)
\( P(A \cap B) = P(A)P(B|A) \)

\( P(A|B) = \frac{P(B|A)P(A)}{P(B)} \)
\( P(B) = P(B|A)P(A) + P(B|A^c)P(A^c) \)

\( A \) and \( B \) independent if \( P(A \cap B) = P(A)P(B) \)
\( P(A|B) = P(A) \) if independent

\( E[X] = \sum x_i p_i \) (discrete)
\( E[X] = \int_{-\infty}^{\infty} x f(x) dx \) (continuous)
\( \text{Var}(X) = E[(X-\mu)^2] = E[X^2] - (E[X])^2 \)
\( \text{SD}(X) = \sqrt{\text{Var}(X)} \)

\( M_X(t) = E[e^{tX}] \)
\( E[X^n] = M_X^{(n)}(0) \)

Weak: \( \frac{1}{n}\sum_{i=1}^n X_i \xrightarrow{P} \mu \)
Strong: \( \frac{1}{n}\sum_{i=1}^n X_i \xrightarrow{a.s.} \mu \)

\( \frac{\bar{X} - \mu}{\sigma/\sqrt{n}} \xrightarrow{d} N(0,1) \)

Rectangular: \( z = x + iy \)
Polar: \( z = r(\cos\theta + i\sin\theta) = re^{i\theta} \)
Multiplication: \( r_1e^{i\theta_1} \cdot r_2e^{i\theta_2} = r_1r_2e^{i(\theta_1+\theta_2)} \)
De Moivre: \( (re^{i\theta})^n = r^ne^{in\theta} \)

\( e^{i\theta} = \cos\theta + i\sin\theta \)
\( \cos\theta = \frac{e^{i\theta} + e^{-i\theta}}{2} \)
\( \sin\theta = \frac{e^{i\theta} - e^{-i\theta}}{2i} \)

\( f(z) = u(x,y) + iv(x,y) \) analytic if:
\( \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \), \( \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \)

Cauchy's Theorem: \( \oint_\gamma f(z)dz = 0 \) if f analytic inside γ
Cauchy's Formula: \( f(a) = \frac{1}{2\pi i} \oint_\gamma \frac{f(z)}{z-a}dz \)
\( f^{(n)}(a) = \frac{n!}{2\pi i} \oint_\gamma \frac{f(z)}{(z-a)^{n+1}}dz \)

Simple pole: \( \text{Res}(f, a) = \lim_{z\to a} (z-a)f(z) \)
Pole order m: \( \text{Res}(f, a) = \frac{1}{(m-1)!} \lim_{z\to a} \frac{d^{m-1}}{dz^{m-1}}[(z-a)^m f(z)] \)
Residue Theorem: \( \oint_\gamma f(z)dz = 2\pi i \sum \text{Res}(f, a_k) \)

Permutations: \( P(n,r) = \frac{n!}{(n-r)!} \)
Combinations: \( C(n,r) = \binom{n}{r} = \frac{n!}{r!(n-r)!} \)
With repetition: \( \binom{n+r-1}{r} \)
Binomial: \( (x+y)^n = \sum_{k=0}^n \binom{n}{k} x^{n-k}y^k \)

\( |A \cup B| = |A| + |B| - |A \cap B| \)
\( |A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C| \)

Handshaking: \( \sum_{v \in V} \deg(v) = 2|E| \)
Complete graph \( K_n \): \( \frac{n(n-1)}{2} \) edges
Tree with n vertices: n-1 edges
Planar: \( |V| - |E| + |F| = 2 \) (Euler)

Hamiltonian path: visits each vertex once
Eulerian path: visits each edge once
Dijkstra's algorithm for shortest path

De Morgan: \( \neg(p \land q) \equiv \neg p \lor \neg q \)
\( \neg(p \lor q) \equiv \neg p \land \neg q \)
Implication: \( p \rightarrow q \equiv \neg p \lor q \)
Contrapositive: \( p \rightarrow q \equiv \neg q \rightarrow \neg p \)

Direct proof
Proof by contradiction
Proof by induction: Base case + inductive step
Proof by contrapositive

Gradient: \( \nabla f = \left\langle \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right\rangle \)
Divergence: \( \nabla \cdot \vec{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} \)
Curl: \( \nabla \times \vec{F} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ F_x & F_y & F_z \end{vmatrix} \)

Divergence Theorem: \( \iiint_V (\nabla \cdot \vec{F}) dV = \oiint_S \vec{F} \cdot d\vec{S} \)
Stokes' Theorem: \( \iint_S (\nabla \times \vec{F}) \cdot d\vec{S} = \oint_C \vec{F} \cdot d\vec{r} \)
Green's Theorem: \( \oint_C (P dx + Q dy) = \iint_D \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) dA \)

\( f(x) = \frac{a_0}{2} + \sum_{n=1}^\infty \left[a_n \cos\left(\frac{2\pi nx}{T}\right) + b_n \sin\left(\frac{2\pi nx}{T}\right)\right] \)
\( a_n = \frac{2}{T} \int_0^T f(x) \cos\left(\frac{2\pi nx}{T}\right) dx \)
\( b_n = \frac{2}{T} \int_0^T f(x) \sin\left(\frac{2\pi nx}{T}\right) dx \)

\( \hat{f}(\xi) = \int_{-\infty}^\infty f(x) e^{-2\pi i x \xi} dx \)
Inverse: \( f(x) = \int_{-\infty}^\infty \hat{f}(\xi) e^{2\pi i x \xi} d\xi \)
Convolution: \( (f * g)(x) = \int_{-\infty}^\infty f(y)g(x-y) dy \)

Bisection: \( x_{n+1} = \frac{a_n + b_n}{2} \)
Newton-Raphson: \( x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \)
Secant: \( x_{n+1} = x_n - f(x_n)\frac{x_n - x_{n-1}}{f(x_n) - f(x_{n-1})} \)

\( x_{n+1} = g(x_n) \)
Convergence if \( |g'(x)| < 1 \) near root

Trapezoidal: \( \int_a^b f(x)dx \approx \frac{b-a}{2}[f(a) + f(b)] \)
Simpson's 1/3: \( \int_a^b f(x)dx \approx \frac{b-a}{6}[f(a) + 4f(\frac{a+b}{2}) + f(b)] \)
Simpson's 3/8: \( \int_a^b f(x)dx \approx \frac{b-a}{8}[f(a) + 3f(\frac{2a+b}{3}) + 3f(\frac{a+2b}{3}) + f(b)] \)

Composite Trapezoidal: \( \frac{h}{2}[f(x_0) + 2\sum_{i=1}^{n-1} f(x_i) + f(x_n)] \)
Composite Simpson: \( \frac{h}{3}[f(x_0) + 4\sum_{i=1,3,5}^{n-1} f(x_i) + 2\sum_{i=2,4,6}^{n-2} f(x_i) + f(x_n)] \)

Forward: \( f'(x) \approx \frac{f(x+h) - f(x)}{h} \)
Backward: \( f'(x) \approx \frac{f(x) - f(x-h)}{h} \)
Central: \( f'(x) \approx \frac{f(x+h) - f(x-h)}{2h} \)

\( f''(x) \approx \frac{f(x+h) - 2f(x) + f(x-h)}{h^2} \)
\( f^{(4)}(x) \approx \frac{f(x+2h) - 4f(x+h) + 6f(x) - 4f(x-h) + f(x-2h)}{h^4} \)

Wave: \( u_{tt} = c^2 u_{xx} \) (Hyperbolic)
Heat: \( u_t = \alpha u_{xx} \) (Parabolic)
Laplace: \( u_{xx} + u_{yy} = 0 \) (Elliptic)
General: \( A u_{xx} + B u_{xy} + C u_{yy} + \dots = 0 \)

Assume \( u(x,t) = X(x)T(t) \)
Substitute into PDE and separate variables
Solve resulting ODEs with boundary conditions

Fundamental: \( u(x,t) = \frac{1}{\sqrt{4\pi\alpha t}} e^{-x^2/(4\alpha t)} \)
Fourier series solution for bounded domains
Error function solutions for semi-infinite domains

d'Alembert: \( u(x,t) = f(x-ct) + g(x+ct) \)
Fourier series for vibrating string
Spherical waves in 3D

Closure: \( a,b \in G \Rightarrow ab \in G \)
Associativity: \( (ab)c = a(bc) \)
Identity: \( \exists e \in G: ea = ae = a \)
Inverse: \( \forall a \in G, \exists a^{-1}: aa^{-1} = a^{-1}a = e \)

Lagrange: \( |H| \) divides \( |G| \) for subgroup H
Cayley: Every group isomorphic to permutation group
Sylow: Existence of p-subgroups

Additive group: \( (R,+) \) abelian group
Multiplication: associative, distributive
Commutative ring: multiplication commutative
Field: commutative ring with multiplicative inverses

Ideal: subgroup closed under multiplication by ring elements
Principal ideal: \( (a) = \{ra : r \in R\} \)
Maximal ideal: not contained in any proper ideal
Prime ideal: \( ab \in P \Rightarrow a \in P \) or \( b \in P \)

Degree: \( [F(\alpha):F] = \deg(\text{minimal polynomial}) \)
Finite extension: \( [K:F] < \infty \)
Algebraic extension: every element algebraic over F
Transcendental extension: contains transcendental element

Galois group: \( \text{Gal}(K/F) = \{\sigma: K \to K \mid \sigma|_F = id\} \)
Fundamental theorem: Correspondence between subgroups and intermediate fields
Solvability by radicals related to solvable Galois group

Collection of open sets satisfying:
1. ∅ and X are open
2. Union of open sets is open
3. Finite intersection of open sets is open

f continuous if \( f^{-1}(U) \) open for all open U
Homeomorphism: bijective continuous map with continuous inverse
Topological invariant: property preserved by homeomorphism

T2: ∀ distinct points x,y, ∃ disjoint open sets U,V with x∈U, y∈V
Regular: T1 + points and closed sets separable
Normal: T1 + disjoint closed sets separable
Completely regular: T1 + points and closed sets separated by continuous functions

Every open cover has finite subcover
Continuous image of compact is compact
Closed subset of compact is compact
Heine-Borel: In ℝⁿ, compact = closed and bounded

Connected: cannot be partitioned into two nonempty disjoint open sets
Path-connected: any two points connected by continuous path
Components: maximal connected subsets
Continuous image of connected is connected

Lagrangian: \( L = T - V \)
Euler-Lagrange: \( \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_i}\right) - \frac{\partial L}{\partial q_i} = 0 \)
Action: \( S = \int_{t_1}^{t_2} L dt \)

Hamiltonian: \( H = \sum p_i \dot{q}_i - L \)
Hamilton's equations: \( \dot{q}_i = \frac{\partial H}{\partial p_i}, \dot{p}_i = -\frac{\partial H}{\partial q_i} \)
Poisson bracket: \( \{f,g\} = \sum \left(\frac{\partial f}{\partial q_i}\frac{\partial g}{\partial p_i} - \frac{\partial f}{\partial p_i}\frac{\partial g}{\partial q_i}\right) \)

Time-dependent: \( i\hbar\frac{\partial \psi}{\partial t} = \hat{H}\psi \)
Time-independent: \( \hat{H}\psi = E\psi \)
Hamiltonian: \( \hat{H} = -\frac{\hbar^2}{2m}\nabla^2 + V(\vec{r}) \)

Position: \( \hat{x}\psi = x\psi \)
Momentum: \( \hat{p} = -i\hbar\frac{\partial}{\partial x} \)
Commutator: \( [\hat{A},\hat{B}] = \hat{A}\hat{B} - \hat{B}\hat{A} \)
Uncertainty: \( \sigma_A \sigma_B \geq \frac{1}{2}|\langle[\hat{A},\hat{B}]\rangle| \)

Gauss: \( \nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0} \)
Gauss (mag): \( \nabla \cdot \vec{B} = 0 \)
Faraday: \( \nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t} \)
Ampère-Maxwell: \( \nabla \times \vec{B} = \mu_0\vec{J} + \mu_0\epsilon_0\frac{\partial \vec{E}}{\partial t} \)

Wave equation: \( \nabla^2 \vec{E} = \mu_0\epsilon_0\frac{\partial^2 \vec{E}}{\partial t^2} \)
Speed of light: \( c = \frac{1}{\sqrt{\mu_0\epsilon_0}} \)
Poynting vector: \( \vec{S} = \frac{1}{\mu_0}\vec{E} \times \vec{B} \)

Future value: \( FV = PV(1 + r)^n \)
Present value: \( PV = \frac{FV}{(1 + r)^n} \)
Continuous compounding: \( FV = PVe^{rt} \)
Effective annual rate: \( \left(1 + \frac{r}{m}\right)^m - 1 \)

Ordinary annuity: \( PV = PMT\frac{1 - (1+r)^{-n}}{r} \)
Annuity due: \( PV = PMT\frac{1 - (1+r)^{-n}}{r}(1+r) \)
Perpetuity: \( PV = \frac{PMT}{r} \)

Call option: \( C = S_0N(d_1) - Ke^{-rT}N(d_2) \)
Put option: \( P = Ke^{-rT}N(-d_2) - S_0N(-d_1) \)
\( d_1 = \frac{\ln(S_0/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}} \)
\( d_2 = d_1 - \sigma\sqrt{T} \)

Delta: \( \Delta = \frac{\partial C}{\partial S} \)
Gamma: \( \Gamma = \frac{\partial^2 C}{\partial S^2} \)
Theta: \( \Theta = \frac{\partial C}{\partial t} \)
Vega: \( \nu = \frac{\partial C}{\partial \sigma} \)
Rho: \( \rho = \frac{\partial C}{\partial r} \)

Expected return: \( E[R_p] = \sum w_i E[R_i] \)
Portfolio variance: \( \sigma_p^2 = \sum\sum w_i w_j \sigma_{ij} \)
Sharpe ratio: \( \frac{E[R_p] - R_f}{\sigma_p} \)
Capital Market Line: \( E[R] = R_f + \frac{E[R_m] - R_f}{\sigma_m} \sigma \)

\( E[R_i] = R_f + \beta_i(E[R_m] - R_f) \)
\( \beta_i = \frac{\text{Cov}(R_i, R_m)}{\text{Var}(R_m)} \)
Security Market Line

Maximize \( c^T x \)
Subject to \( Ax \leq b \)
\( x \geq 0 \)
Dual problem: Minimize \( b^T y \) subject to \( A^T y \geq c, y \geq 0 \)

Basic feasible solution
Pivot operations
Reduced costs: \( c_j - c_B^T B^{-1} A_j \)
Optimality condition: all reduced costs ≤ 0

Arrival rate: λ, Service rate: μ
Utilization: \( \rho = \frac{\lambda}{\mu} \)
Average number in system: \( L = \frac{\rho}{1-\rho} \)
Average wait time: \( W = \frac{1}{\mu-\lambda} \)

\( L = \lambda W \)
\( L_q = \lambda W_q \)
Applies to any stable queueing system

Dijkstra's algorithm
Bellman-Ford algorithm
Floyd-Warshall algorithm
A* search algorithm

Ford-Fulkerson algorithm
Max-flow min-cut theorem
Residual networks
Augmenting paths

Survival probability: \( _tp_x = \frac{S(x+t)}{S(x)} \)
Force of mortality: \( \mu_x = -\frac{d}{dx}\ln S(x) \)
Life expectancy: \( \overset{\circ}{e}_x = \int_0^\infty {}_tp_x dt \)

Whole life: \( A_x = \sum_{k=0}^\infty v^{k+1} {}_kp_x q_{x+k} \)
Term insurance: \( A^1_{x:\angl{n}} = \sum_{k=0}^{n-1} v^{k+1} {}_kp_x q_{x+k} \)
Endowment: \( A_{x:\angl{n}} = A^1_{x:\angl{n}} + v^n {}_np_x \)

Whole life annuity: \( \ddot{a}_x = \sum_{k=0}^\infty v^k {}_kp_x \)
Temporary annuity: \( \ddot{a}_{x:\angl{n}} = \sum_{k=0}^{n-1} v^k {}_kp_x \)
Deferred annuity: \( _{m|\ddot{a}_x} = v^m {}_mp_x \ddot{a}_{x+m} \)

Equivalence principle: \( E[PV benefits] = E[PV premiums] \)
Whole life: \( P_x = \frac{A_x}{\ddot{a}_x} \)
n-year term: \( P^1_{x:\angl{n}} = \frac{A^1_{x:\angl{n}}}{\ddot{a}_{x:\angl{n}}} \)

Key generation: Choose primes p,q, n = pq, φ(n) = (p-1)(q-1)
Public key: (n,e) where gcd(e,φ(n)) = 1
Private key: d where ed ≡ 1 mod φ(n)
Encryption: c ≡ m^e mod n
Decryption: m ≡ c^d mod n

Public parameters: prime p, generator g
Alice: A = g^a mod p
Bob: B = g^b mod p
Shared secret: K = B^a mod p = A^b mod p = g^{ab} mod p

Curve: y² = x³ + ax + b mod p
Point addition: P + Q = R
Point doubling: P + P = 2P
Scalar multiplication: kP = P + P + ... + P (k times)

ECDH: Similar to Diffie-Hellman using elliptic curves
ECDSA: Digital signature algorithm using elliptic curves
Smaller key sizes with equivalent security to RSA

Strategy profile where no player can improve payoff by unilaterally changing strategy
Pure strategy Nash equilibrium
Mixed strategy Nash equilibrium
Existence guaranteed by Nash's theorem

Payoff matrix analysis
Dominant strategy equilibrium
Pareto efficiency vs. Nash equilibrium
Iterated prisoner's dilemma

Solve game tree from terminal nodes backward
Subgame perfect equilibrium
Sequential rationality
Perfect information games

Games with incomplete information
Type spaces and beliefs
Bayesian Nash equilibrium
Signaling games

\( H(X) = -\sum_{i=1}^n p(x_i) \log_2 p(x_i) \)
Joint entropy: \( H(X,Y) = -\sum_{x,y} p(x,y) \log_2 p(x,y) \)
Conditional entropy: \( H(Y|X) = -\sum_{x,y} p(x,y) \log_2 p(y|x) \)

\( I(X;Y) = H(X) - H(X|Y) = H(Y) - H(Y|X) \)
\( I(X;Y) = \sum_{x,y} p(x,y) \log_2 \frac{p(x,y)}{p(x)p(y)} \)
Measures shared information between X and Y

Channel capacity: \( C = \max_{p(x)} I(X;Y) \)
Shannon's theorem: Rates below C are achievable
Noisy channel coding theorem
Error-correcting codes

Binary symmetric channel: \( C = 1 - H(p) \)
Gaussian channel: \( C = \frac{1}{2} \log_2(1 + \text{SNR}) \)
Bandlimited channel: \( C = B \log_2(1 + \frac{S}{N_0B}) \)

\( \Gamma(z) = \int_0^\infty t^{z-1} e^{-t} dt \)
\( \Gamma(n+1) = n! \) for integer n
\( \Gamma(z+1) = z\Gamma(z) \)
\( \Gamma(\frac{1}{2}) = \sqrt{\pi} \)

\( B(x,y) = \int_0^1 t^{x-1}(1-t)^{y-1} dt \)
\( B(x,y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)} \)
Symmetry: \( B(x,y) = B(y,x) \)

\( x^2 y'' + x y' + (x^2 - \nu^2)y = 0 \)
First kind: \( J_\nu(x) = \sum_{m=0}^\infty \frac{(-1)^m}{m!\Gamma(m+\nu+1)}\left(\frac{x}{2}\right)^{2m+\nu} \)
Second kind: \( Y_\nu(x) \) (Neumann function)

Recurrence: \( J_{\nu-1}(x) + J_{\nu+1}(x) = \frac{2\nu}{x} J_\nu(x) \)
Derivatives: \( \frac{d}{dx}[x^\nu J_\nu(x)] = x^\nu J_{\nu-1}(x) \)
Orthogonality: \( \int_0^1 x J_\nu(\alpha x) J_\nu(\beta x) dx = 0 \) for α≠β

Rodrigues: \( P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n}[(x^2-1)^n] \)
Generating: \( \frac{1}{\sqrt{1-2xt+t^2}} = \sum_{n=0}^\infty P_n(x) t^n \)
Orthogonality: \( \int_{-1}^1 P_m(x) P_n(x) dx = \frac{2}{2n+1} \delta_{mn} \)

Rodrigues: \( H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2} \)
Generating: \( e^{2xt-t^2} = \sum_{n=0}^\infty H_n(x) \frac{t^n}{n!} \)
Orthogonality: \( \int_{-\infty}^\infty H_m(x) H_n(x) e^{-x^2} dx = \sqrt{\pi} 2^n n! \delta_{mn} \)