Mathematical Formulas

Algebra
1	2
$a x^{2} + b x + c = 0$	$x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$
$\| a + b \| \leq \| a \| + \| b \|$	$\sum a_{n} b_{n} \leq {(\sum a_{n}^{2})}^{½} {(\sum b_{n}^{2})}^{½}$

Trigonometry and Hyperbolics
1	2	3	4
Given a right triangle with sides a, b, and c and opposite angles A, B, and C, C being the right angle.
$A + B + C = π$ $C = \frac{π}{2}$	$a = opp$ $b = adj$ $c = hyp$	$c^{2} = a^{2} + b^{2}$
$\sin A = \frac{a}{c} = \frac{opp}{hyp}$	$\sin - x = - \sin x$ (i.e. sin is odd)	$\sin (u + v) = \sin u \cos v + \cos u \sin v$ $\sin (u - v) = \sin u \cos v - \cos u \sin v$	$\sin u + \sin v = 2 \sin ½ (u + v) \cos ½ (u - v)$ $\sin u - \sin v = 2 \sin ½ (u - v) \cos ½ (u + v)$
$\cos A = \frac{b}{c} = \frac{adj}{hyp}$	$\cos - x = \cos x$ (i.e. cos is even)	$\cos (u + v) = \cos u \cos v - \sin u \sin v$ $\cos (u - v) = \cos u \cos v + \sin u \sin v$	$\cos u + \cos v = 2 \cos ½ (u + v) \cos ½ (u - v)$ $\cos u - \cos v = - 2 \sin ½ (u + v) \sin ½ (u - v)$
$\sin^{2} x + \cos^{2} x = 1$		$\sin u \sin v = ½ \cos (u - v) - ½ \cos (u + v)$ $\cos u \cos v = ½ \cos (u - v) + ½ \cos (u + v)$	$\sin u \cos v = ½ \sin (u + v) + ½ \sin (u - v)$ $\cos u \sin v = ½ \sin (u + v) - ½ \sin (u - v)$
$\sin (u + \frac{π}{2}) = \cos u$	$\sin (u - \frac{π}{2}) = - \cos u$	$\cos (u + \frac{π}{2}) = - \sin u$	$\cos (u - \frac{π}{2}) = \sin u$
$\tan A = \frac{a}{b} = \frac{opp}{adj} = \frac{\sin A}{\cos A}$
$\csc A = \frac{c}{a} = \frac{1}{\sin A}$	$\sin \frac{π}{6} = \frac{1}{2} = 0.5$	$\sin \frac{π}{4} = \frac{\sqrt{2}}{2} = 0.707107$	$\sin \frac{π}{3} = \frac{\sqrt{3}}{2} = 0.866025$
$\sec A = \frac{c}{b} = \frac{1}{\cos A}$	$\cos \frac{π}{6} = \frac{\sqrt{3}}{2} = 0.866025$	$\cos \frac{π}{4} = \frac{\sqrt{2}}{2} = 0.707107$	$\cos \frac{π}{3} = \frac{1}{2} = 0.5$
$\cot A = \frac{b}{a} = \frac{1}{\tan A} = \frac{\cos A}{\sin A}$	$\tan \frac{π}{3} = \sqrt{3} = 1.73205$	$\tan \frac{π}{6} = \frac{\sqrt{3}}{3} = 0.577350$	$\tan \frac{π}{4} = 1$
$\sinh x = \frac{ⅇ^{x} - ⅇ^{- x}}{2} = \pm {(\cosh^{2} x - 1)}^{½}$	$\sinh - x = - \sinh x$ (i.e. sinh is odd)	$\sinh (u + v) = \sinh u \cosh v + \cosh u \sinh v$ $\sinh (u - v) = \sinh u \cosh v - \cosh u \sinh v$	$\sinh u + \sinh v = 2 \sinh ½ (u + v) \cosh ½ (u - v)$ $\sinh u - \sinh v = 2 \cosh ½ (u + v) \sinh ½ (u - v)$
$\cosh x = \frac{ⅇ^{x} + ⅇ^{- x}}{2} = {(\sinh^{2} x + 1)}^{½}$	$\cosh - x = \cosh x$ (i.e. cosh is even)	$\cosh (u + v) = \cosh u \cosh v + \sinh u \sinh v$ $\cosh (u - v) = \cosh u \cosh v - \sinh u \sinh v$	$\cosh u + \cosh v = 2 \cosh ½ (u + v) \cosh ½ (u - v)$ $\cosh u - \cosh v = 2 \sinh ½ (u + v) \sinh ½ (u - v)$
$\cosh^{2} x - \sinh^{2} x = 1$	$\sinh x + \cosh x = ⅇ^{x}$ $\sinh x - \cosh x = ⅇ^{- x}$	$\sinh u \sinh v = ½ \cosh (u + v) - ½ \cosh (u - v)$ $\cosh u \cosh v = ½ \cosh (u + v) + ½ \cosh (u - v)$	$\sinh u \cosh v = ½ \sinh (u + v) + ½ \sinh (u - v)$ $\cosh u \sinh v = ½ \sinh (u + v) - ½ \sinh (u - v)$
$\tanh x = \frac{\sinh x}{\cosh x} = \frac{ⅇ^{x} - ⅇ^{- x}}{ⅇ^{x} + ⅇ^{- x}} = \frac{ⅇ^{2 x} - 1}{ⅇ^{2 x} + 1}$	$\tanh - x = - \tanh x$ (i.e. tanh is odd)
$csch x = \frac{1}{\sinh x}$
$sech x = \frac{1}{\cosh x}$
$\coth x = \frac{1}{\tanh x} = \frac{\cosh x}{\sinh x}$
$ⅇ^{ⅈ x} = \cos x + ⅈ \sin x$	$\sin x = \frac{ⅇ^{ⅈ x} - ⅇ^{- ⅈ x}}{2 ⅈ}$	$\cos x = \frac{ⅇ^{ⅈ x} + ⅇ^{- ⅈ x}}{2}$
$\sinh ⅈ x = ⅈ \sin x$	$\cosh ⅈ x = \cos x$

Coordinate transformations
Spherical coordinates		Cylindrical coordinates
$x = r \sin θ \cos φ$	$r = {(x^{2} + y^{2} + z^{2})}^{½}$	$x = r \cos θ$	$r = {(x^{2} + y^{2})}^{½}$
$y = r \sin θ \sin φ$	$θ = atan (\frac{{(x^{2} + y^{2})}^{½}}{z})$	$y = r \sin θ$	$θ = atan \frac{y}{x}$
$z = r \sin θ$	$φ = atan \frac{y}{z}$	$z = z$	$z = z$

Integration and Differentiation
1	2	3	4
$\frac{ⅆ}{ⅆ x} f (g (x)) = f^{'} (g (x)) g^{'} (x)$	$\frac{ⅆ}{ⅆ q} \int_{p}^{q} f (x) ⅆ x = f (q)$ [p constant]	$\frac{ⅆ}{ⅆ p} \int_{p}^{q} f (x) ⅆ x = - f (p)$ [q constant]	$\frac{ⅆ}{ⅆ a} \int_{p}^{q} f (x, a) ⅆ x = \int_{p}^{q} \frac{\partial}{\partial a} [f (x, a)] ⅆ x + f (q, a) \frac{ⅆ q}{ⅆ a} - f (p, a) \frac{ⅆ p}{ⅆ a}$
$\frac{ⅆ}{ⅆ x} (u v) = u \frac{ⅆ v}{ⅆ x} + v \frac{ⅆ u}{ⅆ x}$	$\frac{ⅆ}{ⅆ x} (\frac{u}{v}) = \frac{1}{u} \frac{ⅆ u}{ⅆ x} - \frac{u}{v^{2}} \frac{ⅆ v}{ⅆ x}$	$\frac{ⅆ}{ⅆ x} (u v w) = u v \frac{ⅆ w}{ⅆ x} + v w \frac{ⅆ u}{ⅆ x} + u w \frac{ⅆ v}{ⅆ x}$
$\frac{ⅆ}{ⅆ x} (u^{n} v^{m}) = u^{n - 1} v^{m - 1} (n v \frac{ⅆ u}{ⅆ x} + m u \frac{ⅆ v}{ⅆ x})$	$\frac{ⅆ}{ⅆ x} (\frac{u^{n}}{v^{m}}) = \frac{u^{n - 1}}{v^{m + 1}} (n v \frac{ⅆ u}{ⅆ x} - m u \frac{ⅆ v}{ⅆ x})$
$\frac{ⅆ}{ⅆ x} (x^{n}) = n x^{n - 1}$	$\int x^{n} ⅆ x = \frac{1}{n + 1} x^{n + 1}$	$\int \frac{1}{x} ⅆ x = \ln x$	$\int a^{x} ⅆ x = a^{x} \ln a$
$\frac{ⅆ}{ⅆ x} \ln x = \frac{1}{x}$	$\int \ln x ⅆ x = x \ln x - x$	$\frac{ⅆ}{ⅆ x} ⅇ^{a x} = a ⅇ^{a x}$	$\int ⅇ^{a x} ⅆ x = \frac{1}{a} ⅇ^{a x}$
$\frac{ⅆ}{ⅆ x} \sin x = \cos x$	$\int \sin x ⅆ x = - \cos x$	$\frac{ⅆ}{ⅆ x} \sinh x = \cosh x$	$\int \sinh x ⅆ x = \cosh x$
$\frac{ⅆ}{ⅆ x} \cos x = - \sin x$	$\int \cos x ⅆ x = \sin x$	$\frac{ⅆ}{ⅆ x} \cosh x = \sinh x$	$\int \cosh x ⅆ x = \sinh x$
$\frac{ⅆ}{ⅆ x} \tan x = \sec^{2} x = \frac{1}{\cos^{2} x}$	$\int \tan x ⅆ x = \ln \sec x$	$\frac{ⅆ}{ⅆ x} \tanh x = {sech}^{2} x = \frac{1}{\cosh^{2} x}$	$\int \tanh x ⅆ x = \ln \cosh x$
$\frac{ⅆ}{ⅆ x} asin a x = \frac{a}{{(1 - a^{2} x^{2})}^{½}}$	$\int asin a x ⅆ x = x asin a x + \frac{1}{a} {(1 - a^{2} x^{2})}^{½}$	$\frac{ⅆ}{ⅆ x} asinh a x = \frac{a}{{(a^{2} x^{2} + 1)}^{½}}$	$\int asinh a x ⅆ x = x asinh a x - \frac{1}{a} {(a^{2} x^{2} + 1)}^{½}$
$\frac{ⅆ}{ⅆ x} acos a x = - \frac{a}{{(1 - a^{2} x^{2})}^{½}}$	$\int acos a x ⅆ x = x acos a x - \frac{1}{a} {(1 - a^{2} x^{2})}^{½}$	$\frac{ⅆ}{ⅆ x} acosh a x = \frac{a}{{(a^{2} x^{2} - 1)}^{½}}$	$\int acosh a x ⅆ x = x acosh a x - \frac{1}{a} {(a^{2} x^{2} - 1)}^{½}$
$\frac{ⅆ}{ⅆ x} atan a x = \frac{a}{(1 + a^{2} x^{2})}$	$\int atan a x ⅆ x = x atan a x - \frac{\ln (1 + a^{2} x^{2})}{2 a}$	$\frac{ⅆ}{ⅆ x} atanh a x = \frac{a}{(1 - a^{2} x^{2})}$	$\int atanh a x ⅆ x = x atanh a x + \frac{\ln (1 - a^{2} x^{2})}{2 a}$
$f : U ⟶ V$ $U \subseteq ℛ^{n}$ $V \subseteq ℛ^{m}$ $g : V ⟶ ℛ$	$D f (x) = (\begin{matrix} \frac{\partial f_{1}}{\partial x_{1}} (x) & \frac{\partial f_{1}}{\partial x_{2}} (x) & \dots & \frac{\partial f_{1}}{\partial x_{n}} (x) \\ \frac{\partial f_{2}}{\partial x_{1}} (x) & \frac{\partial f_{2}}{\partial x_{2}} (x) & \dots & \frac{\partial f_{2}}{\partial x_{n}} (x) \\ ⋮ & ⋮ & ⋮ \\ \frac{\partial f_{m}}{\partial x_{1}} (x) & \frac{\partial f_{m}}{\partial x_{2}} (x) & \dots & \frac{\partial f_{m}}{\partial x_{n}} (x) \end{matrix})$	$\int_{V} g = \int_{U} g \circ f \| \det (D f) \|$
$\int u v ⅆ t = u v - \int v ⅆ u$

Transforms
1	2	3
Fourier sine	$f (x) = \sum_{n = 1}^{\infty} A_{n} \sin \frac{n π x}{l}$	$A_{n} = \frac{2}{l} \int_{0}^{l} f (x) \sin \frac{n π x}{l} ⅆ x$
Fourier cosine	$f (x) = \frac{1}{2} A_{0} + \sum_{n = 1}^{\infty} A_{n} \cos \frac{n π x}{l}$	$A_{n} = \frac{2}{l} \int_{0}^{l} f (x) \cos \frac{n π x}{l} ⅆ x$
Fourier	$f (x) = \frac{1}{2} A_{0} + \sum_{n = 1}^{\infty} (A_{n} \cos \frac{n π x}{l} + B_{n} \sin \frac{n π x}{l})$	$A_{n} = \frac{2}{l} \int_{- l}^{l} f (x) \cos \frac{n π x}{l} ⅆ x$	$B_{n} = \frac{2}{l} \int_{- l}^{l} f (x) \sin \frac{n π x}{l} ⅆ x$
Laplace	$F (s) = \int_{0}^{\infty} ⅇ^{- s t} f (t) ⅆ t$	$f (t) = \frac{1}{2 π} \int_{0}^{\infty} ⅇ^{s t} F (s) ⅆ s$

Series
1	2	3
$\sum_{i = 1}^{n} i = \frac{n (n + 1)}{2}$	$\sum_{i = 1}^{n} i^{2} = \frac{n (n + 1) (2 n + 1)}{6}$	$\sum_{i = 1}^{n} i^{3} = \frac{n^{2} {(n + 1)}^{2}}{4}$
$\sum_{i = 0}^{n} x^{i} = \frac{x^{n + 1} - 1}{x - 1}$	$\sum_{i = 0}^{\infty} x^{i} = \frac{1}{1 - x}$	$\sum_{i = 1}^{\infty} x^{i} = \frac{x}{1 - x}$

Linear Algebra
$A^{- 1} B^{- 1} = {(B A)}^{- 1}$	$A^{T} B^{T} = {(B A)}^{T}$	${(A^{- 1})}^{T} = {(A^{T})}^{- 1}$	$Tr (A B) = Tr (B A)$
$\det (A B) = \det (A) \det (B)$	$\det (A^{- 1}) = \frac{1}{\det (A)}$	$\det (\begin{matrix} a_{1, 1} & a_{1, 2} & \dots & a_{1, n} \\ a_{2, 1} & a_{2, 2} & \dots & a_{2, n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ a_{n, 1} & a_{n, 2} & \dots & a_{n, n} \end{matrix}) = \sum {(- 1)}^{σ} a_{1, i_{1}} a_{2, i_{2}} \dots a_{n, i_{n}}$ $for all permutations i_{1}, i_{2}, \dots i_{n} of 1, 2, \dots, n where σ is the degree of permutation$
Characteristic equation	$\det (A - λ I) = 0$

QM Linear Algebra
$⟨A\| = (\begin{matrix} A_{1}^{} & A_{2}^{} & \dots & A_{n}^{*} \end{matrix})$	$\|B⟩ = (\begin{matrix} B_{1} \\ B_{2} \\ ⋮ \\ B_{n} \end{matrix})$	$\|B⟩ ⟨A\| = (\begin{matrix} B_{1} A_{1}^{} & B_{1} A_{2}^{} & \dots & B_{1} A_{n}^{} \\ B_{2} A_{1}^{} & B_{2} A_{2}^{} & \dots & B_{2} A_{n}^{} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ B_{n} A_{1}^{} & B_{n} A_{2}^{} & \dots & B_{n} A_{n}^{*} \end{matrix})$
$⟨A\| B⟩ = A_{1}^{} B_{1} + A_{2}^{} B_{2} + \dots + A_{N}^{*} B_{N}$		${⟨A\|}^{†} = \|A⟩$	${\|A⟩}^{†} = ⟨A\|$	${⟨φ\| ψ⟩}^{*} = ⟨ψ\| φ⟩$
		${⟨φ\| A\| ψ⟩}^{*} = ⟨ψ\| A^{†}\| φ⟩$	${⟨φ\| A^{†} B^{†}\| ψ⟩}^{*} = ⟨ψ\| B A\| φ⟩$
For self-adjoint operators (Hermitian matrices) and lie rings:
$[A, B] = A B - B A$	$[A, A] = 0$	$[A, B] = - [B, A]$	$[A, B C] = [A, B] C + B [A, C]$
$[A, [B, C]] + [B, [C, A]] + [C, [A, B]] = 0$		$A^{†} = A$	$[A B, C] = A [B, C] + [A, C] B$

n-dimensional spheres
Dimension (n)	Shape	(n-1)-Content (Boundary)	n-Content (Volume)
2	Circle	$Circumference = 2 π R$	$Area = π R^{2}$
3	Sphere	$Surface Area = 4 π R^{2}$	$Volume = \frac{4}{3} π R^{3}$
4	4-Sphere	$2 π^{2} R^{3}$	$\frac{1}{2} π^{2} R^{4}$
5	5-Sphere	$\frac{8}{3} π^{2} R^{4}$	$\frac{8}{15} π^{2} R^{5}$
6	6-Sphere	$π^{3} R^{5}$	$\frac{1}{6} π^{3} R^{6}$
even	n-Sphere	$\frac{n}{(n / 2)!} π^{n / 2} R^{n - 1}$	$\frac{1}{(n / 2)!} π^{n / 2} R^{n}$
odd	n-Sphere	$\frac{2^{(n + 1) / 2}}{(n - 2) (n - 4) \dots 3} π^{(n - 1) / 2} R^{n - 1}$	$\frac{2^{(n + 1) / 2}}{n (n - 2) (n - 4) \dots 3} π^{(n - 1) / 2} R^{n}$

volume
$The volume of a k -dimensional parallelopiped in ℛ^{n} specified by the vectors x_{1}, x_{2}, \dots, x_{k}$ $is given by v (x_{1}, x_{2}, \dots, x_{k}) = {\det (X^{T} X)}^{½}$ $where X is the n \times k matrix with x_{1}, x_{2}, \dots, x_{k} as its columns$
$Let A be a n \times n matrix. Let h : ℛ^{n} ⟶ ℛ^{n} be the linear transformation h (x) = A x .$ $Let S be a rectifiable set in ℛ^{n}, and let T = h (S) . Then v (T) = \| \det (A) \| v (S)$

Earl Killian <earl at killian.com>