Calculus & analysis symbols

July 15, 2014 Mathematical Symbols

Calculus:
The branch of mathematics that deals with the finding and properties of derivatives and integrals of functions, by methods originally based on the summation of infinitesimal differences. The two main types are differential calculus and integral calculus.

Symbol	Symbol Name	Meaning / definition	Example
$\lim_{x\to x0}f(x)$	limit	limit value of a function
ε	epsilon	represents a very small number, near zero	ε → 0
e	e constant / Euler’s number	e = 2.718281828…	e = lim (1+1/x)^x ,x→∞
y ‘	derivative	derivative – Leibniz’s notation	(3x³)’ = 9x²
y ”	second derivative	derivative of derivative	(3x³)” = 18x
y⁽ⁿ⁾	nth derivative	n times derivation	(3x³)⁽³⁾ = 18
$\frac{dy}{dx}$	derivative	derivative – Lagrange’s notation	d(3x³)/dx = 9x²
$\frac{d^2y}{dx^2}$	second derivative	derivative of derivative	d²(3x³)/dx² = 18x
$\frac{d^ny}{dx^n}$	nth derivative	n times derivation
$\dot{y}$	time derivative	derivative by time – Newton notation
	time second derivative	derivative of derivative
$\frac{\partial f(x,y)}{\partial x}$	partial derivative		∂(x²+y²)/∂x = 2x
∫	integral	opposite to derivation
∬	double integral	integration of function of 2 variables
∭	triple integral	integration of function of 3 variables
∮	closed contour / line integral
∯	closed surface integral
∰	closed volume integral
[a,b]	closed interval	[a,b] = {x \| a ≤ x ≤ b}
(a,b)	open interval	(a,b) = {x \| a < x < b}
i	imaginary unit	i ≡ √-1	z = 3 + 2i
z*	complex conjugate	z = a+bi → z*=a–bi	z* = 3 + 2i
z	complex conjugate	z = a+bi → z = a–bi	z = 3 + 2i
∇	nabla / del	gradient / divergence operator	∇f (x,y,z)
	vector
	unit vector
x * y	convolution	y(t) = x(t) * h(t)
	Laplace transform	F(s) = {f (t)}
	Fourier transform	X(ω) = {f (t)}
δ	delta function
∞	lemniscate	infinity symbol