Calculus and Analysis symbols and Signs

Details: Category: Mathematics notes and reviews

Calculus & analysis symbols

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 - Lagrange'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 - Leibniz'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's notation
	time second derivative	derivative of derivative
D_xy	derivative	derivative - Euler's notation
D_x²y	second derivative	derivative of derivative
$\frac{\partial f(x,y)}{\partial x}$	partial derivative		∂(x²+y²)/∂x = 2x
∫	integral	opposite to derivation	∫ f(x)dx
∫∫	double integral	integration of function of 2 variables	∫∫ f(x,y)dxdy
∫∫∫	triple integral	integration of function of 3 variables	∫∫∫ f(x,y,z)dxdydz
∮	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
Re(z)	real part of a complex number	z = a+bi → Re(z)=a	Re(3 - 2i) = 3
Im(z)	imaginary part of a complex number	z = a+bi → Im(z)=b	Im(3 - 2i) = -2
\| z \|	absolute value/magnitude of a complex number	\|z\| = \|a+bi\| = √(a²+b²)	\|3 - 2i\| = √13
arg(z)	argument of a complex number	The angle of the radius in the complex plane	arg(3 + 2i) = 33.7°
∇	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