Calculus and Analysis Math Symbols

\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

Dx y

derivative

derivative - Euler's notation

Dx2y

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

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