Mathematical notations in physics

Physics is a science that is essentially based on measurement and modeling. The physicist’s background must therefore contain the tools and methods that allow him to:

derive rational information from his measurements.
approach the behavior of a model system using mathematical analysis.

Notations

Tab. 1 Mathematical notations

Symbol	Signification
def =	equal by definition
≃	approximately equal to
∼	equal in order of magnitude
A≫B	A very large in front of B
A≪B	A very small in front of B
AB^⌢	arc length

[G]	dimension of the quantity G

	statistical mean of a random quantity or time mean of a signal
σ_s	standard deviation associated with a random quantity
s_rms	effective value of a signal
(ν) = TF[s(t)]	Fourier transform of the signal s(t)
(ν_x,ν_y) = TF_2D[s(x,y)]	two-dimensional Fourier transform of the signal s(x,y)
δ(t)	Dirac impulse

z	complex magnitude
z*	conjugate complex
Re(z)	real part of z
Im(z)	imaginary part of z
∣z∣	modulus of z
arg(z)	argument of z

(_x, y, z)	Cartesian base
(_r, _θ, _z)	cylindrical base
(_r, _φ, _z)	spherical base
∥∥ or A	norm of vector
	scalar product of two vectors
∧	vector product of two vectors
A_z	component along the axis (O_z) = A_z = ⋅ _z

= dy/dt	first derivative with respect to time
= d²y/dt²	second derivative with respect to time
∂f(x,y,z,t)/∂x or f′_x	partial derivative of f with respect to the variable x
Df(x,y,z,t)/Dt	particular derivative of f with respect to the variable t
∮_C(M,t)⋅d	circulation of a vector field along the closed-circuit C
∬_S(M,t)⋅dS	Flow of a vector field
f or f	gradient of a scalar field f
div or ⋅	divergence of a vector field
or ∧	rotational of a vector field
Δf = ⋅f = ∇²f	Laplacian of a scalar field f
Δ	Laplacian of a vector field