About This Page
This page introduces advanced mathematical symbols for science and engineering students.
For basic symbols, please see the university-level mathematics page.
Table of Contents
Vector Calculus
Nabla Operator
Code:
$\nabla$
Display:
$\nabla$
Laplacian
Code:
$\nabla^2 f = \Delta f$
Display:
$\nabla^2 f = \Delta f$
Complex Analysis
Complex Function
Code:
$f(z) = u(x,y) + iv(x,y)$
Display:
$f(z) = u(x,y) + iv(x,y)$
Cauchy-Riemann Equations
Code:
$\displaystyle\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$
Display:
$\displaystyle\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$
Residue
Code:
$\text{Res}(f, z_0)$
Display:
$\text{Res}(f, z_0)$
Contour Integral
Code:
$\displaystyle\oint_C f(z) \, dz$
Display:
$\displaystyle\oint_C f(z) \, dz$
Residue Theorem
Code:
$\displaystyle\oint_C f(z) dz = 2\pi i \sum \text{Res}(f, z_k)$
Display:
$\displaystyle\oint_C f(z) dz = 2\pi i \sum \text{Res}(f, z_k)$
Fourier Analysis
Fourier Series
Code:
$\displaystyle f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} (a_n \cos nx + b_n \sin nx)$
Display:
$\displaystyle f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} (a_n \cos nx + b_n \sin nx)$
Fourier Transform
Code:
$\displaystyle F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-i\omega t} dt$
Display:
$\displaystyle F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-i\omega t} dt$
Inverse Fourier Transform
Code:
$\displaystyle f(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) e^{i\omega t} d\omega$
Display:
$\displaystyle f(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) e^{i\omega t} d\omega$
Convolution
Code:
$\displaystyle (f * g)(t) = \int_{-\infty}^{\infty} f(\tau) g(t - \tau) d\tau$
Display:
$\displaystyle (f * g)(t) = \int_{-\infty}^{\infty} f(\tau) g(t - \tau) d\tau$
Laplace Transform
Laplace Transform
Code:
$\displaystyle \mathcal{L}[f(t)] = F(s) = \int_0^{\infty} f(t) e^{-st} dt$
Display:
$\displaystyle \mathcal{L}[f(t)] = F(s) = \int_0^{\infty} f(t) e^{-st} dt$
Inverse Laplace Transform
Code:
$\mathcal{L}^{-1}[F(s)] = f(t)$
Display:
$\mathcal{L}^{-1}[F(s)] = f(t)$
Differentiation Theorem
Code:
$\mathcal{L}[f'(t)] = sF(s) - f(0)$
Display:
$\mathcal{L}[f'(t)] = sF(s) - f(0)$
Partial Differential Equations
Wave Equation
Code:
$\displaystyle\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$
Display:
$\displaystyle\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$
Heat Equation
Code:
$\displaystyle\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}$
Display:
$\displaystyle\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}$
Laplace's Equation
Code:
$\displaystyle\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$
Display:
$\displaystyle\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$
Poisson's Equation
Code:
$\nabla^2 u = f$
Display:
$\nabla^2 u = f$
Tensors
Tensor Notation
Code:
$T^{ij}$ or $T_{ij}$
Display:
$T^{ij}$ or $T_{ij}$
Kronecker Delta
Code:
$\delta_{ij} = \begin{cases} 1 & (i = j) \\ 0 & (i \neq j) \end{cases}$
Display:
$\delta_{ij} = \begin{cases} 1 & (i = j) \\ 0 & (i \neq j) \end{cases}$
Levi-Civita Symbol
Code:
$\epsilon_{ijk}$
Display:
$\epsilon_{ijk}$
Metric Tensor
Code:
$g_{ij}$
Display:
$g_{ij}$
Special Functions
Gamma Function
Code:
$\displaystyle\Gamma(n) = \int_0^{\infty} t^{n-1} e^{-t} dt$
Display:
$\displaystyle\Gamma(n) = \int_0^{\infty} t^{n-1} e^{-t} dt$
Beta Function
Code:
$\displaystyle B(p,q) = \int_0^1 t^{p-1}(1-t)^{q-1} dt$
Display:
$\displaystyle B(p,q) = \int_0^1 t^{p-1}(1-t)^{q-1} dt$
Bessel Function
Code:
$J_n(x)$
Display:
$J_n(x)$
Legendre Polynomial
Code:
$P_n(x)$
Display:
$P_n(x)$
Usage Examples in Reading Notes
Example 2: Fourier Transform
Code:
The Fourier transform of a rectangular wave $f(t)$ is $\displaystyle F(\omega) = \frac{2\sin(\omega T)}{\omega}$
Display:
The Fourier transform of a rectangular wave $f(t)$ is $\displaystyle F(\omega) = \frac{2\sin(\omega T)}{\omega}$
Example 3: Partial Differential Equation
Code:
The solution to the 1D wave equation $\displaystyle\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$ is $u(x,t) = f(x-ct) + g(x+ct)$
Display:
The solution to the 1D wave equation $\displaystyle\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$ is $u(x,t) = f(x-ct) + g(x+ct)$
Commonly Used Symbols
\nabla→ Nabla operator ∇\Delta→ Laplacian Δ\mathcal{L}→ Script L (for Laplace transform, etc.)\mathcal{F}→ Script F (for Fourier transform, etc.)\Gamma→ Gamma function Γ\epsilon→ Epsilon ε\delta→ Delta δ (Kronecker delta, etc.)\omega→ Angular frequency ω\tau→ Tau τ (time delay, etc.)\quad→ Space^{ij}→ Superscript (contravariant tensor)_{ij}→ Subscript (covariant tensor)