Math Equations

From TechWiki

Contents

Notation

  • Gradient: \vec  \nabla f (\vec x) = \mbox{grad } f (\vec x) = \left(\frac{\partial f}{\partial x_1 }, \dots,  \frac{\partial f}{\partial x_n }  \right)^T, where \nabla is called nabla
  • Divergence: \nabla\cdot\vec F (\vec x) = \mbox{div } \vec F (\vec x)= \sum_{i=1}^n \frac{\partial F_i}{\partial x_i}
  • Laplacian: \Delta = \vec \nabla^2 = \vec \nabla \cdot \vec \nabla = \sum_{i=1}^n \frac {\partial^2}{\partial x^2_i}
  • Curl (rot): \mbox{curl }(\vec F) = \nabla \times \vec F

Derivatives

  • F = F(x,y,z) \Rightarrow  dF = \frac{\partial F}{\partial x} dx +  \cdots + \frac{\partial F}{\partial z} dz
  • F = F (x, \phi(x)) \Rightarrow \frac{d F}{dx} = \frac{\partial F}{\partial x} + \frac{\partial F}{\partial \phi} \frac{d \phi}{dx}
  • F = F(x(y,z),y,z) \Rightarrow  \frac{dF}{dy}\bigg|_z = \frac{\partial F}{\partial x} \frac{\partial x}{\partial y} + \frac{\partial F}{\partial y} + \cdots
  • L = L (y(x), \dot y(x), x)) \Rightarrow   \frac{dL}{dx} +  \frac{\partial L}{\partial y} \dot y + \frac{\partial L}{\partial \dot y} \ddot y + \frac{\partial L}{\partial x}
  • x=x(t)\Rightarrow \frac{dx}{d \tau} = \frac{dx}{dx} \frac{dt}{d \tau}

Integrals

  • \int_a^b  \vec{\nabla} \phi d \vec s = \phi(b) - \phi(a)
  • Partial integration:
\int u^{\prime}  v = uv|_a^b - \int_a^b u v^{\prime}
  • Substitution: x=g(z), dx/dz=g^{\prime}
\int_a^b f(g(z)) dz = \int_{g(a)}^{g(b)}f(x) dx
  • Variable transformation: \vec u \mapsto \vec x := g(\vec u)
\int f(\vec x) d \vec x = \int f (\vec u) \mathcal{J}_g (\vec u) d \vec u,
where

\mathcal{J}_g (\vec u) = \big| \det dg(\vec u) \big|,  dg(\vec u)= \left( \frac{\partial x_i}{\partial x_j} \right)

Fundamental Theorem of Calculus

  • \frac{d}{dx} \int_a^x f(t) dt = f(x)
  • \int_a^b \frac{d}{dx} f(x) dx = f(x) |_a^b

Vector Calculus

  • Gradient theorem: \int_L \nabla\phi\cdot d\vec r = \phi\left(\vec q\right)-\phi\left(\vec p\right)
  • Stoke's theorem: \int_{\Sigma} \nabla \times \vec F \cdot d\vec f = \oint_{\partial\Sigma} \vec F \cdot d \vec s
  • Gauss' theorem (divergence theorem): \int\limits_V\left(\nabla\cdot\vec F\right)d^3x=\int\limits_{\part V}\vec F\cdot d\vec s,

Relations

  • \operatorname{rot } \; \operatorname{grad} \;f = 0
  • \operatorname{div} \; \operatorname{rot} \; \vec F = 0
  • \operatorname{div} \; \operatorname{grad} \; f = \Delta f
  • \operatorname{rot} \; \operatorname{rot} \; \vec F = \operatorname{grad} \; \operatorname{div} \; \vec F - \Delta \vec F

Wavelets

Linear Algebra

General

  • Space: \mathcal R^n
  • Scalar product: \langle . , .\rangle : \mathcal R^n \times \mathcal R^n \to \mathcal R
  • Orthonormal basis \{ \vec e_i \}: \langle \vec e_i , \vec e_j \rangle = \delta_{ij} \vec e_i


Relations

  • \vec X = \sum_i a_i \vec e_i
  • a_i = \langle \vec X, \vec e_i \rangle

 

Fourier

General

  • Function space: \mathbb{L}_2 ( \mathcal R)
  • Scalar product: \langle f , g\rangle := \int f(t) \overline{g(t)} dt
  • Orthonormal basis: {eiωt}


Relations

  • f(t) = \frac{1}{2 \pi} \int_{-\infty}^{\infty} f (\omega) e^{i\omega t} d \omega
  • f(\omega) = \langle f(t),  e^{i\omega t}\rangle = \frac{1}{2 \pi} \int_{-\infty}^{\infty} f (t) e^{-i\omega t} d t


Properties

  • Transform: F : f(t) \to f(\omega); [Ff](t) = f(ω)
  • Interpretation: \textrm{f} = \frac{1}{t}=\frac{\omega}{2 \pi}
  • Global
  • Scaling in frequency

 

Wavelets

General

  • \psi_{j,k} (t) := 2^{j/2} \psi(2^j t-k) ; \psi \in \mathbb L_2( \mathcal R)
  • Scalar product: \langle f , g\rangle := \int f(t) \overline{g(t)} dt
  • Orthon. basis: j,k}: \langle \psi_{j,k}, \psi_{l,m}\rangle = \delta_{jl} \delta_{km}


Relations

  • f(t) = cj,kψj,k(t)
    j,k
  • c_{j,k} = \langle  f(t) ,\psi_{j,k} (t) \rangle = \int_{- \infty}^{\infty} f(t) \overline{\psi_{j,k} (t)} dt


Properties

  • Transform: [W_{\psi} f ](a,b) := \frac{1}{\sqrt a} \int_{- \infty}^{\infty} f(t) \overline{\psi (\frac{t-b}{a})} dt
  • cj,k = [Wψf](2j,k2j)
  • f_a (t) = \int_{- \infty}^{\infty}  [W_{\psi} f ] (a,b) \psi_{a,b}(t)db
  • Scale a or frequency band [\frac{1}{a},\frac{2}{a}]
  • Local projection
  • Scaling in time