Math Equations

From TechWiki

1 Notation
2 Derivatives
3 Integrals
4 Fundamental Theorem of Calculus
5 Vector Calculus
- 5.1 Relations
6 Wavelets

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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: $\Delta = \vec \nabla^2 = \vec \nabla \cdot \vec \nabla = \sum_{i=1}^n \frac {\partial^2}{\partial x^2_i}$

d'Alembertian: generilzation of the Laplacian to Minkowski space (see special relativity): $\square$

Curl (rot): $\mbox{curl }(\vec F) = \nabla \times \vec F$

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Derivatives

$F = F(x,y,z) \Rightarrow dF = \frac{\partial F}{\partial x} dx + \cdots + \frac{\partial F}{\partial z} dz$
$F = F (x, \phi(x)) \Rightarrow \frac{d F}{dx} = \frac{\partial F}{\partial x} + \frac{\partial F}{\partial \phi} \frac{d \phi}{dx}$
$F = F(x(y,z),y,z) \Rightarrow \frac{dF}{dy}\bigg|_z = \frac{\partial F}{\partial x} \frac{\partial x}{\partial y} + \frac{\partial F}{\partial y} + \cdots$

$L = L (y(x), \dot y(x), x)) \Rightarrow \frac{dL}{dx} + \frac{\partial L}{\partial y} \dot y + \frac{\partial L}{\partial \dot y} \ddot y + \frac{\partial L}{\partial x}$

$x=x(t)\Rightarrow \frac{dx}{d \tau} = \frac{dx}{dx} \frac{dt}{d \tau}$

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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

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

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Fundamental Theorem of Calculus

$\frac{d}{dx} \int_a^x f(t) dt = f(x)$

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

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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): $\int\limits_V\left(\nabla\cdot\vec F\right)d^3x=\int\limits_{\part V}\vec F\cdot d\vec s,$

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Relations

$\operatorname{rot } \; \operatorname{grad} \;f = 0$

$\operatorname{div} \; \operatorname{rot} \; \vec F = 0$

$\operatorname{div} \; \operatorname{grad} \; f = \Delta f$

$\operatorname{rot} \; \operatorname{rot} \; \vec F = \operatorname{grad} \; \operatorname{div} \; \vec F - \Delta \vec F$

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Wavelets

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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

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

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Fourier

General

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

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

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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: ${ψ j, k}$ : $\langle \psi_{j,k}, \psi_{l,m}\rangle = \delta_{jl} \delta_{km}$

Relations

$f (t) = \sum c j, k ψ j, k (t) j, k$
$c_{j,k} = \langle f(t) ,\psi_{j,k} (t) \rangle = \int_{- \infty}^{\infty} f(t) \overline{\psi_{j,k} (t)} dt$

Properties

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