Function \(\textbf{f}: X^n \to \mathbb{R}\) that assigns a scalar to each point in a space
\(f ( \textbf{x} ) \)
Function \(\textbf{r} : X^n \to \mathbb{R}^m, m \geq 2\) that takes parameter vector or scalar as input and outputs a cartesian vector
Vector-valued function \(\textbf{F} : X^n \to \mathbb{R}^n\) that assigns a cartesian vector with same dimension as the space, to each point in said space \(X^n\)
\(\textbf{F} ( \textbf{x} )\)
Vector valued function \(\textbf{r}: [t_0 , t_1] \to \mathbb{R}^n \) that takes a scalar parameter \(t\) as input and outputs a cartesian vector, essentially representing a path in a space
Differential operator \(\nabla\) used as a notation for vector operators and hints towards their methods of calculation
\( \displaystyle \nabla = \begin{pmatrix} \frac{\partial}{\partial x} \\ \frac{\partial}{\partial y} \\ \frac{\partial}{\partial z} \end{pmatrix} \)
Vector operator that returns the vector of maximum change of a point in a scalar field \(f\)
\(\displaystyle \nabla f = \begin{pmatrix} \frac{\partial f }{\partial x} \\ \frac{\partial f}{\partial y} \end{pmatrix}\)
See Mathematics 2
Vector operator that returns the scalar quantity of flow in and out of a point in a vector field \(\textbf{F}\)
To calculate based on intuition, look to the top, bottom, left and right of the point and note how regarding these adjacent vectors the point absorbs and emits
\(\displaystyle \nabla \cdot \textbf{F} = \begin{pmatrix} \frac{\partial}{\partial x} \\ \frac{\partial}{\partial y} \\ \frac{\partial}{\partial z} \end{pmatrix} \cdot \textbf{F} = \frac{\partial F_{x}}{\partial x}+\frac{\partial F_{y}}{\partial y}+\frac{\partial F_{z}}{\partial z}\)
Points in vector fields with more inward flow
\((x,y) \text{ is a sink } \iff \nabla \cdot f(x,y) \lt 0\)
Points in vector fields with more outward flow
\((x,y) \text{ is a source } \iff \nabla \cdot f(x,y) \gt 0\)
Vector operator that returns the vector normal to the direction of counterclockwise rotation with its magnitude representing the intensity of the rotation at a point in vector field \(\textbf{F}\)
\(\nabla \times \textbf{F} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ F_x & F_y & F_z \end{vmatrix} = \begin{pmatrix} \frac{\partial F_{z}}{\partial y} - \frac{\partial F_{y}}{\partial z} \\ \frac{\partial F_{z}}{\partial x} - \frac{\partial F_{x}}{\partial z} \\ \frac{\partial F_{y}}{\partial x} - \frac{\partial F_{x}}{\partial y} \end{pmatrix}\)
Vector operator that returns the scalar quantity of 'curvature' at a point in the scalar field \(f\)
This works by capturing the gradient of the scalar field and finding the divergence of this gradient at some point
\(\displaystyle \nabla^2 f = \nabla \cdot (\nabla f) = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2}\)
System of variables used to define a set of points in a space
Each coordinate system has a set of equations to translate points in another system to said coordinate system
Each coordinate system has an orthonormal basis (frame) relative to a point in space that represents any vector from that point, e see Linear Algebra
Note that some orthonormal basises may be dependent on some \(\theta\) or \(\phi\) relational to the vector's base from the origin
Ordered 3-tuple \( (x,y,z) \) that represents a point in a 3D space
\( \text{span}\{ \hat{i}, \hat{j}, \hat{k}\} = \mathbb{R}^3 \)
\( dV = dxdydz \)
Ordered pair \( (r,\theta) \) that represents any point and vector in a 2D space
For some chosen angle \(\theta\)
\( \text{span}\{ \hat{r}, \hat{\theta}\} = \mathbb{R}^2 \)
\( P_{ (r,\theta) \to (x,y)} = \begin{bmatrix} \cos (\theta) & -\sin(\theta) \\ \sin (\theta) & \cos (\theta) \end{bmatrix}\)
\( dA = rdrd\theta \)
Ordered 3-tuple \( (r,\theta,z) \) that represents any point and vector in a 3D space
For some chosen angle \(\theta\)
\( \text{span}\{ \hat{r}, \hat{\theta}, \hat{z}\} = \mathbb{R}^3 \)
\( P_{ (r,\theta, z) \to (x,y,z)} = \begin{bmatrix} \cos (\theta) & -\sin(\theta) & 0 \\ \sin (\theta) & \cos (\theta) & 0 \\ 0 & 0 & 1 \end{bmatrix}\)
\( dV = rdrd\theta dz \)
Ordered 3-tuple \( (\rho,\theta,\phi) \) that represents any point and vector in a 3D space
\( \forall \theta,\phi\in [0,2\pi] ,\text{span}\{ \hat{\rho}, \hat{\theta}, \hat{\varphi}\} = \mathbb{R}^3 \)
\( P_{ (\rho,\theta, \phi) \to (x,y,z)} = \begin{bmatrix} \sin (\theta) \cos (\phi) & \cos(\theta) \cos(\phi) & -\sin(\phi) \\ \sin(\theta) \sin(\phi) & \cos(\theta) \sin(\phi) & \cos( \phi ) \\ \cos(\theta) & -\sin (\theta) & 0 \end{bmatrix}\)
\( dV = \rho ^2 \sin ( \theta ) d\rho d\theta d\phi \)
Scalar quantity \(ds\) representing an infinitesimal distance traversed in a space
\( ds= \sqrt{ (\frac{d r_x}{dt})^2 + (\frac{d r_y}{dt})^2 + (\frac{d r_z}{dt})^2 } \)
\( ds= |d\textbf{r}| = |\textbf{r}'(t)|dt \)
Scalar quantity \(s\) representing total distance traversed on a curve to get from one point to another
\( \displaystyle s(t) = \int_{\mathcal{C}} ds = \int_{t_0}^{t} |\textbf{r}'(t')|dt' \)
Set of connected points in \(\mathbb{R}^n\) defined by the image of a differentiable position function
\( \mathcal{C} = \{ \textbf{r}(t) : t \in [t_0,t_1] \}\)
\(\mathcal{C} \text{ is closed } \iff \textbf{r}(t_0) = \textbf{r}(t_1) \)
Class of integrals returning scalar quantities, related to weighting a curve in field
Line integral where each point of the curve is weighted by multiplication with the scalar field's intensity at that point
\(\displaystyle \int_{\mathcal{C}} f(\textbf{x})ds = \int_{t_0}^{t_1} f(\textbf{r}(t))|\textbf{r}'(t)|dt\)
\(\displaystyle \int_{\mathcal{C}} f(\textbf{x})ds = \lim_{\Delta s \to 0, n \to \infty} \sum^{n}_{i=1} f(\textbf{x}_i) \Delta s_i\)
Line integral where each point of the curve is weighted by the dot product with the vector field at that point and the curve's tangent at that point. This is essentially the 'work' of a vector field
\(\displaystyle \int_{\mathcal{C}} \textbf{F}(\textbf{x}) \cdot d\textbf{r} = \int_{\textbf{r}(t_0)}^{\textbf{r}(t_1)} \textbf{F}(\textbf{r}(t)) \cdot \textbf{r}'(t)dt\)
\(\displaystyle \int_{\mathcal{C}} \textbf{F}(\textbf{x}) \cdot d\textbf{r} = \lim_{\Delta s \to 0, n \to \infty} \sum_{i=1}^{n} \textbf{F}(\textbf{x}_i) \cdot \hat{T}\Delta s_i \)
\(\displaystyle \int_{\textbf{r}_0}^{\textbf{r}_1} \nabla f \cdot d\textbf{r} = f(\textbf{r}_1) - f(\textbf{r}_0)\)
Vector field such that the line integral result is purely dependent on the endpoints of the line
\(\textbf{F} \text{ is conservative } \iff\)
\(f \text{ is the potential function of } \textbf{F} \iff \nabla f = \textbf{F}\)
Set of connected points in \(\mathbb{R}^n\) defined by either:
\(S = \{ (x,y,z) : z=g(x,y) \} \)
\(S = \{ (x,y,z) : g(x,y,z)=0 \} \)
\( S = \{ \textbf{r}(u,v) : (u,v) \in [u_0,u_1] \times [v_0 , v_1] \}\)
The normal of a level set is the gradient
Take a level set \(g=0\), these connected points represent a path/surface where the output of \(g\) is the same, therefore there is no rate of change along the level set. the gradient therefore represents vectors perpendicular to the level set
\(\textbf{n} = \nabla (z-g) \)
\(\textbf{n} = \nabla g \)
\(\textbf{n} = \frac{d \textbf{r}}{du} \times \frac{d \textbf{r}}{dv} \)
Vector quantity \(d\textbf{S}\) representing an infinitesimal of surface, where it has:
\(d\textbf{S} = \hat{n} dS\)
\(dS = \| d\textbf{S} \| = \frac{dxdy}{|\hat{n} \cdot \hat{k}|} \)
\(dS = \| \frac{\partial \textbf{r}}{\partial u} \times \frac{\partial \textbf{r}}{\partial v} \| du dv \)
Class of integrals returning scalar quantities, related to weighting a surface in field
Surface integral where each point of the surface is weighted by multiplication with the scalar field's intensity at that point
\( \displaystyle \iint_{S} f(\textbf{x}) dS = \iint_{S} f(\textbf{r}(u,v)) \| \frac{\partial \textbf{r}}{\partial u} \times \frac{\partial \textbf{r}}{\partial v} \| dudv \)
\( \displaystyle \iint_{S} f(\textbf{x}) dS = \lim_{\Delta S \to 0 } \sum_{\forall \Delta S} f(\textbf{x}) \Delta S \)
Surface integral where each point of the curve is weighted by the dot product with the vector field at that point and the surface's normal at that point.
\( \displaystyle \iint_{S} \textbf{F}(\textbf{x}) \cdot d\textbf{S} = \iint_{S} \textbf{F}(\textbf{x}) \cdot \hat{n} dS \)
\( \displaystyle \iint_{S} f(\textbf{x}) dS = \lim_{\Delta S \to 0 } \sum_{\forall \Delta S} \textbf{F}(\textbf{x}) \cdot \hat{n} \Delta S \)
\(\displaystyle \nabla \times \textbf{F} = \hat{n} \lim_{\Delta s \to 0}\frac{1}{\Delta s} \oint_{\mathcal{C}} \textbf{F} \cdot d\textbf{r}\)
\(\displaystyle \nabla \cdot \textbf{F} = \lim_{\Delta V \to 0}\iint_{S} \textbf{F} \cdot d\textbf{S}\)
Theorem asserting that the divergence of all points in a volume equals the flux integral of the volume's closed surface.
Intuitively, this is because by thinking of the divergence of each infinitesimal volume element bounded by the surface, each infinitesimal volume elements has its flux 'cancelled out' by adjacent volume elements
\(\displaystyle \iiint_{V} (\nabla \cdot \textbf{F}) dV = \iint_{S} \textbf{F} \cdot d\textbf{S} \)
Theorem asserting that the curl of all points on an open surface equals the line integral of the open surface's edge.
Intuitively, this is because by thinking of the curl of each infinitesimal surface element, each infinitesimal volume elements has its curl 'cancelled out' by adjacent surface elements
\(\displaystyle \iint_{S} (\nabla \times \textbf{F}) \cdot d\textbf{S} = \oint_{\mathcal{C}} \textbf{F} \cdot d\textbf{r} \)
Corrolary of Stoke's theorem, form of Stoke's theorem of a function projected in the xy-plane.
\(\displaystyle \iint_{S} ( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} ) dxdy = \oint_{\mathcal{\partial S}} P(x,y)dx + Q(x,y)dy \)
Set of differential equations governing classical electromagnetism by stating the properties of electic and magnetic fields that are either derived from Coulomb's law and Biot-Savart law and an empirical observation (Faraday's law).
See Lebesgue integration fourier analysis
Hilbert space of functions whose square are integrable on some domain, the space has infinite dimension. The \(L^2\) space defines the iner product and norm as such.
\( \displaystyle \langle f, g \rangle = \int_{D} \overline{f(x)} g(x) w(x) dx\)
\( \| f \| = \sqrt{ \langle f,f \rangle }\)
See Differential Equations
Differential operator \( \mathcal{L} \) of the following form
\(\mathcal{L}f = -\frac{1}{w(x)} [ \frac{d}{dx} p(x) \frac{d f}{dx} -q(x)f ] \)
\(\mathcal{L} \phi = \lambda \phi \)
Boundary conditions applies to this problem determine the interval of the \(L^2\) space
Expansion of some function \(f\) for some weighted \(L^2\) space \( L^2_{w}([a,b])\) as the linear combination of an orthogonal basis' functions
\( f(x) = \sum^{\infty}_{n=0} c_n \phi_n (x) \)
\( f(x) = \sum^{\infty}_{m=0} \frac{ \langle \phi_m , f \rangle}{\| \phi_m \|^2 } \phi_m (x) \)
The constant \( c_m = \frac{ \langle \phi_m , f \rangle}{\| \phi_m \|^2} \) is derived by noting that \( \langle \phi_m , f \rangle = \sum^{\infty}_{n=0} c_n \langle \phi_m , \phi_n \rangle = c_m \langle \phi_m , \phi_m \rangle = c_m \| \phi_m \|^2\)
\( ( \cos (nx) )_{n\in \mathbb{N}} \) is an orthogonal basis for periodic \(L^{2}([-\pi,\pi])\)
\( ( \sin (nx) )_{n\in \mathbb{N}} \) is an orthogonal basis for periodic \(L^{2}([-\pi,\pi])\)
\( L^2 ([-\pi,\pi]) \implies \| \sin (nx) \|^2 = \| \cos( n x ) \|^2 = \pi \)
Second order linear PDE that is an SL eigenvalue problem in \(\mathbb{R}\), it usually models time independent waves as well as other natural phenomenon. it can be solved using
\( \nabla^2 f = -k^2 f \)
\( f (x) = e^{i k x} \)
\( f (r, \theta ) = J_m( k r) e^{i m x} \)
\( A(x,y) \frac{\partial^2 f}{\partial x^2} + 2B(x,y) \frac{\partial^2 f}{\partial x \partial y} + C(x,y) \frac{\partial^2 f}{\partial y^2} = F(x,y,f,\frac{\partial f}{\partial x} , \frac{\partial f}{\partial y} )\)
Second order linear PDE, it models mechanical waves
\(\nabla^2 \psi - \frac{1}{c^2} \frac{\partial^2 \psi}{\partial t^2} = 0 \)
Second order linear PDE that is an SL eigenvalue problem in \(\mathbb{R}\), it models static electromagnetic fields with no charges
\(\nabla^2 f = 0 \)
\( f_{k}(x, y ) = (e^{i k x} + e^{-i k x})(e^{ k y} + e^{- k y}) \)
\( k \) is a constant in terms of the eigenvalue
\( f_{k}(r, \theta ) = e^{i k \theta } \begin{cases} A + B \ln |r| \\ A r^{k} \end{cases}\)
\( k \) is a constant in terms of the eigenvalue
\( f_{m, \ell}(\rho, \theta , \phi ) = Y_{\ell , m} (\theta , \phi) [ r^{\ell} + r^{-\ell -1 } ]\)
\( m \) is a constant in terms of the primary eigenvalue
\( \ell \) is a constant in terms of the secondary eigenvalue
Second order linear PDE, it models heat distribution in a medium
\(\frac{\partial^2 u}{\partial x^2} = \frac{1}{\kappa} \frac{\partial u}{\partial t} \)
In \(\mathbb{R}^2\), a domain boundary may be parametrized as \(\textbf{r}(u)\), and conditions of this boundary may take several forms
See Differential Equations
\( \sum^{n}_{i=0} k_i x^{i} y^{(i)} = 0\)
\( y(x)= \sum^{n}_{i=1} c_i x^{m_i} \)
See Differential Equations
Sequence polynomials that satisfy the following ODE
\( (1-x^2)y''-2xy'+[\ell (\ell +1)-\frac{m^2}{1-x^2}]y=0 \)
\(P^{m}_{\ell}(x)\)
They form an orthonormal basis for \(L^2 [-1,1]\)
Complete set of functions representing the 'angular' separated term from solving Laplace's eqaution in spherical coordinates by separatio of variables
\( Y_{\ell, m} (\theta , \phi ) = e^{i m \phi } P^{m}_{\ell} ( \cos ( \theta )) \)
\( \frac{d }{dr}(r^2 \frac{d R}{d r}) = -k^2 R \)
\( R(r) = \frac{e^{i k r} }{ r }\)