Vector operators
- \(\nabla f = \frac{\partial f}{\partial r} \hat{r} + \frac{1}{r}\frac{\partial f}{\partial \theta} \hat{\theta} \)
- \(\nabla \cdot \textbf{F} = \frac{1}{r}\frac{\partial ( r F_{r})}{\partial r}+ \frac{1}{r} \frac{\partial F_{\theta}}{\partial \theta}\)
- \(\nabla^2 f = \frac{1}{r} \frac{\partial }{ \partial r} (r \frac{\partial f}{\partial r}) + \frac{1}{r^2}\frac{\partial^2 f }{\partial \theta^2}\)
Vector operators
- \(\nabla f = \frac{\partial f}{\partial r} \hat{r} + \frac{1}{r}\frac{\partial f}{\partial \theta} \hat{\theta}+\frac{\partial f}{\partial z} \hat{z}\)
- \(\nabla \cdot \textbf{F} = \frac{1}{r}\frac{\partial ( r F_{r})}{\partial r}+ \frac{1}{r} \frac{\partial F_{\theta}}{\partial \theta}+\frac{\partial F_{z}}{\partial z}\)
- \(\nabla \times \textbf{F} = ( \frac{1}{r} \frac{\partial F_{z}}{\partial \theta} - \frac{\partial F_{\theta}}{\partial z}) \hat{r} + ( \frac{\partial F_{r}}{\partial z} - \frac{\partial F_{z}}{\partial r}) \hat{\theta} + \frac{1}{r} (\frac{\partial (r F_{\theta})}{\partial r} - \frac{\partial F_{r}}{\partial \theta}) \hat{z}\)
- \(\nabla^2 f = \frac{1}{r} \frac{\partial }{ \partial r} (r \frac{\partial f}{\partial r}) + \frac{1}{r^2}\frac{\partial^2 f }{\partial \theta^2} + \frac{\partial^2 f}{ \partial z^2}\)
Vector operators
\[ \nabla f = \frac{\partial f}{\partial \rho} \hat{\rho} + \frac{1}{\rho}\frac{\partial f}{\partial \theta} \hat{\theta}+ \frac{1}{\rho \sin (\theta)}\frac{\partial f}{\partial \varphi} \hat{\varphi}\]
\[ \nabla \cdot \textbf{F} = \frac{1}{{\rho}^2}\frac{\partial ( {\rho} ^2 F_{\rho})}{\partial \rho}+ \frac{1}{\rho \sin (\theta) } \frac{\partial ( \sin (\theta) F_{\theta} )}{\partial \theta}+ \frac{1}{\rho \sin (\theta)} \frac{\partial F_{\varphi}}{\partial \varphi}\]
\[ \nabla \times \textbf{F} = \frac{1}{\rho \sin (\theta)} ( \frac{\partial ( \sin (\theta) F_{\varphi}) }{\partial \theta} - \frac{\partial F_{\theta}}{\partial \varphi}) \hat{\rho} + \frac{1}{\rho} ( \frac{1}{\sin (\theta)}\frac{\partial F_{\rho}}{\partial \varphi} - \frac{\partial (\rho F_{\varphi}) }{\partial \rho}) \hat{\theta} + \frac{1}{\rho} (\frac{\partial (\rho F_{\theta})}{\partial \rho} - \frac{\partial F_{\rho}}{\partial \theta}) \hat{\varphi}\]
\[ \nabla^2 f = \frac{1}{\rho^2} \frac{\partial }{ \partial \rho } (\rho^2 \frac{\partial f}{\partial \rho}) + \frac{1}{\rho^2 \sin ( \theta )} \frac{\partial }{\partial \theta } ( \sin ( \theta ) \frac{\partial f }{\partial \theta } ) + \frac{1}{\rho^2 \sin ^2 (\theta )} \frac{\partial ^2 f }{ \partial \phi} \]
Volume element
\( dV = \rho ^2 \sin ( \theta ) d\rho d\theta d\phi \)
Vector operators
Nabla symbol
Differential operator \(\nabla\) used as a notation for vector operators and hints towards their methods of calculation
\( \displaystyle \nabla = \begin{pmatrix} \frac{\partial}{\partial x} \\ \frac{\partial}{\partial y} \\ \frac{\partial}{\partial z} \end{pmatrix} \)
Gradient Gradiente 勾配
Vector operator that returns the vector of maximum change of a point in a scalar field \(f\)
\(\displaystyle \nabla f = \begin{pmatrix} \frac{\partial f }{\partial x} \\ \frac{\partial f}{\partial y} \end{pmatrix}\)
- \(f\) is the scalar field
See Mathematics 2
Divergence Divergenza 発散
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
\(\displaystyle \nabla \cdot \textbf{F} = \begin{pmatrix} \frac{\partial}{\partial x} \\ \frac{\partial}{\partial y} \\ \frac{\partial}{\partial z} \end{pmatrix} \cdot \textbf{F} = \frac{\partial F_{x}}{\partial x}+\frac{\partial F_{y}}{\partial y}+\frac{\partial F_{z}}{\partial z}\)
- \(\textbf{F}\) is the vector field
- \(F_{n}\) is the vector field's \(n\) argument
Sink
Points in vector fields with more inward flow
\((x,y) \text{ is a sink } \iff \nabla \cdot f(x,y) \lt 0\)
Source
Points in vector fields with more outward flow
\((x,y) \text{ is a source } \iff \nabla \cdot f(x,y) \gt 0\)
Curl Rotore 回転
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 \(\textbf{F}\)
\(\nabla \times \textbf{F} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ F_x & F_y & F_z \end{vmatrix} = \begin{pmatrix} \frac{\partial F_{z}}{\partial y} - \frac{\partial F_{y}}{\partial z} \\ \frac{\partial F_{z}}{\partial x} - \frac{\partial F_{x}}{\partial z} \\ \frac{\partial F_{y}}{\partial x} - \frac{\partial F_{x}}{\partial y} \end{pmatrix}\)
- \(\textbf{F}\) is the vector field
- \(F_{n}\) is the vector field's \(n\) argument
Laplacian Laplaciano ラプラス作用素
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
\(\displaystyle \nabla^2 f = \nabla \cdot (\nabla f) = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2}\)
Vector-valued function
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 field Campo vettoriale
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 \(X^n\)
\(\textbf{F} ( \textbf{x} )\)
Position function
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
Vector operators
Nabla symbol
Differential operator \(\nabla\) used as a notation for vector operators and hints towards their methods of calculation
\( \displaystyle \nabla = \begin{pmatrix} \frac{\partial}{\partial x} \\ \frac{\partial}{\partial y} \\ \frac{\partial}{\partial z} \end{pmatrix} \)
Gradient Gradiente 勾配
Vector operator that returns the vector of maximum change of a point in a scalar field \(f\)
\(\displaystyle \nabla f = \begin{pmatrix} \frac{\partial f }{\partial x} \\ \frac{\partial f}{\partial y} \end{pmatrix}\)
- \(f\) is the scalar field
See Mathematics 2
Divergence Divergenza 発散
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
\(\displaystyle \nabla \cdot \textbf{F} = \begin{pmatrix} \frac{\partial}{\partial x} \\ \frac{\partial}{\partial y} \\ \frac{\partial}{\partial z} \end{pmatrix} \cdot \textbf{F} = \frac{\partial F_{x}}{\partial x}+\frac{\partial F_{y}}{\partial y}+\frac{\partial F_{z}}{\partial z}\)
- \(\textbf{F}\) is the vector field
- \(F_{n}\) is the vector field's \(n\) argument
Sink
Points in vector fields with more inward flow
\((x,y) \text{ is a sink } \iff \nabla \cdot f(x,y) \lt 0\)
Source
Points in vector fields with more outward flow
\((x,y) \text{ is a source } \iff \nabla \cdot f(x,y) \gt 0\)
Curl Rotore 回転
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 \(\textbf{F}\)
\(\nabla \times \textbf{F} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ F_x & F_y & F_z \end{vmatrix} = \begin{pmatrix} \frac{\partial F_{z}}{\partial y} - \frac{\partial F_{y}}{\partial z} \\ \frac{\partial F_{z}}{\partial x} - \frac{\partial F_{x}}{\partial z} \\ \frac{\partial F_{y}}{\partial x} - \frac{\partial F_{x}}{\partial y} \end{pmatrix}\)
- \(\textbf{F}\) is the vector field
- \(F_{n}\) is the vector field's \(n\) argument
Laplacian Laplaciano ラプラス作用素
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
\(\displaystyle \nabla^2 f = \nabla \cdot (\nabla f) = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2}\)
Coordinate system
System of variables used to define a set of points in a space
Point representation
Each coordinate system has a set of equations to translate points in another system to said coordinate system
Vector representation
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
\(\mathbb{R}^2\)
- Cartesian
- Polar
- Parabolic
- Bipolar
- Elliptic
\(\mathbb{R}^3\)
- Cartesian
- Cylindrical
- Spherical
Cartesian coordinates
Ordered 3-tuple \( (x,y,z) \) that represents a point in a 3D space
- \( x \in \mathbb{R}\) is the translation along the x-axis
- \( y \in \mathbb{R}\) is the horizontal translation perpendicular to the x-axis
- \( z \in \mathbb{R}\) is the vertical translation
Vector transform
\( \text{span}\{ \hat{i}, \hat{j}, \hat{k}\} = \mathbb{R}^3 \)
- \( \hat{i} = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} \)
- \( \hat{j} = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} \)
- \( \hat{k} = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} \)
Volume element
\( dV = dxdydz \)
Polar coordinates
Ordered pair \( (r,\theta) \) that represents any point and vector in a 2D space
- \(r \in \mathbb{R}_{+}\) represents the modulus
- \(\theta \in [0,2\pi] \) represents the azimuthal angle (angle from x to y)
Point transfom
- \(r=\sqrt{x^2 + y^2}\)
- \(\theta = \arctan (\frac{y}{x})\)
- \(x=r\cos (\theta)\)
- \(y=r\sin (\theta)\)
Vector transform
For some chosen angle \(\theta\)
\( \text{span}\{ \hat{r}, \hat{\theta}\} = \mathbb{R}^2 \)
- \( \hat{r} = \begin{pmatrix} \cos (\theta) \\ \sin (\theta) \end{pmatrix} \)
- \( \hat{\theta} = \begin{pmatrix} -\sin (\theta) \\ \cos (\theta) \end{pmatrix} \)
\( P_{ (r,\theta) \to (x,y)} = \begin{bmatrix} \cos (\theta) & -\sin(\theta) \\ \sin (\theta) & \cos (\theta) \end{bmatrix}\)
Vector operators
- \(\nabla f = \frac{\partial f}{\partial r} \hat{r} + \frac{1}{r}\frac{\partial f}{\partial \theta} \hat{\theta} \)
- \(\nabla \cdot \textbf{F} = \frac{1}{r}\frac{\partial ( r F_{r})}{\partial r}+ \frac{1}{r} \frac{\partial F_{\theta}}{\partial \theta}\)
- \(\nabla^2 f = \frac{1}{r} \frac{\partial }{ \partial r} (r \frac{\partial f}{\partial r}) + \frac{1}{r^2}\frac{\partial^2 f }{\partial \theta^2}\)
Area element
\( dA = rdrd\theta \)
Cylindrical coordinates
Ordered 3-tuple \( (r,\theta,z) \) that represents any point and vector in a 3D space
- \(r \in \mathbb{R}_{+}\) represents the modulus projected along the x-y plane
- \(\theta \in [0,2\pi]\) represents the azimuthal angle (horizontally from +x-axis to +y-axis)
- \(z \in \mathbb{R}\) represents vertical translation
Point transform
- \(r=\sqrt{x^2 + y^2}\)
- \(\theta = \arctan (\frac{y}{x})\)
- \(z=z\)
- \(x=r\cos (\theta)\)
- \(y=r\sin (\theta)\)
- \(z=z\)
Vector transform
For some chosen angle \(\theta\)
\( \text{span}\{ \hat{r}, \hat{\theta}, \hat{z}\} = \mathbb{R}^3 \)
- \( \hat{r} = \begin{pmatrix} \cos (\theta) \\ \sin (\theta) \\ 0 \end{pmatrix} \)
- \( \hat{\theta} = \begin{pmatrix} -\sin (\theta) \\ \cos (\theta) \\ 0 \end{pmatrix} \)
- \( \hat{z} = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} \)
\( P_{ (r,\theta, z) \to (x,y,z)} = \begin{bmatrix} \cos (\theta) & -\sin(\theta) & 0 \\ \sin (\theta) & \cos (\theta) & 0 \\ 0 & 0 & 1 \end{bmatrix}\)
Vector operators
- \(\nabla f = \frac{\partial f}{\partial r} \hat{r} + \frac{1}{r}\frac{\partial f}{\partial \theta} \hat{\theta}+\frac{\partial f}{\partial z} \hat{z}\)
- \(\nabla \cdot \textbf{F} = \frac{1}{r}\frac{\partial ( r F_{r})}{\partial r}+ \frac{1}{r} \frac{\partial F_{\theta}}{\partial \theta}+\frac{\partial F_{z}}{\partial z}\)
- \(\nabla \times \textbf{F} = ( \frac{1}{r} \frac{\partial F_{z}}{\partial \theta} - \frac{\partial F_{\theta}}{\partial z}) \hat{r} + ( \frac{\partial F_{r}}{\partial z} - \frac{\partial F_{z}}{\partial r}) \hat{\theta} + \frac{1}{r} (\frac{\partial (r F_{\theta})}{\partial r} - \frac{\partial F_{r}}{\partial \theta}) \hat{z}\)
- \(\nabla^2 f = \frac{1}{r} \frac{\partial }{ \partial r} (r \frac{\partial f}{\partial r}) + \frac{1}{r^2}\frac{\partial^2 f }{\partial \theta^2} + \frac{\partial^2 f}{ \partial z^2}\)
Volume element
\( dV = rdrd\theta dz \)
Spherical coordinates
Ordered 3-tuple \( (\rho,\theta,\phi) \) that represents any point and vector in a 3D space
- \(\rho \in \mathbb{R}_{+}\) represents the modulus
- \(\theta \in [0,\pi] \) represents the inclination angle (vertically from +z-axis to x-y-plane)
- \(\phi \in [0,2\pi]\) represents the azimuthal angle (horizontally from +x-axis to +y-axis)
Point transform
- \(\rho=\sqrt{x^2 + y^2 + z^2}\)
- \(\theta= \arccos (\frac{z}{\rho})\)
- \(\phi= \arctan (\frac{y}{x})\)
- \(x=\rho \sin(\theta) \cos (\phi) \)
- \(y=\rho \sin (\theta)\sin (\phi) \)
- \(z=\rho \cos (\theta)\)
Vector transform
\( \forall \theta,\phi\in [0,2\pi] ,\text{span}\{ \hat{\rho}, \hat{\theta}, \hat{\varphi}\} = \mathbb{R}^3 \)
- \( \hat{\rho} = \begin{pmatrix} \sin (\theta) \cos (\phi) \\ \sin (\theta) \sin (\phi) \\ \cos (\theta) \end{pmatrix} \)
- \( \hat{\theta} = \begin{pmatrix} \cos(\theta) \cos (\varphi) \\ \cos(\theta) \sin (\varphi) \\ -\sin (\theta) \end{pmatrix} \)
- \( \hat{\varphi} = \begin{pmatrix} -\sin (\varphi) \\ \cos (\varphi) \\ 0 \end{pmatrix} \)
\( P_{ (\rho,\theta, \phi) \to (x,y,z)} = \begin{bmatrix} \sin (\theta) \cos (\phi) & \cos(\theta) \cos(\phi) & -\sin(\phi) \\ \sin(\theta) \sin(\phi) & \cos(\theta) \sin(\phi) & \cos( \phi ) \\ \cos(\theta) & -\sin (\theta) & 0 \end{bmatrix}\)
Vector operators
\(\nabla f = \frac{\partial f}{\partial \rho} \hat{\rho} + \frac{1}{\rho}\frac{\partial f}{\partial \theta} \hat{\theta}+ \frac{1}{\rho \sin (\theta)}\frac{\partial f}{\partial \varphi} \hat{\varphi}\)
\(\nabla \cdot \textbf{F} = \frac{1}{{\rho}^2}\frac{\partial ( {\rho} ^2 F_{\rho})}{\partial \rho}+ \frac{1}{\rho \sin (\theta) } \frac{\partial ( \sin (\theta) F_{\theta} )}{\partial \theta}+ \frac{1}{\rho \sin (\theta)} \frac{\partial F_{\varphi}}{\partial \varphi}\)
\(\nabla \times \textbf{F} = \frac{1}{\rho \sin (\theta)} ( \frac{\partial ( \sin (\theta) F_{\varphi}) }{\partial \theta} - \frac{\partial F_{\theta}}{\partial \varphi}) \hat{\rho} + \frac{1}{\rho} ( \frac{1}{\sin (\theta)}\frac{\partial F_{\rho}}{\partial \varphi} - \frac{\partial (\rho F_{\varphi}) }{\partial \rho}) \hat{\theta} + \frac{1}{\rho} (\frac{\partial (\rho F_{\theta})}{\partial \rho} - \frac{\partial F_{\rho}}{\partial \theta}) \hat{\varphi}\)
\(\nabla^2 = \frac{1}{\rho^2} \frac{\partial }{ \partial \rho } (\rho^2 \frac{\partial f}{\partial \rho}) + \frac{1}{\rho^2 \sin ( \theta )} \frac{\partial }{\partial \theta } ( \sin ( \theta ) \frac{\partial f }{\partial \theta } ) + \frac{1}{\rho^2 \sin ^2 (\theta )} \frac{\partial ^2 f }{ \partial \phi} \)
Volume element
\( dV = \rho ^2 \sin ( \theta ) d\rho d\theta d\phi \)