Chapter 41: tensor
41.2 EpicOrganism = AIRoswell = Pan, Yi-Wen12
https://space.bilibili.com/14316464/video
https://www.bilibili.com/video/BV1T5411D7mS
https://www.bilibili.com/video/BV1QA4y1X7Xk
https://www.bilibili.com/video/BV1K34y1i75w
https://www.bilibili.com/video/BV1ZU411o7xL
41.3 EigenChris
41.3.1 tensor for beginner
https://www.youtube.com/playlist?list=PLJHszsWbB6hrkmmq57lX8BV-o-YIOFsiG
41.3.2 tensor calculus
https://www.youtube.com/playlist?list=PLJHszsWbB6hpk5h8lSfBkVrpjsqvUGTCx
41.4 Elliot Schneider
- Elliot Schneider: Physics with Elliot
41.4.1 Fundamentals of Cartesian Tensors
vector notation
\[ \vec{V}=\overrightarrow{V}=\overset{\backslash\text{harpoon}}{V}=\mathbf{V}=\boldsymbol{V} \]
\boldsymbol{} is relatively more elegant.
\del or \nabla
vector del = vector nabla
\[ \vec{\nabla}=\overrightarrow{\nabla}=\overset{\backslash\text{harpoon}}{\nabla}=\mathbf{\nabla}=\boldsymbol{\nabla} \]
scalar del = scalar nabla
\[ \nabla \]
vector field
e.g. [magnetic] vector potential
\[ \begin{aligned} \boldsymbol{A}\left(\boldsymbol{r}\right)=\vec{A}\left(\vec{r}\right)= & \boldsymbol{A}\left(\left(x_{{\scriptscriptstyle 1}},x_{{\scriptscriptstyle 2}},x_{{\scriptscriptstyle 3}}\right)\right)=\boldsymbol{A}\left(x_{{\scriptscriptstyle 1}},x_{{\scriptscriptstyle 2}},x_{{\scriptscriptstyle 3}}\right)=\begin{pmatrix}A_{{\scriptscriptstyle 1}}\left(x_{{\scriptscriptstyle 1}},x_{{\scriptscriptstyle 2}},x_{{\scriptscriptstyle 3}}\right)\\ A_{{\scriptscriptstyle 2}}\left(x_{{\scriptscriptstyle 1}},x_{{\scriptscriptstyle 2}},x_{{\scriptscriptstyle 3}}\right)\\ A_{{\scriptscriptstyle 3}}\left(x_{{\scriptscriptstyle 1}},x_{{\scriptscriptstyle 2}},x_{{\scriptscriptstyle 3}}\right) \end{pmatrix}\\ = & \boldsymbol{A}\left(\begin{pmatrix}x_{{\scriptscriptstyle 1}}\\ x_{{\scriptscriptstyle 2}}\\ x_{{\scriptscriptstyle 3}} \end{pmatrix}\right)=\boldsymbol{A}\begin{pmatrix}x_{{\scriptscriptstyle 1}}\\ x_{{\scriptscriptstyle 2}}\\ x_{{\scriptscriptstyle 3}} \end{pmatrix}=\begin{pmatrix}A_{{\scriptscriptstyle 1}}\left(\begin{pmatrix}x_{{\scriptscriptstyle 1}}\\ x_{{\scriptscriptstyle 2}}\\ x_{{\scriptscriptstyle 3}} \end{pmatrix}\right)\\ A_{{\scriptscriptstyle 2}}\left(\begin{pmatrix}x_{{\scriptscriptstyle 1}}\\ x_{{\scriptscriptstyle 2}}\\ x_{{\scriptscriptstyle 3}} \end{pmatrix}\right)\\ A_{{\scriptscriptstyle 3}}\left(\begin{pmatrix}x_{{\scriptscriptstyle 1}}\\ x_{{\scriptscriptstyle 2}}\\ x_{{\scriptscriptstyle 3}} \end{pmatrix}\right) \end{pmatrix}=\begin{pmatrix}A_{{\scriptscriptstyle 1}}\begin{pmatrix}x_{{\scriptscriptstyle 1}}\\ x_{{\scriptscriptstyle 2}}\\ x_{{\scriptscriptstyle 3}} \end{pmatrix}\\ A_{{\scriptscriptstyle 2}}\begin{pmatrix}x_{{\scriptscriptstyle 1}}\\ x_{{\scriptscriptstyle 2}}\\ x_{{\scriptscriptstyle 3}} \end{pmatrix}\\ A_{{\scriptscriptstyle 3}}\begin{pmatrix}x_{{\scriptscriptstyle 1}}\\ x_{{\scriptscriptstyle 2}}\\ x_{{\scriptscriptstyle 3}} \end{pmatrix} \end{pmatrix} \end{aligned} \]
scalar field
e.g. temperature
\[ \begin{aligned} T\left(\boldsymbol{r}\right)=T\left(\vec{r}\right)= & T\left(\left(x_{{\scriptscriptstyle 1}},x_{{\scriptscriptstyle 2}},x_{{\scriptscriptstyle 3}}\right)\right)=T\left(x_{{\scriptscriptstyle 1}},x_{{\scriptscriptstyle 2}},x_{{\scriptscriptstyle 3}}\right)\\ = & T\left(\begin{pmatrix}x_{{\scriptscriptstyle 1}}\\ x_{{\scriptscriptstyle 2}}\\ x_{{\scriptscriptstyle 3}} \end{pmatrix}\right)=T\begin{pmatrix}x_{{\scriptscriptstyle 1}}\\ x_{{\scriptscriptstyle 2}}\\ x_{{\scriptscriptstyle 3}} \end{pmatrix} \end{aligned} \]
or generally
\[ f\left(\boldsymbol{r}\right)=f\left(\vec{r}\right)=f\left(x_{{\scriptscriptstyle 1}},x_{{\scriptscriptstyle 2}},x_{{\scriptscriptstyle 3}}\right)=f\left(\cdots,x_{{\scriptscriptstyle i}},\cdots\right) \]
Definition 40.1 directional derivative
\[ \begin{aligned} \nabla_{{\scriptscriptstyle \boldsymbol{V}}}f\left(\boldsymbol{r}\right)= & \lim\limits _{\epsilon\rightarrow0}\dfrac{f\left(\boldsymbol{r}+\epsilon\boldsymbol{V}\right)-f\left(\boldsymbol{r}\right)}{\epsilon}\\ =\nabla_{{\scriptscriptstyle \vec{V}}}f\left(\vec{r}\right)= & \lim\limits _{\epsilon\rightarrow0}\dfrac{f\left(\vec{r}+\epsilon\vec{V}\right)-f\left(\vec{r}\right)}{\epsilon} \end{aligned} \]
1D directional derivative
\[ \begin{aligned} f_{{\scriptscriptstyle x}}\left(\boldsymbol{r}\right)=f^{\prime}\left(x\right)= & \begin{cases} \dfrac{\partial f}{\partial x} & f=f\left(\cdots,x,\cdots\right)\\ \dfrac{\mathrm{d}f}{\mathrm{d}x} & f=f\left(x\right) \end{cases}\\ = & \nabla_{{\scriptscriptstyle \frac{\vec{x}}{\left\Vert \vec{x}\right\Vert }}}f\left(\vec{r}\right)=\nabla_{{\scriptscriptstyle \frac{\boldsymbol{x}}{\left\Vert \boldsymbol{x}\right\Vert }}}f\left(\boldsymbol{r}\right)=\nabla_{{\scriptscriptstyle \frac{\vec{x}}{\left|\vec{x}\right|}}}f\left(\vec{r}\right)=\nabla_{{\scriptscriptstyle \frac{\boldsymbol{x}}{\left|\boldsymbol{x}\right|}}}f\left(\boldsymbol{r}\right)\\ = & \nabla_{{\scriptscriptstyle \hat{x}}}f\left(\vec{r}\right)=\nabla_{{\scriptscriptstyle \hat{\boldsymbol{x}}}}f\left(\boldsymbol{r}\right)=\begin{cases} \lim\limits _{\epsilon\rightarrow0}\dfrac{f\left(\boldsymbol{r}+\epsilon\hat{\boldsymbol{x}}\right)-f\left(\boldsymbol{r}\right)}{\epsilon} & f=f\left(\cdots,x,\cdots\right)\\ \lim\limits _{\epsilon\rightarrow0}\dfrac{f\left(x+\epsilon\right)-f\left(x\right)}{\epsilon} & f=f\left(x\right) \end{cases} \end{aligned} \]
According to \(\mathrm{d}f\)[44],
\[ \mathrm{d}f=f^{\prime}\left(x\right)\mathrm{d}x=\left(\dfrac{\mathrm{d}f}{\mathrm{d}x}\right)\mathrm{d}x=\dfrac{\mathrm{d}f}{\mathrm{d}x}\mathrm{d}x \]
is the differential of \(f\), read as “d \(f\)”.
\[ \begin{aligned} \mathrm{d}f=\left(\dfrac{\mathrm{d}f}{\mathrm{d}x}\right)\mathrm{d}x=\dfrac{\mathrm{d}f}{\mathrm{d}x}\mathrm{d}x= & \left(\dfrac{\mathrm{d}f}{\mathrm{d}x}\hat{\boldsymbol{x}}\right)\cdot\left(\mathrm{d}x\hat{\boldsymbol{x}}\right)\\ = & \boldsymbol{\nabla}_{{\scriptscriptstyle \hat{\boldsymbol{x}}}}f\left(\boldsymbol{r}\right)\cdot\mathrm{d}\hat{\boldsymbol{x}}\\ = & \boldsymbol{\nabla}f\cdot\mathrm{d}\boldsymbol{r}=\vec{\nabla}f\cdot\mathrm{d}\vec{r}\\ \overset{\text{commutative}}{=} & \mathrm{d}\boldsymbol{r}\cdot\boldsymbol{\nabla}f=\mathrm{d}\vec{r}\cdot\vec{\nabla}f \end{aligned} \]
\[ \begin{aligned} \mathrm{d}f=\mathrm{d}\boldsymbol{r}\cdot\boldsymbol{\nabla}f=\mathrm{d}\vec{r}\cdot\vec{\nabla}f \end{aligned} \]
\(\boldsymbol{\nabla}f\) is the gradient of \(f\), read as “del \(f\)”.
According to \(\mathrm{d}f\)[44],
\[ \mathrm{d}f=f^{\prime}\left(x\right)\mathrm{d}x=\left(\dfrac{\mathrm{d}f}{\mathrm{d}x}\right)\mathrm{d}x=\dfrac{\mathrm{d}f}{\mathrm{d}x}\mathrm{d}x \]
is the differential of \(f\), read as “d \(f\)”.
2D directional derivative
According to \(\mathrm{d}f\)[44],
\[ \mathrm{d}f=\dfrac{\partial f}{\partial x}\mathrm{d}x+\dfrac{\partial f}{\partial y}\mathrm{d}y \]
\[ \begin{aligned} \mathrm{d}f= & \dfrac{\partial f}{\partial x}\mathrm{d}x+\dfrac{\partial f}{\partial y}\mathrm{d}y\\ = & \left(\dfrac{\partial f}{\partial x}\hat{\boldsymbol{x}}+\dfrac{\partial f}{\partial y}\hat{\boldsymbol{y}}\right)\cdot\left(\mathrm{d}x\hat{\boldsymbol{x}}+\mathrm{d}y\hat{\boldsymbol{y}}\right)\\ = & \boldsymbol{\nabla}f\cdot\mathrm{d}\boldsymbol{r},\begin{cases} \boldsymbol{\nabla}f=\dfrac{\partial f}{\partial x}\hat{\boldsymbol{x}}+\dfrac{\partial f}{\partial y}\hat{\boldsymbol{y}}\\ \mathrm{d}\boldsymbol{r}=\mathrm{d}x\hat{\boldsymbol{x}}+\mathrm{d}y\hat{\boldsymbol{y}} \end{cases} \end{aligned} \]
example:
\(f\left(\boldsymbol{r}\right)=f\left(x_{{\scriptscriptstyle 1}},x_{{\scriptscriptstyle 2}}\right)=f\left(x,y\right)=\sqrt{x^{2}+y^{2}}\)
using Feynman method of differentiation / derivative technique[47.1],
\[ \begin{aligned} \boldsymbol{\nabla}f=\dfrac{\partial f}{\partial x}\hat{\boldsymbol{x}}+\dfrac{\partial f}{\partial y}\hat{\boldsymbol{y}}= & \dfrac{\partial\sqrt{x^{2}+y^{2}}}{\partial x}\hat{\boldsymbol{x}}+\dfrac{\partial\sqrt{x^{2}+y^{2}}}{\partial y}\hat{\boldsymbol{y}}\\ \overset{\text{Feynman}}{=} & \dfrac{\frac{1}{2}\cdot2x}{\sqrt{x^{2}+y^{2}}}\hat{\boldsymbol{x}}+\dfrac{\frac{1}{2}\cdot2y}{\sqrt{x^{2}+y^{2}}}\hat{\boldsymbol{y}}\\ = & \dfrac{x}{\sqrt{x^{2}+y^{2}}}\hat{\boldsymbol{x}}+\dfrac{y}{\sqrt{x^{2}+y^{2}}}\hat{\boldsymbol{y}}\\ = & \dfrac{x\hat{\boldsymbol{x}}+y\hat{\boldsymbol{y}}}{\sqrt{x^{2}+y^{2}}}=\dfrac{\boldsymbol{r}}{\left|\boldsymbol{r}\right|}=\frac{\boldsymbol{r}}{\left\Vert \boldsymbol{r}\right\Vert }=\frac{\vec{r}}{\left\Vert \vec{r}\right\Vert } \end{aligned} \]
geometric meaning of a gradient
\(\boldsymbol{\nabla}f\) points in the direction of “steepest ascent” of \(f\).
directional derivative: from definition to calculation
\[ \begin{aligned} \mathrm{d}f= & \mathrm{d}\boldsymbol{r}\cdot\boldsymbol{\nabla}f=\mathrm{d}\vec{r}\cdot\vec{\nabla}f\\ = & \epsilon\boldsymbol{V}\cdot\boldsymbol{\nabla}f=\epsilon\vec{V}\cdot\vec{\nabla}f \end{aligned} \]
\[ \mathrm{d}f=f\left(\boldsymbol{r}+\epsilon\boldsymbol{V}\right)-f\left(\boldsymbol{r}\right)=f\left(\vec{r}+\epsilon\vec{V}\right)-f\left(\vec{r}\right) \]
\[ \begin{aligned} & \nabla_{{\scriptscriptstyle \boldsymbol{V}}}f\left(\boldsymbol{r}\right)=\nabla_{{\scriptscriptstyle \vec{V}}}f\left(\vec{r}\right)\\ = & \lim\limits _{\epsilon\rightarrow0}\dfrac{f\left(\boldsymbol{r}+\epsilon\boldsymbol{V}\right)-f\left(\boldsymbol{r}\right)}{\epsilon}=\lim\limits _{\epsilon\rightarrow0}\dfrac{f\left(\vec{r}+\epsilon\vec{V}\right)-f\left(\vec{r}\right)}{\epsilon}\\ = & \lim\limits _{\epsilon\rightarrow0}\dfrac{\mathrm{d}f}{\epsilon}=\lim\limits _{\epsilon\rightarrow0}\dfrac{\mathrm{d}f}{\epsilon}\\ = & \lim\limits _{\epsilon\rightarrow0}\dfrac{\mathrm{d}\boldsymbol{r}\cdot\boldsymbol{\nabla}f}{\epsilon}=\lim\limits _{\epsilon\rightarrow0}\dfrac{\mathrm{d}\vec{r}\cdot\vec{\nabla}f}{\epsilon}\\ = & \lim\limits _{\epsilon\rightarrow0}\dfrac{\epsilon\boldsymbol{V}\cdot\boldsymbol{\nabla}f}{\epsilon}=\lim\limits _{\epsilon\rightarrow0}\dfrac{\epsilon\vec{V}\cdot\vec{\nabla}f}{\epsilon}\\ = & \lim\limits _{\epsilon\rightarrow0}\dfrac{\boldsymbol{V}\cdot\boldsymbol{\nabla}f}{1}=\lim\limits _{\epsilon\rightarrow0}\dfrac{\vec{V}\cdot\vec{\nabla}f}{1}\\ = & \boldsymbol{V}\cdot\boldsymbol{\nabla}f=\vec{V}\cdot\vec{\nabla}f \end{aligned} \]
\[ \nabla_{{\scriptscriptstyle \boldsymbol{V}}}f=\nabla_{{\scriptscriptstyle \boldsymbol{V}}}f\left(\boldsymbol{r}\right)=\boldsymbol{V}\cdot\boldsymbol{\nabla}f=\nabla_{{\scriptscriptstyle \vec{V}}}f=\nabla_{{\scriptscriptstyle \vec{V}}}f\left(\vec{r}\right)=\vec{V}\cdot\vec{\nabla}f \]
\[ \begin{aligned} \nabla_{{\scriptscriptstyle \boldsymbol{V}}}f= & \boldsymbol{V}\cdot\boldsymbol{\nabla}f\\ =\nabla_{{\scriptscriptstyle \vec{V}}}f= & \vec{V}\cdot\vec{\nabla}f \end{aligned} \]
in the view of operator acting on scalar field(s)
\[ \boldsymbol{V}=V_{{\scriptscriptstyle x}}\hat{\boldsymbol{x}}+V_{{\scriptscriptstyle y}}\hat{\boldsymbol{y}} \]
grad or gradient = vector del or vector nabla as a vector operator acting on scalar field(s)
\[ \begin{aligned} \boldsymbol{\nabla}f=\vec{\nabla}f= & \dfrac{\partial f}{\partial x}\hat{\boldsymbol{x}}+\dfrac{\partial f}{\partial y}\hat{\boldsymbol{y}}=\nabla_{{\scriptscriptstyle \hat{\boldsymbol{x}}}}f\hat{\boldsymbol{x}}+\nabla_{{\scriptscriptstyle \hat{\boldsymbol{y}}}}f\hat{\boldsymbol{y}}\\ = & \hat{\boldsymbol{x}}\dfrac{\partial f}{\partial x}+\hat{\boldsymbol{y}}\dfrac{\partial f}{\partial y}=\hat{\boldsymbol{x}}\nabla_{{\scriptscriptstyle \hat{\boldsymbol{x}}}}f+\hat{\boldsymbol{y}}\nabla_{{\scriptscriptstyle \hat{\boldsymbol{y}}}}f\\ = & \left(\hat{\boldsymbol{x}}\dfrac{\partial}{\partial x}+\hat{\boldsymbol{y}}\dfrac{\partial}{\partial y}\right)f=\left(\hat{\boldsymbol{x}}\nabla_{{\scriptscriptstyle \hat{\boldsymbol{x}}}}+\hat{\boldsymbol{y}}\nabla_{{\scriptscriptstyle \hat{\boldsymbol{y}}}}\right)f\\ \Downarrow\\ \boldsymbol{\nabla}=\vec{\nabla}= & \hat{\boldsymbol{x}}\dfrac{\partial}{\partial x}+\hat{\boldsymbol{y}}\dfrac{\partial}{\partial y}=\hat{\boldsymbol{x}}\nabla_{{\scriptscriptstyle \hat{\boldsymbol{x}}}}+\hat{\boldsymbol{y}}\nabla_{{\scriptscriptstyle \hat{\boldsymbol{y}}}} \end{aligned} \]
directional derivative = scalar del or scalar nabla as a scalar operator acting on scalar field(s)
\[ \begin{aligned} \nabla_{{\scriptscriptstyle \boldsymbol{V}}}f=\nabla_{{\scriptscriptstyle \vec{V}}}f= & \boldsymbol{V}\cdot\boldsymbol{\nabla}f=\vec{V}\cdot\vec{\nabla}f\\ = & \left(V_{{\scriptscriptstyle x}}\hat{\boldsymbol{x}}+V_{{\scriptscriptstyle y}}\hat{\boldsymbol{y}}\right)\cdot\left(\nabla_{{\scriptscriptstyle \hat{\boldsymbol{x}}}}f\hat{\boldsymbol{x}}+\nabla_{{\scriptscriptstyle \hat{\boldsymbol{y}}}}f\hat{\boldsymbol{y}}\right)\\ = & V_{{\scriptscriptstyle x}}\nabla_{{\scriptscriptstyle \hat{\boldsymbol{x}}}}f+V_{{\scriptscriptstyle y}}\nabla_{{\scriptscriptstyle \hat{\boldsymbol{y}}}}f=\left(V_{{\scriptscriptstyle x}}\nabla_{{\scriptscriptstyle \hat{\boldsymbol{x}}}}+V_{{\scriptscriptstyle y}}\nabla_{{\scriptscriptstyle \hat{\boldsymbol{y}}}}\right)f\\ = & V_{{\scriptscriptstyle x}}\dfrac{\partial f}{\partial x}+V_{{\scriptscriptstyle y}}\dfrac{\partial f}{\partial y}=\left(V_{{\scriptscriptstyle x}}\dfrac{\partial}{\partial x}+V_{{\scriptscriptstyle y}}\dfrac{\partial}{\partial y}\right)f\\ \Downarrow\\ \nabla_{{\scriptscriptstyle \boldsymbol{V}}}=\nabla_{{\scriptscriptstyle \vec{V}}}= & V_{{\scriptscriptstyle x}}\nabla_{{\scriptscriptstyle \hat{\boldsymbol{x}}}}+V_{{\scriptscriptstyle y}}\nabla_{{\scriptscriptstyle \hat{\boldsymbol{y}}}}=V_{{\scriptscriptstyle x}}\dfrac{\partial}{\partial x}+V_{{\scriptscriptstyle y}}\dfrac{\partial}{\partial y} \end{aligned} \]
product rule
\[ \begin{aligned} \nabla_{{\scriptscriptstyle \boldsymbol{V}}}\left(fg\right)= & \nabla_{{\scriptscriptstyle \vec{V}}}\left(fg\right)=\boldsymbol{V}\cdot\boldsymbol{\nabla}\left(fg\right)=\vec{V}\cdot\vec{\nabla}\left(fg\right)\\ = & \boldsymbol{V}\cdot\left[\left(\boldsymbol{\nabla}f\right)g+f\boldsymbol{\nabla}g\right]=\boldsymbol{V}\cdot\left[g\boldsymbol{\nabla}f+f\boldsymbol{\nabla}g\right]=\boldsymbol{V}\cdot\left[f\boldsymbol{\nabla}g+g\boldsymbol{\nabla}f\right]\\ = & f\left(\boldsymbol{V}\cdot\boldsymbol{\nabla}g\right)+g\left(\boldsymbol{V}\cdot\boldsymbol{\nabla}f\right)=g\left(\boldsymbol{V}\cdot\boldsymbol{\nabla}f\right)+f\left(\boldsymbol{V}\cdot\boldsymbol{\nabla}g\right)\\ & =f\left(\nabla_{{\scriptscriptstyle \boldsymbol{V}}}g\right)+g\left(\nabla_{{\scriptscriptstyle \boldsymbol{V}}}f\right)=g\left(\nabla_{{\scriptscriptstyle \boldsymbol{V}}}f\right)+f\left(\nabla_{{\scriptscriptstyle \boldsymbol{V}}}g\right)\\ = & f\boldsymbol{V}\cdot\boldsymbol{\nabla}g+g\boldsymbol{V}\cdot\boldsymbol{\nabla}f=g\boldsymbol{V}\cdot\boldsymbol{\nabla}f+f\boldsymbol{V}\cdot\boldsymbol{\nabla}g\\ & =f\nabla_{{\scriptscriptstyle \boldsymbol{V}}}g+g\nabla_{{\scriptscriptstyle \boldsymbol{V}}}f=g\nabla_{{\scriptscriptstyle \boldsymbol{V}}}f+f\nabla_{{\scriptscriptstyle \boldsymbol{V}}}g\\ \nabla_{{\scriptscriptstyle \boldsymbol{V}}}\left(fg\right)= & f\nabla_{{\scriptscriptstyle \boldsymbol{V}}}g+g\nabla_{{\scriptscriptstyle \boldsymbol{V}}}f \end{aligned} \]
two kinds of linearities or bilinearity of directional derivative operator
\[ \begin{aligned} \nabla_{{\scriptscriptstyle \boldsymbol{V}}}\left(f+\lambda g\right)= & \nabla_{{\scriptscriptstyle \vec{V}}}\left(f+\lambda g\right)=\boldsymbol{V}\cdot\boldsymbol{\nabla}\left(f+\lambda g\right)=\vec{V}\cdot\vec{\nabla}\left(f+\lambda g\right)\\ = & \boldsymbol{V}\cdot\left[\boldsymbol{\nabla}f+\boldsymbol{\nabla}\left(\lambda g\right)\right]=\boldsymbol{V}\cdot\left[\boldsymbol{\nabla}f+\lambda\boldsymbol{\nabla}g\right]\\ = & \boldsymbol{V}\cdot\boldsymbol{\nabla}f+\lambda\left(\boldsymbol{V}\cdot\boldsymbol{\nabla}g\right)\\ = & \nabla_{{\scriptscriptstyle \boldsymbol{V}}}f+\lambda\nabla_{{\scriptscriptstyle \boldsymbol{V}}}g\\ \nabla_{{\scriptscriptstyle \boldsymbol{V}}}\left(f+\lambda g\right)= & \nabla_{{\scriptscriptstyle \boldsymbol{V}}}f+\lambda\nabla_{{\scriptscriptstyle \boldsymbol{V}}}g \end{aligned} \]
\[ \begin{aligned} \nabla_{{\scriptscriptstyle \boldsymbol{V}+\lambda\boldsymbol{W}}}f= & \nabla_{{\scriptscriptstyle \vec{V}+\lambda\vec{W}}}f=\left(\boldsymbol{V}+\lambda\boldsymbol{W}\right)\cdot\boldsymbol{\nabla}f=\left(\vec{V}+\lambda\vec{W}\right)\cdot\vec{\nabla}f\\ = & \boldsymbol{V}\cdot\boldsymbol{\nabla}f+\lambda\boldsymbol{W}\cdot\boldsymbol{\nabla}f=\vec{V}\cdot\vec{\nabla}f+\lambda\vec{W}\cdot\vec{\nabla}f\\ = & \nabla_{{\scriptscriptstyle \boldsymbol{V}}}f+\lambda\nabla_{{\scriptscriptstyle \boldsymbol{W}}}f=\nabla_{{\scriptscriptstyle \vec{V}}}f+\lambda\nabla_{{\scriptscriptstyle \vec{W}}}f\\ \nabla_{{\scriptscriptstyle \boldsymbol{V}+\lambda\boldsymbol{W}}}f= & \nabla_{{\scriptscriptstyle \boldsymbol{V}}}f+\lambda\nabla_{{\scriptscriptstyle \boldsymbol{W}}}f \end{aligned} \]
\[ \begin{cases} \nabla_{{\scriptscriptstyle \boldsymbol{V}}}\left(f+\lambda g\right) & =\nabla_{{\scriptscriptstyle \boldsymbol{V}}}f+\lambda\nabla_{{\scriptscriptstyle \boldsymbol{V}}}g\\ \nabla_{{\scriptscriptstyle \boldsymbol{V}+\lambda\boldsymbol{W}}}f & =\nabla_{{\scriptscriptstyle \boldsymbol{V}}}f+\lambda\nabla_{{\scriptscriptstyle \boldsymbol{W}}}f \end{cases} \]
\[ \begin{cases} \nabla_{{\scriptscriptstyle \boldsymbol{V}}}\left(f+\lambda g\right)=\nabla_{{\scriptscriptstyle \boldsymbol{V}}}f+\lambda\nabla_{{\scriptscriptstyle \boldsymbol{V}}}g & \Leftrightarrow\nabla_{{\scriptscriptstyle \boldsymbol{V}}}\left(\lambda f+\mu g\right)=\lambda\nabla_{{\scriptscriptstyle \boldsymbol{V}}}f+\mu\nabla_{{\scriptscriptstyle \boldsymbol{V}}}g\\ \nabla_{{\scriptscriptstyle \boldsymbol{V}+\lambda\boldsymbol{W}}}f=\nabla_{{\scriptscriptstyle \boldsymbol{V}}}f+\lambda\nabla_{{\scriptscriptstyle \boldsymbol{W}}}f & \Leftrightarrow\nabla_{{\scriptscriptstyle \lambda\boldsymbol{V}+\mu\boldsymbol{W}}}f=\lambda\nabla_{{\scriptscriptstyle \boldsymbol{V}}}f+\mu\nabla_{{\scriptscriptstyle \boldsymbol{W}}}f \end{cases} \]
\[ \begin{aligned} \nabla_{{\scriptscriptstyle \lambda\boldsymbol{V}+\mu\boldsymbol{W}}}f= & \nabla_{{\scriptscriptstyle \lambda\boldsymbol{V}}}f+\nabla_{{\scriptscriptstyle \mu\boldsymbol{W}}}f\\ = & \lambda\nabla_{{\scriptscriptstyle \boldsymbol{V}}}f+\mu\nabla_{{\scriptscriptstyle \boldsymbol{W}}}f\\ = & \lambda\dfrac{\partial}{\partial V}f+\mu\dfrac{\partial}{\partial W}f\\ = & \lambda\dfrac{\partial f}{\partial V}+\mu\dfrac{\partial f}{\partial W}\\ \nabla_{{\scriptscriptstyle \boldsymbol{V}}}f=\nabla_{{\scriptscriptstyle V_{{\scriptscriptstyle x}}\hat{\boldsymbol{x}}+V_{{\scriptscriptstyle y}}\hat{\boldsymbol{y}}}}f= & \nabla_{{\scriptscriptstyle V_{{\scriptscriptstyle x}}\hat{\boldsymbol{x}}}}f+\nabla_{{\scriptscriptstyle V_{{\scriptscriptstyle y}}\hat{\boldsymbol{y}}}}f\\ = & V_{{\scriptscriptstyle x}}\nabla_{{\scriptscriptstyle \hat{\boldsymbol{x}}}}f+V_{{\scriptscriptstyle y}}\nabla_{{\scriptscriptstyle \hat{\boldsymbol{y}}}}f\\ = & V_{{\scriptscriptstyle x}}\dfrac{\partial}{\partial x}f+V_{{\scriptscriptstyle y}}\dfrac{\partial}{\partial y}f\\ = & V_{{\scriptscriptstyle x}}\dfrac{\partial f}{\partial x}+V_{{\scriptscriptstyle y}}\dfrac{\partial f}{\partial y} \end{aligned} \]
directional derivative of a 3D vector field
\[ \begin{aligned} \nabla_{{\scriptscriptstyle \boldsymbol{V}}}\boldsymbol{A}=\nabla_{{\scriptscriptstyle \vec{V}}}\vec{A}= & \boldsymbol{V}\cdot\boldsymbol{\nabla}\boldsymbol{A}=\vec{V}\cdot\vec{\nabla}\vec{A}\\ = & \left(\hat{\boldsymbol{x}}V_{{\scriptscriptstyle x}}+\hat{\boldsymbol{y}}V_{{\scriptscriptstyle y}}+\hat{\boldsymbol{z}}V_{{\scriptscriptstyle z}}\right)\cdot\left(\hat{\boldsymbol{x}}\dfrac{\partial\boldsymbol{A}}{\partial x}+\hat{\boldsymbol{y}}\dfrac{\partial\boldsymbol{A}}{\partial x}+\hat{\boldsymbol{z}}\dfrac{\partial\boldsymbol{A}}{\partial x}\right)\\ & =\left(\hat{\boldsymbol{x}}V_{{\scriptscriptstyle x}}+\hat{\boldsymbol{y}}V_{{\scriptscriptstyle y}}+\hat{\boldsymbol{z}}V_{{\scriptscriptstyle z}}\right)\cdot\left(\hat{\boldsymbol{x}}\dfrac{\partial\vec{A}}{\partial x}+\hat{\boldsymbol{y}}\dfrac{\partial\vec{A}}{\partial x}+\hat{\boldsymbol{z}}\dfrac{\partial\vec{A}}{\partial x}\right)\\ \boldsymbol{\nabla}\boldsymbol{A}=\vec{\nabla}\vec{A}= & \hat{\boldsymbol{x}}\dfrac{\partial\boldsymbol{A}}{\partial x}+\hat{\boldsymbol{y}}\dfrac{\partial\boldsymbol{A}}{\partial x}+\hat{\boldsymbol{z}}\dfrac{\partial\boldsymbol{A}}{\partial x}=\hat{\boldsymbol{x}}\dfrac{\partial\vec{A}}{\partial x}+\hat{\boldsymbol{y}}\dfrac{\partial\vec{A}}{\partial x}+\hat{\boldsymbol{z}}\dfrac{\partial\vec{A}}{\partial x}\\ = & \hat{\boldsymbol{x}}\dfrac{\partial\left(\hat{\boldsymbol{x}}A_{{\scriptscriptstyle x}}+\hat{\boldsymbol{y}}A_{{\scriptscriptstyle y}}+\hat{\boldsymbol{z}}A_{{\scriptscriptstyle z}}\right)}{\partial x}\\ + & \hat{\boldsymbol{y}}\dfrac{\partial\left(\hat{\boldsymbol{x}}A_{{\scriptscriptstyle x}}+\hat{\boldsymbol{y}}A_{{\scriptscriptstyle y}}+\hat{\boldsymbol{z}}A_{{\scriptscriptstyle z}}\right)}{\partial x}\\ + & \hat{\boldsymbol{z}}\dfrac{\partial\left(\hat{\boldsymbol{x}}A_{{\scriptscriptstyle x}}+\hat{\boldsymbol{y}}A_{{\scriptscriptstyle y}}+\hat{\boldsymbol{z}}A_{{\scriptscriptstyle z}}\right)}{\partial x}\\ = & \hat{\boldsymbol{x}}\hat{\boldsymbol{x}}\dfrac{\partial A_{{\scriptscriptstyle x}}}{\partial x}+\hat{\boldsymbol{x}}\hat{\boldsymbol{y}}\dfrac{\partial A_{{\scriptscriptstyle y}}}{\partial x}+\hat{\boldsymbol{x}}\hat{\boldsymbol{z}}\dfrac{\partial A_{{\scriptscriptstyle z}}}{\partial x}\\ + & \hat{\boldsymbol{y}}\hat{\boldsymbol{x}}\dfrac{\partial A_{{\scriptscriptstyle x}}}{\partial x}+\hat{\boldsymbol{y}}\hat{\boldsymbol{y}}\dfrac{\partial A_{{\scriptscriptstyle y}}}{\partial x}+\hat{\boldsymbol{y}}\hat{\boldsymbol{z}}\dfrac{\partial A_{{\scriptscriptstyle z}}}{\partial x}\\ + & \hat{\boldsymbol{z}}\hat{\boldsymbol{x}}\dfrac{\partial A_{{\scriptscriptstyle x}}}{\partial x}+\hat{\boldsymbol{z}}\hat{\boldsymbol{y}}\dfrac{\partial A_{{\scriptscriptstyle y}}}{\partial x}+\hat{\boldsymbol{z}}\hat{\boldsymbol{z}}\dfrac{\partial A_{{\scriptscriptstyle z}}}{\partial x}\\ = & \hat{\boldsymbol{x}}\otimes\hat{\boldsymbol{x}}\dfrac{\partial A_{{\scriptscriptstyle x}}}{\partial x}+\hat{\boldsymbol{x}}\otimes\hat{\boldsymbol{y}}\dfrac{\partial A_{{\scriptscriptstyle y}}}{\partial x}+\hat{\boldsymbol{x}}\otimes\hat{\boldsymbol{z}}\dfrac{\partial A_{{\scriptscriptstyle z}}}{\partial x}\\ + & \hat{\boldsymbol{y}}\otimes\hat{\boldsymbol{x}}\dfrac{\partial A_{{\scriptscriptstyle x}}}{\partial x}+\hat{\boldsymbol{y}}\otimes\hat{\boldsymbol{y}}\dfrac{\partial A_{{\scriptscriptstyle y}}}{\partial x}+\hat{\boldsymbol{y}}\otimes\hat{\boldsymbol{z}}\dfrac{\partial A_{{\scriptscriptstyle z}}}{\partial x}\\ + & \hat{\boldsymbol{z}}\otimes\hat{\boldsymbol{x}}\dfrac{\partial A_{{\scriptscriptstyle x}}}{\partial x}+\hat{\boldsymbol{z}}\otimes\hat{\boldsymbol{y}}\dfrac{\partial A_{{\scriptscriptstyle y}}}{\partial x}+\hat{\boldsymbol{z}}\otimes\hat{\boldsymbol{z}}\dfrac{\partial A_{{\scriptscriptstyle z}}}{\partial x} \end{aligned} \]
\(\boldsymbol{\nabla}\boldsymbol{A}=\vec{\nabla}\vec{A}\) is a rank-2 tensor.
\(\otimes\) is tensor product.
\(\nabla_{{\scriptscriptstyle \boldsymbol{V}}}\boldsymbol{A}=\nabla_{{\scriptscriptstyle \vec{V}}}\vec{A}=\boldsymbol{V}\cdot\boldsymbol{\nabla}\boldsymbol{A}=\vec{V}\cdot\vec{\nabla}\vec{A}\) yields a vector, instead of a scalar.
Dan Fleisch: What’s a Tensor?
https://www.youtube.com/watch?v=f5liqUk0ZTw
https://www.youtube.com/watch?v=YxXyN2ifK8A&list=PL2aHrV9pFqNTEMuDFre16Wx2SwBCNiR7j&index=1
https://www.youtube.com/watch?v=A95jdIuUUW0&list=PL2aHrV9pFqNTEMuDFre16Wx2SwBCNiR7j&index=2
https://www.youtube.com/watch?v=51ARho2bvQY&list=PL2aHrV9pFqNTEMuDFre16Wx2SwBCNiR7j&index=3
## (-8.0, 8.0, -8.0, 8.0)
## (-8.0, 8.0, -8.0, 8.0)
## (-8.0, 8.0, -8.0, 8.0)
## (-8.0, 8.0, -8.0, 8.0)
## (-8.0, 8.0, -8.0, 8.0)
## (-8.0, 8.0, -8.0, 8.0)
## (-8.0, 8.0, -8.0, 8.0)
## (-8.0, 8.0, -8.0, 8.0)
## (-8.0, 8.0, -8.0, 8.0)
import numpy as np
import matplotlib.pyplot as plt
A = np.array([[ 0,-1],
[ 1, 0]])
B = np.array([[ 1, 0],
[ 0, 1]])
C = np.array([[ 1,-1],
[ 1, 1]])
D = np.array([[ 1, 1],
[ 1, 1]])
E = np.array([[-1, 0],
[ 0,-1]])
F = np.array([[ 3, 1],
[-2, 2]])
t = np.arange(0, 2 * np.pi, np.pi/5)
# Creating arrow
X = np.cos(t)
Y = np.sin(t)
# X, Y = np.meshgrid(x, y)
# V = np.array([X, Y])
# u = x
# v = y
# u, v = np.matmul(A, V)
u, v = np.matmul(A, np.array([X, Y]))
dx, dy = np.matmul(B, np.array([X, Y]))
dx2, dy2 = np.matmul(C, np.array([X, Y]))
dx3, dy3 = np.matmul(D, np.array([X, Y]))
dx4, dy4 = np.matmul(E, np.array([X, Y]))
dx5, dy5 = np.matmul(F, np.array([X, Y]))
# n = -2
# Defining color
# color = np.sqrt(((v-n)/2)*2 + ((u-n)/2)*2)
# color = (u**2+v**2)**(1/2)
color = u*v
# Creating plot
fig, ax = plt.subplots(3,3,figsize =(15,15), gridspec_kw = {'wspace':0.1, 'hspace':0.1})
# ax.quiver(X, Y, u, v, color, alpha = 1)
for i in range(-2,3):
for j in range(-2,3):
# ax.quiver(1.0*X+4*i, 1.0*Y+4*j, u, v)#, color, alpha = 1)
# ax.quiver(1.5*X+4*i, 1.5*Y+4*j, u, v)#, color, alpha = 1)
# ax.quiver(2.0*X+4*i, 2.0*Y+4*j, u, v)#, color, alpha = 1)
ax[0,0].quiver(0.5*X+4*i, 0.5*Y+4*j, u, v, u*v, alpha = 0.50, cmap='spring')
ax[0,0].quiver(1.0*X+4*i, 1.0*Y+4*j, u, v, u*v, alpha = 1.00, cmap='spring')
ax[0,0].quiver(1.5*X+4*i, 1.5*Y+4*j, u, v, u*v, alpha = 0.25, cmap='spring')
ax[0,0].quiver(2.0*X+4*i, 2.0*Y+4*j, u, v, u*v, alpha = 0.10, cmap='spring')
ax[0,1].quiver(0.5*X+4*i, 0.5*Y+4*j, dx, dy, dx*dy, alpha = 1.00, cmap='summer')
# ax[0,1].quiver(1.0*X+4*i, 1.0*Y+4*j, dx, dy, dx*dy, alpha = 0.50, cmap='summer')
ax[0,1].quiver(1.5*X+4*i, 1.5*Y+4*j, dx, dy, dx*dy, alpha = 0.25, cmap='summer')
# ax[0,1].quiver(2.0*X+4*i, 2.0*Y+4*j, dx, dy, dx*dy, alpha = 0.10, cmap='summer')
ax[1,0].quiver(0.5*X+4*i, 0.5*Y+4*j, dx2, dy2, dx2*dy2, alpha = 1.00, cmap='winter')
ax[1,0].quiver(1.0*X+4*i, 1.0*Y+4*j, dx2, dy2, dx2*dy2, alpha = 0.50, cmap='winter')
ax[1,0].quiver(1.5*X+4*i, 1.5*Y+4*j, dx2, dy2, dx2*dy2, alpha = 0.25, cmap='winter')
ax[1,0].quiver(2.0*X+4*i, 2.0*Y+4*j, dx2, dy2, dx2*dy2, alpha = 0.10, cmap='winter')
# ax[2,0].quiver(0.5*X+4*i, 0.5*Y+4*j, dx3, dy3, dx3*dy3, alpha = 0.50, cmap='cool')
ax[2,0].quiver(1.0*X+4*i, 1.0*Y+4*j, dx3, dy3, dx3*dy3, alpha = 1.00, cmap='cool')
# ax[2,0].quiver(1.5*X+4*i, 1.5*Y+4*j, dx3, dy3, dx3*dy3, alpha = 0.25, cmap='cool')
ax[2,0].quiver(2.0*X+4*i, 2.0*Y+4*j, dx3, dy3, dx3*dy3, alpha = 0.10, cmap='cool')
# ax[0,2].quiver(0.5*X+4*i, 0.5*Y+4*j, dx4, dy4, dx4*dy4, alpha = 0.50)
# ax[0,2].quiver(1.0*X+4*i, 1.0*Y+4*j, dx4, dy4, dx4*dy4, alpha = 0.25)
ax[0,2].quiver(1.5*X+4*i, 1.5*Y+4*j, dx4, dy4, dx4*dy4, alpha = 1.00)
ax[0,2].quiver(2.0*X+4*i, 2.0*Y+4*j, dx4, dy4, dx4*dy4, alpha = 0.10)
ax[1,2].quiver(0.5*X+4*i, 0.5*Y+4*j, dx5, dy5, dx5*dy5, alpha = 0.50)
ax[1,2].quiver(1.0*X+4*i, 1.0*Y+4*j, dx5, dy5, dx5*dy5, alpha = 1.00)
ax[1,2].quiver(1.5*X+4*i, 1.5*Y+4*j, dx5, dy5, dx5*dy5, alpha = 0.25)
ax[1,2].quiver(2.0*X+4*i, 2.0*Y+4*j, dx5, dy5, dx5*dy5, alpha = 0.10)
ax[1,1].quiver(0.5*X+4*i, 0.5*Y+4*j, u, v, u*v, alpha = 1.00, cmap='spring')
ax[1,1].quiver(1.0*X+4*i, 1.0*Y+4*j, dx, dy, dx*dy, alpha = 1.00, cmap='summer')
ax[1,1].quiver(1.0*X+4*i, 1.0*Y+4*j, dx2, dy2, dx2*dy2, alpha = 1.00, cmap='winter')
ax[2,1].quiver(0.5*X+4*i, 0.5*Y+4*j, u, v, u*v, alpha = 1.00, cmap='spring')
ax[2,1].quiver(1.0*X+4*i, 1.0*Y+4*j, dx, dy, dx*dy, alpha = 1.00, cmap='summer')
ax[2,1].quiver(1.0*X+4*i, 1.0*Y+4*j, dx2, dy2, dx2*dy2, alpha = 1.00, cmap='winter')
ax[2,1].quiver(1.0*X+4*i, 1.0*Y+4*j, dx3, dy3, dx3*dy3, alpha = 1.00, cmap='cool')
for ax in fig.get_axes():
# ax.xaxis.set_ticks([])
# ax.yaxis.set_ticks([])
ax.axis([-8, 8, -8, 8])
ax.set_aspect('equal')
# show plot
plt.show()
## (-8.0, 8.0, -8.0, 8.0)
## (-8.0, 8.0, -8.0, 8.0)
## (-8.0, 8.0, -8.0, 8.0)
## (-8.0, 8.0, -8.0, 8.0)
## (-8.0, 8.0, -8.0, 8.0)
## (-8.0, 8.0, -8.0, 8.0)
## (-8.0, 8.0, -8.0, 8.0)
## (-8.0, 8.0, -8.0, 8.0)
## (-8.0, 8.0, -8.0, 8.0)