Cover figure is from: https://arxiv.org/abs/1209.1793

此為 https://www.youtube.com/watch?v=Jt5R-Tm8cV8&list=PLMrJAkhIeNNQromC4WswpU1krLOq5Ro6S 的課程筆記

Some Additional Reference Materials:

Change of Variables Theorem and Surface Integral

Input field Output field Formula 二維舉例
Gradient $\nabla$ scalar field
$f:\mathbb{R}^n\rightarrow\mathbb{R}$ vector field $\nabla f=\left[\begin{array}{c}
\partial f/\partial x\\
\partial f/\partial y
\end{array}\right]$
Divergence $\nabla\cdot$ vector field
$\vec{f}:\mathbb{R}^n\rightarrow\mathbb{R}^n$ scalar field $\nabla\cdot f=\left[\begin{array}{c}
\partial /\partial x\\
\partial /\partial y
\end{array}\right] \cdot
\left[\begin{array}{c}
f_1\\
f_2
\end{array}\right]=\frac{\partial f_1}{\partial x}+\frac{\partial f_2}{\partial y}$
Curl $\nabla\times$ vector field
$\vec{f}:\mathbb{R}^n\rightarrow\mathbb{R}^n$ vector field
$\nabla\times\vec{f}=\left
\begin{array}{cc}
\vec{i} && \vec{j} && \vec{k} \\
\partial/\partial x && \partial/\partial y && \partial/\partial z \\
f_1(x,y,z) && f_2(x,y,z) && f_3(x,y,z)
\end{array}
\right \\
=\left(\frac{\partial f_3}{\partial y}-\frac{\partial f_2}{\partial z}\right)\vec{i}
-\left(\frac{\partial f_3}{\partial x}-\frac{\partial f_1}{\partial z}\right)\vec{j}
+\left(\frac{\partial f_2}{\partial x}-\frac{\partial f_1}{\partial y}\right)\vec{k}$
Laplacian $\nabla\cdot\nabla$
或寫成 $\nabla^2$ scalar field
$f:\mathbb{R}^n\rightarrow\mathbb{R}$ scalar field $\nabla\cdot\nabla f=
\left[\begin{array}{c}
\partial /\partial x\\
\partial /\partial y
\end{array}\right] \cdot
\left[\begin{array}{c}
\partial f/\partial x\\
\partial f/\partial y
\end{array}\right] = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} = \nabla^2 f$
vector field
$\vec{f}:\mathbb{R}^n\rightarrow\mathbb{R}^n$ vector field $\nabla\cdot\nabla \vec{f}=
\left[\begin{array}{c}
\nabla\cdot\nabla f_1 \\
\nabla\cdot\nabla f_2
\end{array}\right] =
\left[\begin{array}{c}
\frac{\partial^2 f_1}{\partial x^2} + \frac{\partial^2 f_1}{\partial y^2}\\
\frac{\partial^2 f_2}{\partial x^2} + \frac{\partial^2 f_2}{\partial y^2}
\end{array}\right]$

Steve Brunton Vector Calculus and PDE 筆記

(1/23) Vector Calculus and Partial Differential Equations: Big Picture Overview

(2/23) Div, Grad, and Curl: Vector Calculus Building Blocks for PDEs

(3/23) The Gradient Operator in Vector Calculus: Directions of Fastest Change & the Directional Derivative

(4/23) The Divergence of a Vector Field: Sources and Sinks

(5/23) The Curl of a Vector Field: Measuring Rotation

(6/23) Gauss's Divergence Theorem

(7/23) The Continuity Equation: A PDE for Mass Conservation, from Gauss's Divergence Theorem