https://www.coursera.org/learn/vector-calculus-engineers#modules
Text book: https://www.math.hkust.edu.hk/~machas/vector-calculus-for-engineers.pdf
或實體書: https://www.amazon.com/Vector-Calculus-Engineers-Mathematics/dp/B0BMJMRYBH/
| 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 | $\nabla\cdot\nabla \vec{f}= | |
| \left[\begin{array}{c} | |||
| \nabla\cdot\nabla f_1 \\ | |||
| \nabla\cdot\nabla f_2 | |||
| \end{array}\right] = | |||
| \left[\begin{array}{c} | |||
| \frac{\partial^2 f_1}{\partial x^2} + \frac{\partial^2 f_1}{\partial y^2}\\ | |||
| \frac{\partial^2 f_2}{\partial x^2} + \frac{\partial^2 f_2}{\partial y^2} | |||
| \end{array}\right]$ |
Week 3 Integration and Curvilinear Coordinates