Mathematical Methods for Physics and Engineering

1. Some conclusions:

(1) Ratio theorem:

Line segment AB is divided into AP and PB. AP:PB = $\lambda:\mu$. Then $\mathbf{OP}=\frac{\mu}{\lambda+\mu} \mathbf{OA}+\frac{\lambda}{\lambda+\mu} \mathbf{OB}$.

(2) Centroid G of a triangle:

$$\mathbf{g}=\frac{1}{3}(\mathbf{a}+\mathbf{b}+\mathbf{c})$$.

(3) The cosine of the angle $\theta$ between a and b:

$$\cos\theta=\frac{a_{x}}{a}\frac{b_{x}}{b}+\frac{a_{y}}{a}\frac{b_{y}}{b}+\frac{a_{z}}{a}\frac{b_{z}}{b}$$.

(4) Complex vector is possible:

$\mathbf{a}\cdot\mathbf{b}=a_x^*b_x+a_y^*b_y+a_z^*b_z$ and $\mathbf{a}\cdot\mathbf{b}=(\mathbf{b}\cdot\mathbf{a})^{*}$.

(5) Vector product is non-associative:

$(\mathbf{a}\times\mathbf{b})\times\mathbf{c}\neq\mathbf{a}\times(\mathbf{b}\times\mathbf{c})$.

(6) Scalar triple product:

a. $[\mathbf{a},\mathbf{b},\mathbf{c}]\equiv\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})\equiv(\mathbf{a}\times\mathbf{b})\cdot\mathbf{c}\equiv|\mathbf{a}||\mathbf{b}||\mathbf{c}|\sin\theta\cos\phi$

b. Especially for real vector: $\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})\equiv\mathbf{c}\cdot(\mathbf{a}\times\mathbf{b})$.

(7) Vector triple product:

a. $\mathbf{a}\times(\mathbf{b}\times\mathbf{c})=(\mathbf{a}\cdot\mathbf{c})\mathbf{b}-(\mathbf{a}\cdot\mathbf{b})\mathbf{c}$

b. $(\mathbf{a}\times\mathbf{b})\times\mathbf{c}=(\mathbf{a}\cdot\mathbf{c})\mathbf{b}-(\mathbf{b}\cdot\mathbf{c})\mathbf{a}$

c. $\mathbf{a}\times(\mathbf{b}\times\mathbf{c})+\mathbf{b}\times(\mathbf{c}\times\mathbf{a})+\mathbf{c}\times(\mathbf{a}\times\mathbf{b})=0$

(8) Lagrange’s identity:

$(\mathbf{a}\times\mathbf{b})\cdot(\mathbf{c}\times\mathbf{d})\equiv(\mathbf{a}\cdot\mathbf{c})(\mathbf{b}\cdot\mathbf{d})-(\mathbf{a}\cdot\mathbf{d})(\mathbf{b}\cdot\mathbf{c})$.

2. Definition of basis vectors:

The basis vectors must satisfy two conditions:

(1) The number of basis vectors is the same as the dimensions.

(2) The basis vectors must be linear independent, namely:

$c_1\mathbf{e}_1+c_2\mathbf{e}_2+\cdots+c_N\mathbf{e}_N\neq\mathbf{0}$ except $c_1=c_2=\cdots=c_N=0$.

3. Equations of lines, planes and spheres:

(1) Lines:

a. $\mathbf{r}=\mathbf{a}+\lambda\mathbf{b}$: $\mathbf{a}$ is its start point. By manipulating $\lambda$, it becomes a line having the same direction as $\mathbf{b}$.

b. $\frac{x-a_{x}}{b_{x}}=\frac{y-a_{y}}{b_{y}}=\frac{z-a_{z}}{b_{z}}=\lambda$: This is deduced from a.

c. $(\mathbf{r}-\mathbf{a})\times\mathbf{b}=\mathbf{0}$: This is deduced from a. $\mathbf{a}$ is its start point, and it has the same direction as $\mathbf{b}$.

d. $\mathbf{r}=\mathbf{a}+\lambda(\mathbf{c}-\mathbf{a})$: This is deduced from a. It passes two fixed points $\mathbf{a}$ and $\mathbf{c}$.

e. $\mathbf{r}=\mu\mathbf{a}+\lambda\mathbf{c}$, $\mu+\lambda=1$: This is deduced from d. It passes two fixed points $\mathbf{a}$ and $\mathbf{c}$.

(2) Planes:

a. $(\mathbf{r}-\mathbf{a})\cdot\mathbf{\hat{n}}=0$: point $\mathbf{a}$ and a direction vector $\mathbf{\hat{n}}$ decided this plane.

b. $lx+my+nz=d$: This is deduced from a. $\mathbf{r}=(x,y,z)$ and $\mathbf{\hat{n}}:=(l,m,n)$. Be aware of the distance $d$ between this plane and origin $O$.

c. $\mathbf{r}=\mathbf{a}+\lambda(\mathbf{b}-\mathbf{a})+\mu(\mathbf{c}-\mathbf{a})$: This is deduced from a. It passes three fixed points $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$.

d. $\mathbf{r}=\alpha\mathbf{a}+\beta\mathbf{b}+\gamma\mathbf{c}$, $\alpha+\beta+\gamma=1$: This is deduced from c. It passes three fixed points $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$.

(3) Spheres:

a. $|\mathbf{r}-\mathbf{c}|^{2}=a^{2}$.

Example:

Find the radius $\rho$ of the circle that is the intersection of the plane $\mathbf{\hat{n}}\cdot\mathbf{r}=p$ and the sphere

of radius $a$ centered on the point with position vector $\mathbf{c}$. Answer is on the page 229.

4. Distance:

(1) Distance from a point to a line:

d=$|(\mathbf{p}-\mathbf{a})\times\mathbf{\hat{b}}|$ where $\mathbf{\hat{b}}$ is a unit direction vector of this line and $\mathbf{a}$ is a point on this line.

(2) Distance from a point to a plane:

d=$(\mathbf{a}-\mathbf{p})\cdot\mathbf{\hat{n}}$ where $\mathbf{\hat{n}}$ is a unit direction vector of this plane and $\mathbf{a}$ is a point on this plane.

(3) Distance from a line to a line:

d=$|(\mathbf{p}-\mathbf{q})\cdot\mathbf{\hat{n}}|$ where $\mathbf{\hat{n}}$ is a unit direction vector ($\mathbf{\hat{n}}=\mathbf{\hat{a}}\times\mathbf{\hat{b}}$, $\mathbf{\hat{a}}$, $\mathbf{\hat{b}}$ are unit direction vectors of two lines) and $\mathbf{p}$, $\mathbf{q}$ are two points on two lines.

(4) Distance from a line to a plane:

d=$|(\mathbf{a}-\mathbf{r})\cdot\mathbf{\hat{n}}|$ where $\mathbf{a}$ is a point on this line and $\mathbf{r}$ is a point on this plane. This is valid only when $\mathbf{b}\cdot\mathbf{\hat{n}}=0$ (this line is parallel with the plane).

5. Reciprocal vectors:

Definition:

The two sets of vectors $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$ and $\mathbf{a}^{\prime}$, $\mathbf{b}^{\prime}$, $\mathbf{c}^{\prime}$ are called reciprocal sets if $\mathbf{a}\cdot\mathbf{a}^{\prime}=\mathbf{b}\cdot\mathbf{b}^{\prime}=\mathbf{c}\cdot\mathbf{c}^{\prime}=1$

and $\mathbf{a}^{\prime}\cdot\mathbf{b}=\mathbf{a}^{\prime}\cdot\mathbf{c}=\mathbf{b}^{\prime}\cdot\mathbf{a}=\mathbf{b}^{\prime}\cdot\mathbf{c}=\mathbf{c}^{\prime}\cdot\mathbf{a}=\mathbf{c}^{\prime}\cdot\mathbf{b}=0$.

How to construct:

$$\mathbf{a}^{\prime}=\frac{\mathbf{b}\times\mathbf{c}}{\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})},\\\mathbf{b}^{\prime}=\frac{\mathbf{c}\times\mathbf{a}}{\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})},\\\mathbf{c}^{\prime}=\frac{\mathbf{a}\times\mathbf{b}}{\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})}$$

where $\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})\neq0$ (not coplanar).

Rewrite a vector using reciprocal vectors:

$\mathbf{a}=(\mathbf{a}\cdot\mathbf{e}_1′)\mathbf{e}_1+(\mathbf{a}\cdot\mathbf{e}_2′)\mathbf{e}_2+(\mathbf{a}\cdot\mathbf{e}_3′)\mathbf{e}_3$. Especially for Cartesian basis vectors $\mathbf{i}$, $\mathbf{j}$, $\mathbf{k}$: $\mathbf{a}=(\mathbf{a}\cdot\mathbf{i})\mathbf{i}+(\mathbf{a}\cdot\mathbf{j})\mathbf{j}+(\mathbf{a}\cdot\mathbf{k})\mathbf{k}$.

6. Vector analysis:

(1) Cylinder coordinate:

$$\begin{aligned} &\left\{\begin{array}{c}x=\rho\cos\phi\\y=\rho\sin\phi\\z=z\end{array}\right. \\ &\left.\left\{\begin{array}{c}\rho=\sqrt{x^{2}+y^{2}}\\\phi=\arctan\left(\frac{y}{x}\right)\\z=z\end{array}\right.\right. \\ &dS=\rho\,d\phi\,dz \\ &dV=\rho\,d\rho\,d\phi\,dz \end{aligned}$$

(2) Circular coordinate:

$$\left\{\begin{array}{c}x=r\sin\theta\cos\phi\\y=r\sin\theta\sin\phi\\z=r\cos\theta\end{array}\right.\\ \left\{\begin{array}{c}r=\sqrt{x^{2}+y^{2}+z^{2}}\\\theta=\arctan\left(\frac{\sqrt{x^{2}+y^{2}}}{z}\right)\\\phi=\arctan\left(\frac{y}{x}\right)\end{array}\right.\\ dS=r^{2}\sin\theta\,d\theta\,d\phi \\ dV=r^{2}\sin\theta\,dr\,d\theta\,d\phi$$

(3) Coordinate transition:

$$\begin{gathered} \begin{bmatrix}A_{\rho}\\A_{\phi}\\A_{z}\end{bmatrix}=\begin{bmatrix}\cos\phi&\sin\phi&0\\-\sin\phi&\cos\phi&0\\0&0&1\end{bmatrix}\begin{bmatrix}A_{x}\\A_{y}\\A_{z}\end{bmatrix} \\ \begin{bmatrix}A_{r}\\A_{\theta}\\A_{\phi}\end{bmatrix}=\begin{bmatrix}\sin\theta\cos\phi&\sin\theta\sin\phi&\cos\theta\\\cos\theta\cos\phi&\cos\theta\sin\phi&-\sin\theta\\-\sin\phi&\cos\phi&0\end{bmatrix}\begin{bmatrix}A_{x}\\A_{y}\\A_{z}\end{bmatrix} \\ \begin{bmatrix}A_{r}\\A_{\theta}\\A_{\phi}\end{bmatrix}=\begin{bmatrix}\sin\theta&0&\cos\theta\\\cos\theta&0&-\sin\phi\\0&1&0\end{bmatrix}\begin{bmatrix}A_{\rho}\\A_{\phi}\\A_{z}\end{bmatrix} \end{gathered}$$

(4) Gradient, curl, divergence:

$$\begin{gathered} \nabla f=\frac{\partial f}{\partial x}\vec{e}_{x}+\frac{\partial f}{\partial y}\vec{e}_{y}+\frac{\partial f}{\partial z}\vec{e}_{z} \\ =\frac{\partial f}{\partial\rho}\vec{e}_{\rho}+\frac{1}{\rho}\frac{\partial f}{\partial\phi}\vec{e}_{\phi}+\frac{\partial f}{\partial z}\vec{e}_{z} \\ =\frac{\partial f}{\partial r}\vec{e}_{r}+\frac{1}{r}\frac{\partial f}{\partial\theta}\vec{e}_{\theta}+\frac{1}{r\sin\theta}\frac{\partial f}{\partial\phi}\vec{e}_{\phi} \end{gathered}$$

$$\nabla\times\vec{\boldsymbol{B}}=\begin{vmatrix}\vec{e}_{x}&\vec{e}_{y}&\vec{e}_{z}\\\\\frac{\partial}{\partial x}&\frac{\partial}{\partial y}&\frac{\partial}{\partial z}\\\\\vec{B}_{x}&\vec{B}_{y}&\vec{B}_{z}\end{vmatrix}\\(\frac{\partial B_{z}}{\partial y}-\frac{\partial B_{y}}{\partial z})\overrightarrow{e}_{x}+(\frac{\partial B_{x}}{\partial z}-\frac{\partial B_{z}}{\partial x})\overrightarrow{e}_{y}+(\frac{\partial B_{y}}{\partial x}-\frac{\partial B_{x}}{\partial y})\overrightarrow{e}_{z}\\=\frac{1}{\rho}\begin{vmatrix}\vec{e}_{\rho}&\rho\vec{e}_{\phi}&\vec{e}_{z}\\\\\frac{\partial}{\partial\rho}&\frac{\partial}{\partial\phi}&\frac{\partial}{\partial z}\\\\\vec{B}_{\rho}&\rho\vec{B}_{\phi}&\vec{B}_{z}\end{vmatrix}=\frac{1}{r^{2}\sin\theta}\begin{vmatrix}\vec{e}_{r}&r\vec{e}_{\theta}&r\sin\theta\vec{e}_{\phi}\\\\\frac{\partial}{\partial r}&\frac{\partial}{\partial\theta}&\frac{\partial}{\partial\phi}\\\\\vec{B}_{r}&r\vec{B}_{\theta}&r\sin\theta\vec{B}_{\phi}\end{vmatrix}$$

$$\begin{gathered} \nabla\cdot\vec{\boldsymbol{A}}=\frac{\partial A_{x}}{\partial x}+\frac{\partial A_{y}}{\partial y}+\frac{\partial A_{z}}{\partial z}\\\\=\frac{1}{\rho}\frac{\partial(\rho A_{\rho})}{\partial\rho}+\frac{1}{\rho}\frac{\partial A_{\phi}}{\partial\phi}+\frac{\partial A_{z}}{\partial z} \end{gathered}\\=\frac{1}{\rho^{2}}\frac{\partial(\rho^{2}A_{\rho})}{\partial \rho}+\frac{1}{\rho\sin\theta}\frac{\partial(\sin\theta A_{\theta})}{\partial\theta}+\frac{1}{r\sin\theta}\frac{\partial A_{\phi}}{\partial\phi}$$

(5) Laplace operator:

$$\begin{gathered} \nabla^{2}f=\frac{\partial^{2}f}{\partial x^{2}}+\frac{\partial^{2}f}{\partial y^{2}}+\frac{\partial^{2}f}{\partial z^{2}} \\ =\frac{1}{\rho}\frac{\partial}{\partial\rho}\left(\rho\frac{\partial f}{\partial\rho}\right)+\frac{1}{\rho^{2}}\frac{\partial^{2}f}{\partial\phi^{2}}+\frac{\partial^{2}f}{\partial z^{2}} \end{gathered}\\ =\frac{1}{\rho^{2}}\frac{\partial}{\partial \rho}\left(\rho^{2}\frac{\partial f}{\partial \rho}\right)+\frac{1}{\rho^{2}\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial f}{\partial\theta}\right)+\frac{1}{\rho^{2}\sin^{2}\theta}\frac{\partial^{2}f}{\partial\phi^{2}}$$