# Mathematical Formulas and Identities

Trigonometric Identities , Series , Derivatives , Indefinite Integrals , Fourier Analysis (Series and Transforms) , Laplace Transform , Determinants of Matrices , Vectors , Vector Calculus , Numerical Methods , 

## Trigonometric Identities

### Trigonometric Functions of Sums/Difference of Angles

• $\cos(A \pm B) = \cos A \cos B \mp \sin A \cos B$
• $\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$
• $\tan(A \pm B) = \dfrac{\tan A \pm tan B}{1 \mp \tan A \tan B}$

### Sums of Trigonometric Functions to Product of Trigonometric Functions

• $\sin A + \sin B = 2 \sin \left( \dfrac {A+B}{2} \right) \cos \left( \dfrac {A-B}{2} \right)$
• $\sin A - \sin B = 2 \sin \left( \dfrac {A - B}{2} \right) \cos \left( \dfrac {A + B}{2} \right)$
• $\cos A + \cos B = 2 \cos \left( \dfrac {A+B}{2} \right) \cos \left( \dfrac {A-B}{2} \right)$
• $\cos A - \cos B = - 2 \sin \left( \dfrac {A+B}{2} \right) \sin \left( \dfrac {A-B}{2} \right)$

### Product of Trigonometric Functions to Sum of Trigonometric Functions

• $\sin A \cos B = \dfrac{1}{2} \; [ \sin(A+B) + \sin(A-B) ]$
• $\cos A \cos B = \dfrac{1}{2} \; [ \cos(A+B) + \cos(A-B) ]$
• $\sin A \sin B = - \dfrac{1}{2} \; [ \cos(A+B) - \cos(A-B) ]$

### Trigonometric Functions of Multiple Angles

• $\sin (2 A) = 2 \sin A \cos A$
• $\cos (2 A) = 1 - 2 \sin^2 A = 2 \cos^2 A -1$
• $\sin (3 A) = 3 \sin A - 4 \sin^3 A$
• $\cos (3 A) = 4\cos^3 A - 3 \cos A$

### Power Reducing Identities

• $\sin^2 A = \dfrac{1}{2} [1 - \cos (2A)]$
• $\cos^2 A = \dfrac{1}{2} [1 + \cos (2A)]$

### Half Angle Formula

• $\sin (A/2) = \pm \sqrt {\dfrac{1- \cos A}{2}}$
• $\cos (A/2) = \pm \sqrt {\dfrac{1 + \cos A}{2}}$
• $\tan (A/2) = \dfrac{1 - \cos A}{\sin A} = \dfrac{\sin A}{1+\cos A}$

## Series

### Arithmetic Series

$S_n = a_1 + a_2 +...+ a_n$
$\quad \quad = a + (a + d) + (a + 2d) + ... + a + (n-1)d$
$d$ is the common difference
$a_1 = a$ , first term in the series
$a_n = a + (n - 1)d$ , nth term in the series
$S_n = \dfrac{n}{2} (\text{first term} + \text{last term})$
$\quad \quad = \dfrac{n}{2}(a_1 + a_n)$
$\quad \quad = \dfrac{n}{2}[ 2 a + (n-1)d ]$

### Geometric Series

$S_n = a_1 + a_2 +...+ a_n$
$\quad \quad = a + a \cdot r + a \cdot r^2 + ... + a \cdot r^{n-1}$
$r$ is the common ratio
$a_1 = a$ , first term in the series
$a_n = a \cdot r^{n-1}$ , nth term in the series
$S_n = a_1\dfrac{1 - r^n}{1 - r}$
$\quad \quad = a \dfrac{1 - r^n}{1 - r}$
For an infinite gemetric series and for $|r| \lt 1$
$S_{\infty} = \dfrac{a_1}{1 - r} = \dfrac{a}{1 - r}$

### Integer Series

$\sum_{k=1}^{n} k = 1 + 2 + 3...+n = \dfrac{1}{2} n(n+1)$
$\sum_{k=1}^{n} k^2 = 1^2 + 2^2 + 3^2...+n^2 = \dfrac{1}{6} n(n+1)(2n+1)$
$\sum_{k=1}^{n} k^3 = 1^3 + 2^3 + 3^3...+n^3 = [ \dfrac{1}{2} n(n+1) ] ^2$

### Binomial Theorem (Binomial Expansion)

$\displaystyle (x+y)^n = \sum_{k=0}^{n} {n \choose k} x^{n-k} y^k$
$\displaystyle \quad \quad = {n \choose 0} x^{n} + {n \choose 1} x^{n-1} y^1 + {n \choose 2} x^{n-2} y^2 + ... + {n \choose n} y^{n}$
where the coefficient ${n \choose k}$ represents the number of (distinct) combinations of $k$ object that can be formed from a sample of $n$ objects and is given by
$\displaystyle {n \choose k} = \dfrac{n!}{k!(n-k)!}$

### Taylor Series

A Taylor series expansion of a function for values close to $x = a$ is given by
$f(x) = f(a) + (x-a) f'(a) + \dfrac{(x-a)^2}{2!} f"(a) + \dfrac{(x-a)^3}{3!} f^{(3)}(a) + ... + \dfrac{(x-a)^n}{n!} f^{(n)}(a)+...$

### Maclaurin Series

A Taylor series expansion of a function $f$ for values close to $x = 0$ is called Maclaurin series of function $f$ and is obtained be setting $a = 0$ in Taylor series which gives
$f(x) = f(0) + x f'(0) + \dfrac{x^2}{2!} f"(0) + \dfrac{x^3}{3!} f^{(3)}(0) + ... + \dfrac{x^n}{n!} f^{(n)}(0)+...$

### Maclaurin Series of Some Common Functions

$e^x = 1 + x + \dfrac{x^2}{2!} + ... + \dfrac{x^n}{n!} + ...$       all $x$
$\ln(1+x) = x - \dfrac{x^2}{2} + \dfrac{x^3}{3} - ... + (-1)^{n+1}\dfrac{x^n}{n} + ...$       $| x| \le 1 ; x \ne -1$
$\sin x = \dfrac{e^{jx} - e^{-jx}}{2j} = x - \dfrac{x^3}{3!} + \dfrac{x^5}{5!} - \dfrac{x^7}{7!}...$       all $x$
$\cos x = \dfrac{e^{jx} + e^{-jx}}{2} = 1 - \dfrac{x^2}{2!} + \dfrac{x^4}{4!} - \dfrac{x^6}{6!}...$       all $x$
$\tan x = x + \dfrac{1}{3} x^3 + \dfrac{2}{15} x^5 + \dfrac{17}{315}x^7...$       $|x| \lt \pi/2$
$\sinh x = \dfrac{e^{x} - e^{-x}}{2} = x + \dfrac{x^3}{3!} + \dfrac{x^5}{5!} + \dfrac{x^7}{7!}...$       all $x$
$\cosh x = \dfrac{e^{x} + e^{-x}}{2} = 1 + \dfrac{x^2}{2!} + \dfrac{x^4}{4!} + \dfrac{x^6}{6!}...$       all $x$
$\tanh x = x - \dfrac{1}{3} x^3 + \dfrac{2}{15} x^5 - \dfrac{17}{315}x^7...$       $|x| \lt \pi/2$
$\sin^{-1} x = \arcsin x = x + \dfrac{1 \cdot x^3}{2\cdot3} + \dfrac{1 \cdot 3 x^5}{2 \cdot 4 \cdot 5} + \dfrac{1 \cdot 3 \cdot 5 x^7}{2 \cdot 4 \cdot 6 \cdot 7} ...$       $|x| \lt 1$
$\tan^{-1} x = \arctan x = x - \dfrac{x^3}{3} + \dfrac{x^5}{5} +...$       $|x| \lt 1$
$\sinh^{-1} x = \text{arcsinh} \; x = x - \dfrac{1 x^3}{2\cdot3} + \dfrac{1 \cdot 3 x^5}{2 \cdot 4 \cdot 5} - \dfrac{1 \cdot 3 \cdot 5 x^7}{2 \cdot 4 \cdot 6 \cdot 7} ...$       $|x| \lt 1$
$\tanh^{-1} x = \arctan x = x + \dfrac{x^3}{3} + \dfrac{x^5}{5} +...$       $|x| \lt 1$

## Derivatives

### $\dfrac{d f(x)}{dx}$

$x^n$ $n x^{n-1}$
$e^x$ $e^x$
$b^x$ $\ln b \cdot b^x$
$\ln x$ $\dfrac{1}{x}$
$\log_b x$ $\dfrac{1}{ x \ln b}$
$\sin x$ $\cos x$
$\cos x$ $- \sin x$
$\tan x$ $\sec^2 x$
$\cot x$ $- \csc^2 x$
$\sec x$ $\sec x \tan x$
$\csc x$ $- \csc x \cot x$
$\sin^{-1} x$ $\dfrac{1}{\sqrt{1-x^2}}$
$\cos^{-1} x$ $- \dfrac{1}{\sqrt{1-x^2}}$
$\tan^{-1} x$ $\dfrac{1}{1+x^2}$
$\sinh x$ $\cosh x$
$\cosh x$ $\sinh x$
$\tanh x$ $\text{sech}^2 x$
$\coth x$ $- \text{csch}^2 x$
$\text{sech} \; x$ $-\text{sech} \; x \tanh x$
$\text{csch} \; x$ $- \text{csch} \; x \coth x$
$\sinh^{-1} x$ $\dfrac{1}{\sqrt{x^2+1}}$
$\cosh^{-1} x$ $\dfrac{1}{\sqrt{x^2-1}}$
$\tanh^{-1} x$ $\dfrac{1}{1-x^2}$
$\coth^{-1} x$ $\dfrac{1}{1-x^2}$

## Indefinite Integrals

Note that in all cases of indefinite integrals, the constant of integration is omitted here but should be added whenever necessary.

### $\displaystyle \int f(x) dx$

$x^n$ $\dfrac{x^{n+1}}{n+1}$
$\dfrac{1}{x}$ $\ln |x|$
$e^x$ $e^x$
$\ln x$ $x \ln x - x$
$\sin x$ $-\cos x$
$\cos x$ $\sin x$
$\tan x$ $-\ln |\cos x|$
$\cot x$ $\ln |\sin x|$
$\sec x$ $\ln( \sec x + \tan x )$
$\csc x$ $\ln(\csc x - \cot x)$
$\sinh x$ $\cosh x$
$\cosh x$ $\sinh x$
$\tanh x$ $\ln( \cosh x)$
$\coth x$ $\ln( \sinh x)$
$\text{sech} \; x$ $2 \tan^{-1}(e^x)$
$\text{csch} \; x$ $-\ln (\coth x + \text{csch}\; x)$
$\dfrac{1}{\sqrt{a^2 - x^2}}$ $\sin^{-1} \left(\dfrac{x}{a}\right) \; , |x| \lt a$
$\dfrac{1}{\sqrt{a^2 - x^2}}$ $- \cos^{-1} \left(\dfrac{x}{a}\right) \; , |x| \lt a$
$\dfrac{1}{\sqrt{x^2 + a^2}}$ $\ln(x+\sqrt{x^2 + a^2})$
$\dfrac{1}{\sqrt{x^2 - a^2}}$ $\ln(x+\sqrt{x^2 - a^2})$
$\dfrac{1}{x^2 + a^2}$ $\dfrac{1}{a} \tan^{-1} \left(\dfrac{x}{a} \right)$
$\dfrac{1}{x^2 - a^2}$ $\dfrac{1}{2 a} \ln \left(\dfrac{x-a}{x+a}\right)$
$\dfrac{1}{a^2 - x^2}$ $\dfrac{1}{2 a} \ln \left(\dfrac{a+x}{a-x}\right)$

## Fourier Analysis (Series and Transforms)

### Real Fourier Series

If $f(t)$ is a periodic function with period $T$, then
$\displaystyle f(t) = \dfrac{1}{2} a_0 + \sum_{m=1}^{\infty} a_m \cos \left(\dfrac{2 \pi m}{T} t\right) + \sum_{m=1}^{\infty} b_m \sin \left(\dfrac{2 \pi m}{T} t\right)$

$\displaystyle a_m = \dfrac{2}{T} \int_0^T f(t) \cos \left(\dfrac{2 \pi m}{T} t\right) dt$

$\displaystyle b_m = \dfrac{2}{T} \int_0^T f(t) \sin \left(\dfrac{2 \pi m}{T} t\right) dt$

### Complex Fourier Series

$j = \sqrt{-1}$ is the imaginary unit
If $f(t)$ is a periodic function with period $T$, then
$\displaystyle f(t) = \sum_{m = -\infty}^{\infty} c_m \exp \left( j \dfrac{2 \pi m}{T} t \right)$

$\displaystyle c_m = \dfrac{1}{T} \int_0^T f(t) \exp \left( - j \dfrac{2 \pi m}{T} t\right) dt$

### Relationship Between Real and Complex Coefficients

$c_m = \dfrac{1}{2} (a_m - j b_m) , m \gt 0$
$c_0 = \dfrac{1}{2} a_0$
$c_m = \dfrac{1}{2} (a_{-m} + j b_{-m}) , m \lt 0$

### Fourier Transform Pair

If $f(t)$ is defined in the range $-\infty \lt t \lt +\infty$, then the fourier transform $F(\omega)$ is defined by
$\displaystyle F(\omega) = \int_{-\infty}^{+\infty} f(t) \exp \left( - j \omega t\right) dt$
and
$\displaystyle f(t) = \dfrac{1}{2 \pi} \int_{-\infty}^{+\infty} F(\omega) \exp \left( j \omega t\right) d\omega$

## Laplace Transform

If $f(t)$ is a one sided function such that $f(t) = 0$ for $t \lt 0$ then the Laplace transform $F(s)$ is defined by $F(s) = \int_{0-}^{+\infty} f(t) e^{-st} dt$ where $s$ is allowed to be a complex number for which the above improper integral converges.

### Transform

$f(t)$ $F(s)$
$1$ $\dfrac{1}{s}$
$t^n$ $\dfrac{n!}{s^{n+1}}$
$e^{-at}$ $\dfrac{1}{s+a}$
$t^n e^{-at}$ $\dfrac{n!}{(s+a)^{n+1}}$
$\sin \omega t$ $\dfrac{\omega}{s^2+\omega^2}$
$t \sin \omega t$ $\dfrac{2 \omega s}{(s^2+\omega^2)^2}$
$\cos \omega t$ $\dfrac{s}{s^2+\omega^2}$
$t \cos \omega t$ $\dfrac{s^2 - \omega^2}{(s^2+\omega^2)^2}$
$\sinh \omega t$ $\dfrac{\omega}{s^2 - \omega^2}$
$\cosh \omega t$ $\dfrac{s }{s^2 - \omega^2}$
$\delta( t - \tau)$ $e^{-s \tau}$ , $\tau \ge 0$
$H( t - \tau)$ $\dfrac{1}{s} e^{-s \tau}$ , $\tau \ge 0$
Note
1) $\delta( t )$ is the Dirac delta function also called impulse function in engineering.
2) $H( t)$ is the Heaviside step function.

## Properties of Laplace Transform

In what follows, function $f(t)$ is written in small letters and its corresponding transform in capital letters $F(s)$
1. Linearity
If $g(t) = a f_1(t) + b f_2(t)$, then $G(s) = a F_1(s) + b F_2(s)$ , $a$ and $b$ are constants.
2. Shift in t
If $g(t) = f(t - \tau) H( t - \tau)$, then $G(s) = e^{- s \tau} F(s)$ , $\tau \ge 0$
3. Multiplication by an exponential in $t$ results in a shift in $s$
If $g(t) = e^{-at} f(t)$, then $G(s) = F(s + a)$ , $a \ge 0$
4. Scaling in $t$
If $g(t) = f(k t)$, then $G(s) = \dfrac{1}{k} F(\dfrac{s}{k})$
5. Derivative of $F(s)$ with respect to $s$
If $g(t) = t f(t)$, then $G(s) = - \dfrac{d F(s)}{d s}$
6. Derivative of $f(t)$ with respect to $t$
If $g(t) = \dfrac{df(t)}{dt} = f'(t)$, then $G(s) = s F(s) - f(0)$
7. Second derivative of $f(t)$ with respect to $t$
If $g(t) = \dfrac{df^2(t)}{dt^2} = f''(t)$, then $G(s) = s^2 F(s) - s f(0) - f'(0)$
8. $n$ th derivative of $f(t)$ with respect to $t$
If $g(t) = \dfrac{df^n(t)}{dt^n} = f^{(n)}(t)$,
then $G(s) = s^n F(s) - s^{n-1} f(0) - s^{n-2} f'(0) - ... - s f^{(n-2)}(0) - f^{(n-1)}(0)$
9. Integral of $f(t)$ with respect to $t$
If $\displaystyle g(t) = \int_0^t f(t') dt'$ , then $G(s) = \dfrac{1}{s} F(s)$
10. Convolution integral
If $\displaystyle g(t) = \int_0^t f_1(t')f_2(t-t') dt'$, then $G(s) = F_1(s) F_2(s)$

## Determinants of Matrices

### Determinant of a 2 by 2 Matrix

Let A be 2 by 2 matrix
$A = \begin{bmatrix} a_1 & a_2 \\ b_1 & b_2 \\ \end{bmatrix}$
The determinant of matrix $A$ is denoted by $|A|$ and given by
$|A| = \begin{vmatrix} a_1 & a_2 \\ b_1 & b_2 \\ \end{vmatrix} = a_1 b_2 - a_2 b_1$

### Determinant of a 3 by 3 Matrix

Let a 3 by 3 matrix $A$ given by
$A = \begin{bmatrix} a_1 & a_2 & a_3\\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \\ \end{bmatrix}$
The determinant of matrix $A$ is given by
$|A| = \begin{vmatrix} a_1 & a_2 & a_3\\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \\ \end{vmatrix} = a_1 \begin{vmatrix} b_2 & b_3 \\ c_2 & c_3 \\ \end{vmatrix} - a_2 \begin{vmatrix} b_1 & b_3 \\ c_1 & c_3 \\ \end{vmatrix} + a_3 \begin{vmatrix} b_1 & b_2 \\ c_1 & c_2 \\ \end{vmatrix}$
There are many other forms for the determinant of $n \times n$ matrices for $n \gt 2$.

## Vectors

### Scalar (or Dot) Product

$\vec a$ and $\vec b$ are vectors given by their components as follows
$\vec a = \lt a_1, a_2 , a_3 \gt$ and $\vec b = \lt b_1, b_2 , b_3 \gt$
The scalar (or dot) product of vectors $\vec a$ and $\vec b$ is defined by
$\vec a \cdot \vec b = || \vec a || \; || \vec b || \cos \theta = \sum_{i=1}^{3} a_i b_i = a_1 b_1 + a_2 b_2 + a_3 b_3$
where $\theta$ is the angle between vectors $\vec a$ and $\vec b$ and $|| \vec a ||$ and $|| \vec b ||$ are their magnitudes.

### Vector (or Cross) Product

The vector (or cross ) product of vectors $\vec a$ and $\vec b$ is defined by
$\vec a \times \vec b = || \vec a || \; || \vec b || \sin \theta \; \vec n = \begin{vmatrix} \vec i & \vec j & \vec k \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ \end{vmatrix}$

$\quad \quad = \vec i \begin{vmatrix} a_2 & a_3 \\ b_2 & b_3 \\ \end{vmatrix} - \vec j \begin{vmatrix} a_1 & a_3 \\ b_1 & b_3 \\ \end{vmatrix} + \vec k \begin{vmatrix} a_1 & a_2 \\ b_1 & b_2 \\ \end{vmatrix}$

$\quad \quad = (a_2 b_3 - a_3 b_2) \vec i - (a_1 b_3 - a_3 b_1) \vec j + (a_1b_2 - a_2 b_1) \vec k$
$\vec n$ is a unit vector perpendicular to vectors $\vec a$ and $\vec b$, such that $\vec a$, $\vec b$ and $\vec n$ form a right-handed set of vectors.

### Scalar Triple Product

The scalar triple product of $\vec a$, $\vec b$ and $\vec c$ is given by
$(\vec a \times \vec b) \cdot \vec c = \vec a \cdot (\vec b \times \vec c)$
$\quad \quad = \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \\ \end{vmatrix}$

$= a_1 \begin{vmatrix} b_2 & b_3 \\ c_2 & c_3 \\ \end{vmatrix} - a_2 \begin{vmatrix} b_1 & b_3 \\ c_1 & c_3 \\ \end{vmatrix} + a_3 \begin{vmatrix} b_1 & b_2 \\ c_1 & c_2 \\ \end{vmatrix}$
$= a_1( b_2 c_3 - b_3 c_2) - a_2 (b_1 c_3 -b_3 c_1) + a_3 (b_1 c_2 - b_2 c_1)$

### Vector (Cross) Triple Product

$\vec a \times (\vec b \times \vec c) = (\vec a \cdot \vec c) \vec b - (\vec a \cdot \vec b) \vec c$

## Vector Calculus

The "del" ($\nabla$) operator is defined in terms of partial derivatives as follows
$\nabla = \left( \dfrac{\partial }{\partial x} , \dfrac{\partial }{\partial y} , \dfrac{\partial }{\partial z} \right)$

If $\psi$ is a function of the variables $x , y$ and $z$ , the gradient is a vector defined by
$\text{grad} \psi = \nabla \psi = \left(\dfrac{\partial \psi}{\partial x},\dfrac{\partial \psi }{\partial y},\dfrac{\partial \psi}{\partial z}\right)$

### Divergence

If $\left( v_1(x,y,z) , v_2(x,y,z) , v_3(x,y,z) \right)$ are the components of vector $\vec v$, the divergence of vector $\vec v$ is a scalar defined by
$\text{div} \; \vec v = \nabla \cdot \vec v = \dfrac{\partial v_1}{\partial x} + \dfrac{\partial v_2 }{\partial y} + \dfrac{\partial v_3}{\partial z}$

### Curl

If $\left( v_1(x,y,z) , v_2(x,y,z) , v_3(x,y,z) \right)$ are the components of vector $\vec v$, the curl of vector $\vec v$ is a vector defined by
$\text{curl} \; \vec v = \nabla \times \vec v = \begin{vmatrix} \vec i & \vec j &\vec k\\ \dfrac{\partial }{\partial x} & \dfrac{\partial }{\partial y} & \dfrac{\partial }{\partial z} \\ v_1 & v_2 & v_3 \\ \end{vmatrix}$

$= \left(\dfrac{\partial v_3}{\partial y} - \dfrac{\partial v_2}{\partial z}\right) \vec i - \left(\dfrac{\partial v_3}{\partial x} - \dfrac{\partial v_1}{\partial z}\right) \vec j + \left(\dfrac{\partial v_2}{\partial x} - \dfrac{\partial v_1}{\partial y}\right) \vec k$

### Laplacian for a Scalar

If $\psi$ is a function of the variables $x , y$ and $z$ , the Laplacian of $\psi$ is a scalar defined by
$\nabla^2 \psi = \text{div} \; \text{grad} \; \psi = \nabla \cdot \nabla \psi = \dfrac{\partial^2 \psi}{\partial x^2} + \dfrac{\partial^2 \psi}{\partial y^2} + \dfrac{\partial^2 \psi}{\partial z^2}$

### Laplacian for a Vector

If $\left( v_1(x,y,z) , v_2(x,y,z) , v_3(x,y,z) \right)$ are the components of vector $\vec v$, the Laplacian of vector $\vec v$ is a vector defined by
$\nabla^2 \vec v = \text{grad} \; \text{div} \; \vec v - \text{curl} \; \text{curl} \; \vec v = \nabla (\nabla \cdot \vec v) - \nabla \times (\nabla \times \vec v)$
$\quad = \nabla^2 v_1 \vec i + \nabla^2 v_2 \vec j + \nabla^2 v_3 \vec k$
$\quad = (\dfrac{\partial^2 v_1}{\partial x^2} + \dfrac{\partial^2 v_1}{\partial y^2} + \dfrac{\partial^2 v_1}{\partial z^2}) \vec i$
$\quad \quad +(\dfrac{\partial^2 v_2}{\partial x^2} + \dfrac{\partial^2 v_2}{\partial y^2} + \dfrac{\partial^2 v_2}{\partial z^2}) \vec j$
$\quad \quad \quad \quad + (\dfrac{\partial^2 v_3}{\partial x^2} + \dfrac{\partial^2 v_3}{\partial y^2} + \dfrac{\partial^2 v_3}{\partial z^2}) \vec k$

### Identities

In what follows, $\psi$ is a function and $\vec v$ is a vector
$\text{div} \; \text{curl} \; \vec v = \nabla \cdot (\nabla \times \vec v) = 0$
$\text{curl} \; \text{grad} \; \psi = \nabla \times (\nabla \; \psi) = 0$

### Gradient, Divergence and Curl of Product

In what follows, $\psi$ and $\phi$ are functions and $\vec v$ is a vector
$\text{grad} \; (\psi \phi) = \psi \; (\nabla \phi) + \phi \; (\nabla \psi)$
$\text{div} \; (\psi \vec v) = \psi \; (\nabla \cdot \vec v) + \vec v \cdot (\nabla\psi)$
$\text{curl} \; (\psi \vec v) = \psi \; ( \nabla \times \vec v) + \nabla \psi \times \vec v$
$\text{grad} \; (\vec u \cdot \vec v) = \vec u \times (\nabla \times \vec v) + \vec v \times (\nabla \times \vec u)+ (\vec u \cdot \nabla) \vec v + (\vec v \cdot \nabla ) \vec u$
$\text{div} \; (\vec u \times \vec v) = (\nabla \times \vec u) \cdot \vec v - (\nabla \times \vec v) \cdot \vec u$
$\text{curl} \; (\vec u \times \vec v ) = \vec u (\nabla \cdot \vec v) - \vec v (\nabla \cdot \vec u) + (\vec v \cdot \nabla) \vec u - (\vec u \cdot \nabla) \vec v$
NOTE
In the above formulas, $\vec u \cdot \nabla$ may be considered as a scalar operator and is given by
$\vec u \cdot \nabla = u_1 \dfrac{\partial }{\partial x} + u_2 \dfrac{\partial }{\partial y} + u_3 \dfrac{\partial }{\partial z}\$
If it is applied to scalar function $f$, it gives a scalar
$(\vec u \cdot \nabla) f = u_1 \dfrac{\partial f}{\partial x} + u_2 \dfrac{\partial f}{\partial y} + u_3 \dfrac{\partial f}{\partial z}\$
If it is applied to vector function $\vec v$, it gives a vector
$(\vec u \cdot \nabla) \vec v = (\vec u \cdot \nabla) v_1 \vec i + (\vec u \cdot \nabla) v_2 \vec j + (\vec u \cdot \nabla) v_3 \vec k$
$\quad = ( u_1 \dfrac{\partial v_1}{\partial x} + u_2 \dfrac{\partial v_1}{\partial y} + u_3 \dfrac{\partial v_1}{\partial z}\ ) \vec i$
$\quad\quad + ( u_1 \dfrac{\partial v_2}{\partial x} + u_2 \dfrac{\partial v_2}{\partial y} + u_3 \dfrac{\partial v_2}{\partial z}\ ) \vec j$
$\quad\quad\quad + ( u_1 \dfrac{\partial v_3}{\partial x} + u_2 \dfrac{\partial v_3}{\partial y} + u_3 \dfrac{\partial v_3}{\partial z}\ ) \vec k$

## Numerical Methods

In what follows $f_m = f(x_m)$. Examples $f_0 = f(x_0)$ , $f_1 = f(x_1)$ , $f_2 = f(x_2)$ ...
$h = \dfrac{x_m - x_0}{m}$

### Trapezoidal Rule of Integration

$\displaystyle \int_{x_0}^{x_m} f(x) dx = h \left( \dfrac{f_0}{2} + f_1 + f_2 + f_3 + ... + f_{m-1}+\dfrac{f_m}{2}\right)$

### Simpson Rule of Integration

The number of intervals must be even and is taken to be $2 m$
$\displaystyle \int_{x_0}^{x_{2m}} f(x) dx = \dfrac{h}{3} \left[ f_0 + 4( f_1 + f_3 + f_5 + ...) + 2 ( f_2 + f_4 +f_6 + .... )+ f_{2m} \right]$

### Euler Method for Differential Equations

$y' = f(x,y)$
$y_{n+1} = y_n + h f(x_n , y_n)$

### Second Order Runge-Kutta Method for Differential Equations

$y' = f(x,y)$
$y_{n+1} = y_n + \dfrac{1}{2} (k_1 + k_2)$
where
$k_1 = h f(x_n , y_n)$
$k_2 = h f(x_n+h , y_n+k_1)$