# 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
$$k_1 = h f(x_n , y_n)$$
$$k_2 = h f(x_n+h , y_n+k_1)$$