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

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

Indefinite Integrals

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

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

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.

Function	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 \( F(s) \)

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.
Shift in t
If \( g(t) = f(t - \tau) H( t - \tau) \), then \( G(s) = e^{- s \tau} F(s) \) , \( \tau \ge 0 \)
Multiplication by an exponential in \( t \) results in a shift in \( s \)
If \( g(t) = e^{-at} f(t) \), then \( G(s) = F(s + a) \) , \( a \ge 0 \)
Scaling in \( t \)
If \( g(t) = f(k t) \), then \( G(s) = \dfrac{1}{k} F(\dfrac{s}{k}) \)
Derivative of \( F(s) \) with respect to \( s \)
If \( g(t) = t f(t) \), then \( G(s) = - \dfrac{d F(s)}{d s} \)
Derivative of \( f(t) \) with respect to \( t \)
If \( g(t) = \dfrac{df(t)}{dt} = f'(t)\), then \( G(s) = s F(s) - f(0) \)
Second derivative of \( f(t) \) with respect to \( t \)
If \( g(t) = \dfrac{df^2(t)}{dt^2} = f''(t)\), then \( G(s) = s^2 F(s) - s f(0) - f'(0) \)
\( n \) th derivative of \( f(t) \) with respect to \( t \)
If \( g(t) = \dfrac{df^n(t)}{dt^n} = f^{(n)}(t)\),
then \( G(s) = s^n F(s) - s^{n-1} f(0) - s^{n-2} f'(0) - ... - s f^{(n-2)}(0) - f^{(n-1)}(0) \)
Integral of \( f(t) \) with respect to \( t \)
If \( \displaystyle g(t) = \int_0^t f(t') dt'\) , then \( G(s) = \dfrac{1}{s} F(s) \)
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
\( \nabla = \left( \dfrac{\partial }{\partial x} , \dfrac{\partial }{\partial y} , \dfrac{\partial }{\partial z} \right) \)

Gradient

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 \( f_0 = f(x_0) \) , \( f_1 = f(x_1) \) , \( f_2 = f(x_2) \) ...
\( h = \dfrac{x_m - x_0}{m} \)

numerical integration of functions

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
\( k_1 = h f(x_n , y_n) \)
\( k_2 = h f(x_n+h , y_n+k_1) \)