Mathematical Formulas and Identities
Trigonometric Identities , Series , Derivatives , Indefinite Integrals , Fourier Analysis (Series and Transforms) , Laplace Transform , Matrix Determinants , Vectors , Vector Calculus , Numerical Methods
\( \)\( \)\( \)Trigonometric Identities
Trigonometric Functions of Angle Sums/Differences
- \( \cos(A \pm B) = \cos A \cos B \mp \sin A \sin B \)
- \( \sin(A \pm B) = \sin A \cos B \pm \cos A \sin B \)
- \( \tan(A \pm B) = \dfrac{\tan A \pm \tan B}{1 \mp \tan A \tan B} \)
Sums of Trigonometric Functions into Products 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 into 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 Reduction Identities
- \( \sin^2 A = \dfrac{1}{2} [1 - \cos (2A)] \)
- \( \cos^2 A = \dfrac{1}{2} [1 + \cos (2A)] \)
Half-Angle Formulas
- \( \sin (A/2) = \pm \sqrt {\dfrac{1- \cos A}{2}} \)
- \( \cos (A/2) = \pm \sqrt {\dfrac{1 + \cos A}{2}} \)
- \( \tan (A/2) = \dfrac{1 - \cos A}{\sin A} = \dfrac{\sin A}{1+\cos A} \)
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 of the series
\[ a_n = a + (n - 1)d \], \( n \)-th term of 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 of the series
\[ a_n = a \cdot r^{n-1} \], \( n \)-th term of the series
\[ S_n = a_1\dfrac{1 - r^n}{1 - r} \]
\[ \quad \quad = a \dfrac{1 - r^n}{1 - r} \]
For an infinite geometric series and for \( |r| \lt 1 \)
\[ S_{\infty} = \dfrac{a_1}{1 - r} = \dfrac{a}{1 - r} \]
Integer Series (Power Sums)
\( \sum_{k=1}^{n} k = 1 + 2 + 3...+n = \dfrac{1}{2} n(n+1) \)\( \sum_{k=1}^{n} k^2 = 1^2 + 2^2 + 3^2...+n^2 = \dfrac{1}{6} n(n+1)(2n+1) \)
\( \sum_{k=1}^{n} k^3 = 1^3 + 2^3 + 3^3...+n^3 = [ \dfrac{1}{2} n(n+1) ] ^2 \)
Binomial Theorem
\( \displaystyle (x+y)^n = \sum_{k=0}^{n} {n \choose k} x^{n-k} y^k \)\( \displaystyle \quad \quad = {n \choose 0} x^{n} + {n \choose 1} x^{n-1} y^1 + {n \choose 2} x^{n-2} y^2 + ... + {n \choose n} y^{n} \)
where the coefficient \( {n \choose k} \) represents the number of (distinct) combinations of \( k \) objects 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
The 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 the Maclaurin series of the function \( f \) and is obtained by setting \( a = 0 \) in the Taylor series, yielding\( f(x) = f(0) + x f'(0) + \dfrac{x^2}{2!} f''(0) + \dfrac{x^3}{3!} f^{(3)}(0) + ... + \dfrac{x^n}{n!} f^{(n)}(0)+... \)
Maclaurin Series of 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 \( x \)
\( \tanh x = x - \dfrac{1}{3} x^3 + \dfrac{2}{15} x^5 - \dfrac{17}{315}x^7... \) \( |x| \lt \pi/2 \)
\( \sin^{-1} x = \arcsin x = x + \dfrac{1 \cdot x^3}{2\cdot3} + \dfrac{1 \cdot 3 x^5}{2 \cdot 4 \cdot 5} + \dfrac{1 \cdot 3 \cdot 5 x^7}{2 \cdot 4 \cdot 6 \cdot 7} ... \) \( |x| \lt 1 \)
\( \tan^{-1} x = \arctan x = x - \dfrac{x^3}{3} + \dfrac{x^5}{5} - ... \) \( |x| \lt 1 \)
\( \sinh^{-1} x = \text{arcsinh} \; x = x - \dfrac{1 x^3}{2\cdot3} + \dfrac{1 \cdot 3 x^5}{2 \cdot 4 \cdot 5} - \dfrac{1 \cdot 3 \cdot 5 x^7}{2 \cdot 4 \cdot 6 \cdot 7} ... \) \( |x| \lt 1 \)
\( \tanh^{-1} x = x + \dfrac{x^3}{3} + \dfrac{x^5}{5} + ... \) \( |x| \lt 1 \)
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 unitIf \( f(t) \) is a periodic function with period \( T \), then
\( \displaystyle f(t) = \sum_{m = -\infty}^{\infty} c_m \exp \left( j \dfrac{2 \pi m}{T} t \right) \)
\( \displaystyle c_m = \dfrac{1}{T} \int_0^T f(t) \exp \left( - j \dfrac{2 \pi m}{T} t\right) dt \)
Relation 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 \( s \) can 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 the impulse function in engineering.
2) \( H( t) \) is the Heaviside step function.
Properties of the Laplace Transform
In what follows, the function \( f(t) \) is written in lowercase and its corresponding transform in uppercase \( F(s) \).
- **Linearity**
If \( g(t) = a f_1(t) + b f_2(t) \), then \( G(s) = a F_1(s) + b F_2(s) \), where \( a \) and \( b \) are constants.
- **Time Shifting**
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 \( t \)** (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 \( G(s) = \dfrac{1}{k} F\left(\dfrac{s}{k}\right) \).
- **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 \( g(t) = \dfrac{d^2f(t)}{dt^2} = f''(t) \), then \( G(s) = s^2 F(s) - s f(0) - f'(0) \).
- **\( n \)-th derivative of \( f(t) \) with respect to \( t \)**
If \( g(t) = \dfrac{d^nf(t)}{dt^n} = f^{(n)}(t) \),
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 \( G(s) = F_1(s) F_2(s) \).
Matrix Determinants
Determinant of a 2 by 2 Matrix
Let \( A \) be a 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 is 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 \( A \) be a 3 by 3 matrix 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 methods and formulas for determinants of \( n \times n \) matrices for \( n \gt 2 \).
Vectors
Dot Product
\( \vec a \) and \( \vec b \) are vectors given by their components as follows:\( \vec a = \langle a_1, a_2 , a_3 \rangle \) and \( \vec b = \langle b_1, b_2 , b_3 \rangle \)
The 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.
Cross Product
The 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 of Three Vectors
The scalar triple product of \( \vec a \), \( \vec b \), and \( \vec c \) is defined 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 Triple Product of Three Vectors
\( \vec a \times (\vec b \times \vec c) = (\vec a \cdot \vec c) \vec b - (\vec a \cdot \vec b) \vec c \)Vector Calculus
The "del" operator (\( \nabla \)) 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 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 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 a Product
In what follows, \( \psi \) and \( \phi \) are functions and \( \vec v \) is a vector.\( \text{grad} \; (\psi \phi) = \psi \; (\nabla \phi) + \phi \; (\nabla \psi) \)
\( \text{div} \; (\psi \vec v) = \psi \; (\nabla \cdot \vec v) + \vec v \cdot (\nabla\psi) \)
\( \text{curl} \; (\psi \vec v) = \psi \; ( \nabla \times \vec v) + \nabla \psi \times \vec v \)
\( \text{grad} \; (\vec u \cdot \vec v) = \vec u \times (\nabla \times \vec v) + \vec v \times (\nabla \times \vec u)+ (\vec u \cdot \nabla) \vec v + (\vec v \cdot \nabla ) \vec u \)
\( \text{div} \; (\vec u \times \vec v) = (\nabla \times \vec u) \cdot \vec v - (\nabla \times \vec v) \cdot \vec u \)
\( \text{curl} \; (\vec u \times \vec v ) = \vec u (\nabla \cdot \vec v) - \vec v (\nabla \cdot \vec u) + (\vec v \cdot \nabla) \vec u - (\vec u \cdot \nabla) \vec v \)
Remarks
In the formulas above, \( \vec u \cdot \nabla \) can be considered as a scalar operator and is given by:
\( \vec u \cdot \nabla = u_1 \dfrac{\partial }{\partial x} + u_2 \dfrac{\partial }{\partial y} + u_3 \dfrac{\partial }{\partial z} \)
When applied to a scalar function \( f \), it yields a scalar:
\( (\vec u \cdot \nabla) f = u_1 \dfrac{\partial f}{\partial x} + u_2 \dfrac{\partial f}{\partial y} + u_3 \dfrac{\partial f}{\partial z} \)
When applied to a vector function \( \vec v \), it yields a vector:
\( (\vec u \cdot \nabla) \vec v = (\vec u \cdot \nabla) v_1 \vec i + (\vec u \cdot \nabla) v_2 \vec j + (\vec u \cdot \nabla) v_3 \vec k \)
\( \quad = \left( u_1 \dfrac{\partial v_1}{\partial x} + u_2 \dfrac{\partial v_1}{\partial y} + u_3 \dfrac{\partial v_1}{\partial z} \right) \vec i \)
\( \quad \quad + \left( u_1 \dfrac{\partial v_2}{\partial x} + u_2 \dfrac{\partial v_2}{\partial y} + u_3 \dfrac{\partial v_2}{\partial z} \right) \vec j \)
\( \quad \quad \quad + \left( u_1 \dfrac{\partial v_3}{\partial x} + u_2 \dfrac{\partial v_3}{\partial y} + u_3 \dfrac{\partial v_3}{\partial z} \right) \vec k \)
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 for 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's Rule for Integration
The number of intervals must be even and is taken as \( 2 m \)\( \displaystyle \int_{x_0}^{x_{2m}} f(x) dx = \dfrac{h}{3} \left[ f_0 + 4( f_1 + f_3 + f_5 + ...) + 2 ( f_2 + f_4 +f_6 + .... )+ f_{2m} \right] \)
Euler's 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
\( k_1 = h f(x_n , y_n) \)
\( k_2 = h f(x_n+h , y_n+k_1) \)
More References and Links
Handbook of Mathematical Functions
Mathematics for Engineers with Examples and Solutions