Mathematical Formula Reference

This page collects the mathematical identities used repeatedly in the worked solutions. A solution links to the relevant identity before applying it, so the algebra can be followed without guessing which rule was used. Symbols are defined locally in each exercise; the symbols on this page are generic.

Algebra and dimensional checks

An equation may be rearranged by applying the same invertible operation to both sides. For nonzero \(a\), the linear and quadratic solutions are

(1)\[ax+b=0\Rightarrow x=-\frac{b}{a},\qquad ax^2+bx+c=0\Rightarrow x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}.\]

Every additive term in a physical equation must have the same dimensions. The final unit is therefore an independent check on an algebraic result.

Product, quotient, and chain rules

For differentiable functions \(f(x)\) and \(g(x)\),

(2)\[\frac{d}{dx}(fg)=f'g+fg',\qquad \frac{d}{dx}\!\left(\frac{f}{g}\right) =\frac{f'g-fg'}{g^2},\qquad \frac{d}{dx}f(g(x))=f'(g(x))g'(x).\]

The last identity is the chain rule. Applying it to a square root gives

(3)\[\frac{d}{dx}\sqrt{u(x)} =\frac{u'(x)}{2\sqrt{u(x)}}.\]

Stationary-value condition

If a differentiable scalar function \(F(x)\) is stationary at an interior point \(x=x_0\), its first derivative vanishes there:

(4)\[\left.\frac{dF}{dx}\right|_{x=x_0}=0.\]

The condition identifies minima, maxima, and stationary inflection points; the second derivative or physical context distinguishes them. For a constraint \(g(\boldsymbol{x})=0\), the equivalent Lagrange-multiplier condition is

(5)\[\nabla F+\lambda\nabla g=0,\qquad g=0.\]

Optical path and Fermat’s principle

The optical path length (OPL) along a piecewise homogeneous path is

(6)\[\mathcal L=\int_A^B n(\boldsymbol r)\,ds \quad\longrightarrow\quad \mathcal L=\sum_i n_i\ell_i.\]

Fermat’s principle states that the physical ray makes the OPL stationary: \(\delta\mathcal L=0\). It need not be a strict minimum.

Trigonometry and small-angle formulas

For a right triangle with opposite side \(a\), adjacent side \(b\), and hypotenuse \(r=\sqrt{a^2+b^2}\),

(7)\[\sin\theta=\frac{a}{r},\qquad \cos\theta=\frac{b}{r},\qquad \tan\theta=\frac{a}{b},\qquad \sin^2\theta+\cos^2\theta=1.\]

The first nonzero Taylor terms about \(x=0\) are

(8)\[\sin x=x+O(x^3),\quad \tan x=x+O(x^3),\quad \cos x=1-\frac{x^2}{2}+O(x^4),\quad \sqrt{1+x}=1+\frac{x}{2}+O(x^2).\]

These approximations require a dimensionless \(|x|\ll1\) (angles in radians).

Solid angle and circular cones

On a unit sphere, polar angle \(\theta\) and azimuth \(\phi\) give the differential solid angle \(d\Omega=\sin\theta\,d\theta\,d\phi\). The solid angle of a circular cone with half-angle \(\alpha\) is therefore

(9)\[\Omega_{\mathrm{cone}} =\int_0^{2\pi}\!\int_0^\alpha \sin\theta\,d\theta\,d\phi =2\pi(1-\cos\alpha).\]

The full sphere contains \(4\pi\) steradians. For an isotropic source, the fraction of its power in any non-overlapping collection of directions is the collection’s total solid angle divided by \(4\pi\).

Exponentials, logarithms, and complex phasors

Euler’s identity and the elementary exponential derivatives are

(10)\[e^{j\phi}=\cos\phi+j\sin\phi,\qquad \frac{d}{dx}e^{ax}=ae^{ax},\qquad \frac{d}{dx}\ln x=\frac1x.\]

Consequently, multiplying a phasor by \(e^{j\phi}\) adds phase \(\phi\), while its magnitude is unchanged.

Integration identities

Integration by parts, substitution, and the Gaussian integral are

(11)\[\int u\,dv=uv-\int v\,du,\qquad \int f(g(x))g'(x)\,dx=\int f(u)\,du,\qquad \int_{-\infty}^{\infty}e^{-ax^2}\,dx=\sqrt{\frac{\pi}{a}} \quad(\Re a>0).\]

For a normalized density \(p(x)\), always check \(\int p(x)\,dx=1\) before using it in an expectation.

Fourier transform, convolution, and correlation

With the frequency convention used here,

(12)\[\widetilde f(\nu)=\int_{-\infty}^{\infty}f(t)e^{-j2\pi\nu t}\,dt, \qquad f(t)=\int_{-\infty}^{\infty}\widetilde f(\nu)e^{j2\pi\nu t}\,d\nu.\]

The convolution and correlation theorems are

(13)\[\mathcal F\{f*g\}=\widetilde f\,\widetilde g,qquad \mathcal F\{f\star g\}=\widetilde f^{\,*}\widetilde g.\]

Matrices, determinants, and eigenvalues

For compatible matrices, multiplication is row by column and is generally not commutative:

(14)\[(AB)_{ij}=\sum_k A_{ik}B_{kj},\qquad AB\ne BA\ \text{in general}.\]

An eigenvalue \(\lambda\) satisfies

(15)\[A\boldsymbol v=\lambda\boldsymbol v,qquad \det(A-\lambda I)=0.\]

For a lossless paraxial \(2\times2\) ray matrix with \(\det M=1\), bounded periodic rays require \(|\operatorname{tr}M/2|<1\).

Vector-calculus identities

The gradient, divergence, Laplacian, and curl identity used in the field solutions are

(16)\[\nabla f=\left(\partial_xf,\partial_yf,\partial_zf\right),\quad \nabla^2f=\nabla\!\cdot\nabla f,\quad \nabla\times(\nabla\times\boldsymbol A) =\nabla(\nabla\!\cdot\boldsymbol A)-\nabla^2\boldsymbol A.\]

Common differential equations

The constant-coefficient first- and second-order equations have solutions

(17)\[y'=ay\Rightarrow y=Ce^{ax},\qquad y''+\omega^2y=0\Rightarrow y=C_1\cos(\omega x)+C_2\sin(\omega x).\]

Boundary or initial conditions determine the constants. Substitution into the original differential equation is the quickest verification.

Probability, expectation, and variance

For a discrete random variable \(N\),

(18)\[\mathbb E[N]=\sum_n nP(n),\qquad \operatorname{var}N=\mathbb E[N^2]-\mathbb E[N]^2.\]

For independent variables, variances add. A Poisson random variable with mean \(\mu\) obeys

(19)\[P(N=n)=e^{-\mu}\frac{\mu^n}{n!},\qquad \mathbb E[N]=\operatorname{var}N=\mu.\]

Power, intensity, and decibels

Power ratios use \(10\log_{10}\); field-amplitude ratios use \(20\log_{10}\) when power is proportional to amplitude squared:

(20)\[G_{\mathrm{dB}}=10\log_{10}\!\left(\frac{P_2}{P_1}\right) =20\log_{10}\!\left|\frac{A_2}{A_1}\right|.\]

Independent losses add in decibels and multiply as linear power ratios.

Verification methods

Every quantitative solution should use at least one of these checks:

• substitute the result into the governing equation;

• verify dimensions and units term by term;

• test a known limit or symmetry;

• differentiate or integrate the proposed result;

• compare with an independent conservation law;

• reproduce the numerical value with unrounded intermediate quantities.

These are mathematical checks, not replacements for the derivation.