physica.
37 formulas

Mathematical Physics

Tensors, groups, PDEs. Every formula below opens into a live, hands-on simulation.

vector calculus
∇f=∂f∂x x^+∂f∂y y^+∂f∂z z^\nabla f = \frac{\partial f}{\partial x}\,\hat{\mathbf{x}} + \frac{\partial f}{\partial y}\,\hat{\mathbf{y}} + \frac{\partial f}{\partial z}\,\hat{\mathbf{z}}

Gradient

The gradient packages all the partial derivatives into one vector that points in the direction of steepest ascent, with length equal to that maximum slope.

vector calculus
∇⋅F=∂Fx∂x+∂Fy∂y+∂Fz∂z\nabla \cdot \mathbf{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}

Divergence

Divergence measures the net outward flux per unit volume from an infinitesimal box — positive at a source, negative at a sink, zero where flow just passes through.

vector calculus
∇×F=(∂Fz∂y−∂Fy∂z, ∂Fx∂z−∂Fz∂x, ∂Fy∂x−∂Fx∂y)\nabla \times \mathbf{F} = \left(\frac{\partial F_z}{\partial y}-\frac{\partial F_y}{\partial z},\ \frac{\partial F_x}{\partial z}-\frac{\partial F_z}{\partial x},\ \frac{\partial F_y}{\partial x}-\frac{\partial F_x}{\partial y}\right)

Curl

Curl measures the local spin of a field — the circulation per unit area around an infinitesimal loop, with direction along the axis of rotation by the right-hand rule.

vector calculus
∇2f=∇⋅∇f=∂2f∂x2+∂2f∂y2+∂2f∂z2\nabla^2 f = \nabla\cdot\nabla f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}

Laplacian

The Laplacian compares the value at a point to the average of its neighbors: positive where the point sits in a valley, negative where it caps a hill, zero where it equals the surrounding average (harmonic).

vector calculus
∮∂VF⋅dA=∫V(∇⋅F) dV\oint_{\partial V} \mathbf{F}\cdot d\mathbf{A} = \int_V (\nabla\cdot\mathbf{F})\, dV

Divergence (Gauss) Theorem

The total outward flux through a closed surface equals the sum of all the sources and sinks enclosed — surface bookkeeping equals volume bookkeeping.

vector calculus
∮∂SF⋅dr=∫S(∇×F)⋅dA\oint_{\partial S} \mathbf{F}\cdot d\mathbf{r} = \int_S (\nabla\times\mathbf{F})\cdot d\mathbf{A}

Stokes' Theorem

The circulation of a field around a closed loop equals the total curl 'flux' threading any surface that the loop bounds — boundary swirl equals interior swirl.

complex analysis
eiθ=cos⁡θ+i sin⁡θe^{i\theta} = \cos\theta + i\,\sin\theta

Euler's Formula

Multiplying by e^{iθ} rotates the complex plane by angle θ. The exponential of an imaginary number is a point on the unit circle, decomposing naturally into cosine (real) and sine (imaginary) parts.

complex analysis
∂u∂x=∂v∂y,∂u∂y=−∂v∂x\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \qquad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}

Cauchy–Riemann Equations

For a complex function f = u + iv to be differentiable (analytic), the rate of change must be the same in every direction. That single requirement collapses into two coupled equations relating the partials of the real and imaginary parts.

fourier analysis
f(x)=a02+∑n=1∞[ancos⁡nπxL+bnsin⁡nπxL]f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty}\left[a_n\cos\frac{n\pi x}{L} + b_n\sin\frac{n\pi x}{L}\right]

Fourier Series

Any reasonable periodic function is an infinite weighted sum of sines and cosines (or complex exponentials). The coefficients are the amounts of each pure frequency present — the function's recipe in tones.

fourier analysis
f^(k)=∫−∞∞f(x) e−ikx dx\hat{f}(k) = \int_{-\infty}^{\infty} f(x)\, e^{-i k x}\, dx

Fourier Transform

The Fourier transform measures how much of each frequency e^{ikx} a signal contains, by correlating the signal against every pure wave. It turns convolution into multiplication and derivatives into multiplication by ik.

fourier analysis
F{f∗g}=f^(k) g^(k)\mathcal{F}\{f * g\} = \hat{f}(k)\,\hat{g}(k)

Convolution Theorem

Convolution in one domain is plain multiplication in the other. A smearing operation that costs an integral over all shifts becomes a single product once you switch to the frequency domain.

partial differential equations
∂2u∂t2=c2 ∇2u\frac{\partial^2 u}{\partial t^2} = c^2\,\nabla^2 u

Wave Equation

Wherever the field curves in space (nonzero ∇²u), it accelerates in time — restoring toward its neighbors' average. That feedback launches disturbances that travel at speed c without changing shape.

partial differential equations
∂u∂t=D ∇2u\frac{\partial u}{\partial t} = D\,\nabla^2 u

Heat Equation

The temperature at a point rises in proportion to how much colder it is than its neighbors' average (the Laplacian). Heat flows down gradients, smoothing bumps and erasing fine detail irreversibly.

special functions
∫−∞∞f(x) δ(x−a) dx=f(a)\int_{-\infty}^{\infty} f(x)\,\delta(x-a)\,dx = f(a)

Dirac Delta Function

The delta function is an idealized spike with zero width and unit area. Its defining job is the sifting property: integrated against any function, it samples that function at a single point.

Probability & Stochastic Processes
P(A∣B)=P(B∣A) P(A)P(B)P(A\mid B)=\frac{P(B\mid A)\,P(A)}{P(B)}

Bayes' Theorem

Bayes' theorem is how you update a belief when new evidence arrives. Start with a prior — how likely a hypothesis was before — multiply by how well the hypothesis predicts the new data (the likelihood), and renormalize. The counterintuitive punchline: even a very accurate test for a rare condition mostly returns false positives, because the tiny prior overwhelms the likelihood. It flips 'probability of data given hypothesis' into the thing you actually want: 'probability of hypothesis given data.'

Theorems & Identities
(x+y)n=∑k=0n(nk)xn−kyk(x+y)^n=\sum_{k=0}^{n}\binom{n}{k}x^{n-k}y^{k}

Binomial Theorem

Expanding (x+y)ⁿ, each term picks either x or y from each of the n factors; the coefficient counts how many ways to pick k y's — that's 'n choose k', Pascal's triangle. Newton's leap was letting the exponent be any real number, turning the finite sum into an infinite series that gives (1+x)^α. Physicists live on the first two terms: (1+x)^α ≈ 1+αx for small x, the workhorse approximation behind nearly every 'to first order' calculation.

Theorems & Identities
∣⟨u,v⟩∣≤∥u∥ ∥v∥|\langle u,v\rangle|\le \lVert u\rVert\,\lVert v\rVert

Cauchy-Schwarz Inequality

The overlap of two vectors can never exceed the product of their lengths — geometrically because ⟨u,v⟩ = ‖u‖‖v‖cosθ and cosθ ≤ 1. Equality happens only when the vectors are parallel. This humble bound is secretly everywhere: it guarantees correlation coefficients stay within ±1, it underlies the triangle inequality, and in quantum mechanics it's the step that turns operator commutators into the Heisenberg uncertainty principle.

Probability & Stochastic Processes
1n∑i=1nXi−μσ  →d  N(0,1)\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\frac{X_i-\mu}{\sigma}\;\xrightarrow{d}\;\mathcal{N}(0,1)

Central Limit Theorem

Add up many small independent random effects — no matter what weird shape each one has — and their sum looks like a bell curve. This is why the normal distribution is everywhere: measurement errors, heights, thermal noise, diffusion. The individual quirks wash out; only the mean and variance survive. The averaged quantity's spread shrinks as 1/√n, which is why more measurements give proportionally better precision.

Probability & Stochastic Processes
P(k)=(nk)pk(1−p)n−kP(k)=\binom{n}{k}p^{k}(1-p)^{n-k}

Binomial Distribution

Flip a biased coin n times; the binomial distribution gives the probability of exactly k heads. Each specific sequence with k heads has probability p^k(1−p)^{n−k}, and there are C(n,k) such sequences. The distribution peaks near np, its mean, and spreads as √(np(1−p)). It's the discrete ancestor of the bell curve: for large n it becomes Gaussian (de Moivre–Laplace), and for rare events (small p, large n) it becomes Poisson.

Complex Analysis
f(z0)=12πi∮Cf(z)z−z0 dzf(z_0)=\frac{1}{2\pi i}\oint_{C}\frac{f(z)}{z-z_0}\,dz

Cauchy's Integral Formula

This is complex analysis's near-magical result: if you know a holomorphic function only on a closed loop, you know it everywhere inside. The value at any interior point is a weighted average around the boundary. It means analytic functions are absurdly rigid — pin down a curve and the interior is forced. It also hands you all the derivatives from the same boundary data, proving holomorphic functions are infinitely differentiable automatically.

Complex Analysis
∮Cf(z) dz=0\oint_{C}f(z)\,dz=0

Cauchy's Integral Theorem

If a complex function is holomorphic (smooth in the complex sense) throughout a region, then integrating it around any closed loop gives exactly zero. Equivalently, the integral between two points doesn't care which path you take. This is the complex cousin of a conservative force field having zero circulation. It's the foundation stone: from it flow Cauchy's integral formula, the residue theorem, and the whole contour-integration toolkit physicists use to crack hard integrals.

Complex Analysis
∂u∂x=∂v∂y,∂u∂y=−∂v∂x\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y},\quad \frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}

Analytic (Holomorphic) Functions

A complex function is analytic if it's differentiable in the complex sense — and that single requirement is astonishingly strong. Complex differentiability forces the real and imaginary parts to satisfy the Cauchy–Riemann equations, which make each part harmonic (a solution of Laplace's equation). Analytic functions are automatically infinitely differentiable and equal to their own Taylor series. It's a world where 'differentiable once' secretly means 'perfect forever' — nothing like real calculus.

Linear Algebra
p(λ)=det⁡(A−λI)=0p(\lambda)=\det(A-\lambda I)=0

Characteristic Polynomial

To find a matrix's eigenvalues — the special scaling factors along which it merely stretches vectors — you ask when A−λI collapses a direction to zero, i.e. when its determinant vanishes. That determinant, as a function of λ, is the characteristic polynomial. Its roots are the eigenvalues. Physicists call it the 'secular equation' because it first arose computing slow (secular) perturbations of planetary orbits. Its coefficients encode invariants: the trace and determinant of A.

Special Functions
x2y′′+xy′+(x2−ν2)y=0,y=Jν(x)x^2 y'' + x y' + (x^2-\nu^2)y=0,\quad y=J_\nu(x)

Bessel Functions

Bessel functions are what sine and cosine become in cylindrical worlds. Whenever a problem has circular symmetry — a vibrating drumhead, waves in a round pipe, light through a circular aperture — the radial part obeys Bessel's equation and the solutions J_ν(x) appear. They wiggle like damped sinusoids whose amplitude fades as 1/√x and whose zeros aren't evenly spaced. Those zeros set the drum's overtones and the diffraction rings you see around a point of light.

asymptotics perturbation
f(x)=∑n=0∞f(n)(a)n!(x−a)nf(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n

Taylor Expansion

Near a point a, a smooth function looks like its tangent line; add curvature and it looks like a parabola; add the next derivative and it hugs the curve further out. Each term corrects the previous approximation using one more derivative.

theorems identities
∫abf(x) dx=F(b)−F(a),F′(x)=f(x)\int_a^b f(x)\,dx = F(b) - F(a), \quad F'(x)=f(x)

Fundamental Theorem of Calculus

The area accumulated under f up to x is a function A(x). Nudging x by dx adds a sliver of area f(x)dx, so dA/dx = f(x). Accumulation and rate-of-change are inverse operations.

theorems identities
f′(c)=f(b)−f(a)b−a,c∈(a,b)f'(c) = \frac{f(b)-f(a)}{b-a}, \quad c \in (a,b)

Mean Value Theorem

The average rate of change over an interval is the slope of the secant. If the curve is smooth, its instantaneous slope must equal that average at least once — the tangent runs parallel to the secant somewhere inside.

probability stochastic
p(x)=1σ2π e−(x−μ)22σ2p(x) = \frac{1}{\sigma\sqrt{2\pi}}\,e^{-\frac{(x-\mu)^2}{2\sigma^2}}

Gaussian (Normal) Distribution

The exponential of a negative square gives a symmetric hump centered at μ. σ sets the spread, and the prefactor is exactly what makes the total probability one. Squaring the deviation punishes outliers smoothly, producing the ubiquitous bell.

linear algebra
Av=λvA\mathbf{v} = \lambda\mathbf{v}

Eigenvalue Equation

An eigenvector of A is a direction the matrix only stretches (by λ), never rotates. Those directions are the coordinate system in which A acts as simple scaling — the skeleton of the transformation.

linear algebra
det⁡ ⁣(abcd)=ad−bc\det\!\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc

Determinant

The determinant is the signed volume-scaling factor of the linear map. |det| tells you how much areas/volumes grow; its sign tells you whether orientation is preserved or flipped; zero means the map crushes space into a lower dimension.

symmetry algebra
R(θ)=(cos⁡θ−sin⁡θsin⁡θcos⁡θ)R(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}

Rotation Matrices

A rotation keeps lengths and angles fixed and the origin put — so its columns must be perpendicular unit vectors. In 2D those columns are (cosθ, sinθ) and its 90°-turned partner, giving the familiar cosine–sine block.

special functions
Γ(z)=∫0∞tz−1e−t dt\Gamma(z) = \int_0^{\infty} t^{z-1} e^{-t}\,dt

Gamma Function

The integral weighs powers of t against an exponential cutoff. Integration by parts reproduces the factorial recursion Γ(z+1)=zΓ(z), so the smooth curve inherits the factorial's stair-step growth while filling in every value between.

special functions
Pn(x)=12nn!dndxn(x2−1)nP_n(x) = \frac{1}{2^n n!}\frac{d^n}{dx^n}(x^2-1)^n

Legendre Polynomials

Legendre polynomials are what you get by orthogonalizing 1, x, x², … on the interval [-1,1]. They are the angular building blocks of any problem with spherical symmetry — each degree n captures one more level of angular detail.

calculus of variations
ddt ⁣(∂L∂q˙)−∂L∂q=0\frac{d}{dt}\!\left(\frac{\partial L}{\partial \dot q}\right) - \frac{\partial L}{\partial q} = 0

Euler–Lagrange Equation

A path that minimizes (or extremizes) an integral can't be improved by any tiny wiggle — the first-order change must vanish. Demanding that for every possible wiggle forces the integrand's derivatives into this exact balance, the equation of motion.

vector calculus
∮∂D(P dx+Q dy)=∬D ⁣(∂Q∂x−∂P∂y)dA\oint_{\partial D}(P\,dx + Q\,dy) = \iint_D\!\left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)dA

Green's Theorem

The circulation of a vector field around a closed curve counts the net swirl it encloses. Interior swirls of adjacent tiny loops cancel on shared edges, leaving only the outer boundary — so the boundary integral equals the area integral of the curl.

numerical methods
xn+1=xn−f(xn)f′(xn)x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}

Newton–Raphson Method

Near a root the curve looks like its tangent. So follow the tangent to where it crosses zero and use that as the next guess. Each step squares the error, giving quadratic convergence once you're close.

ordinary differential equations
dydx=f(x) g(y)  ⇒  ∫dyg(y)=∫f(x) dx\frac{dy}{dx} = f(x)\,g(y) \;\Rightarrow\; \int\frac{dy}{g(y)} = \int f(x)\,dx

Separable ODE

If the rate factorizes into an x-only piece times a y-only piece, you can herd all the y's to one side and all the x's to the other, then integrate each side independently. One integration on each side solves it.