Calculus

Differential Forms

Цели урока

Understand k-forms as the unified framework for functions, covectors, and volume elements
Master the exterior derivative d and its key property d^2 = 0
Know the generalized Stokes theorem and identify all classical theorems as special cases
Understand pullback as the operation that makes forms work on manifolds

Предварительные знания

Divergence Theorem
Classical Stokes Theorem
Basic manifolds

Divergence Theorem

Why are Green's theorem, Stokes' theorem, and Gauss's theorem all the same formula - and can one operator replace gradient, curl, and divergence?

Maxwell's equations: dF = 0 and d*F = J encode all of electromagnetism in two lines using 2-forms on 4D spacetime
PyTorch autograd: backpropagation is pullback of the loss gradient through layers - F*(d L) = d(F* L) is the chain rule
General relativity: Einstein's equations G = 8*pi*T involve curvature forms on a 4D Lorentzian manifold
Geometric machine learning: discrete exterior calculus on triangle meshes computes discrete versions of d and Laplacian for Graph Neural Networks

Cartan and the birth of exterior calculus

Elie Cartan (1869-1951) spent much of his career outside the academic mainstream. His theory of differential forms and moving frames was considered too abstract for decades. In 1901 he introduced the exterior derivative d with the fundamental property d^2 = 0. Poincare used forms in topology. Levi-Civita and Ricci built tensor calculus. But Cartan unified them. By 1945 exterior calculus was standard language in differential geometry. Today it underlies string theory, topological quantum field theory, and geometric deep learning on manifolds.

k-Forms and the Exterior Product

Elie Cartan in 1901 built exterior calculus: the wedge product, the exterior derivative d, and the fundamental identity d^2 = 0. In 1865 Maxwell needed 20 equations to describe electromagnetism. With differential forms: four equations, then two, then ultimately dF = 0 and d*F = J. One notation that survived 120 years because it captures what matters.

On R^3: gradient = d on 0-forms, curl = *d on 1-forms, divergence = *d* on 1-forms. Three distinct vector calculus operations are all the same operator d, just in different degrees. The Hodge star * connects complementary degrees.

The electromagnetic 2-form

Maxwell's equations in forms

The Faraday 2-form on 4D spacetime: F = E_x dx^dt + E_y dy^dt + E_z dz^dt + B_x dy^dz + B_y dz^dx + B_z dx^dy. The equation dF = 0 encodes two Maxwell equations: div B = 0 and curl E = -dB/dt. The equation d*F = J encodes the other two: div E = rho/eps_0 and curl B = mu_0 J + ... Eight equations become two lines.

The wedge product dx wedge dy gives an oriented area element: swapping to dy wedge dx reverses the sign. When integrating a 2-form over a surface, orientation matters. This is why area integrals pick up a sign under coordinate changes.

The 2-form dx wedge dy on R^2, integrated over the unit square [0,1]^2, equals:

iint_{D} dx wedge dy = iint_D 1 dA = area(D). For the unit square = 1.

Exterior Derivative and d^2 = 0

d^2 = 0. Three characters encoding all of topology. The boundary of a boundary is empty: partial(partial M) = 0. This is not just an analogy - it is a precise duality. Closed forms (d omega = 0) and exact forms (omega = d alpha) differ by exactly the topological obstructions - the holes in the space. de Rham cohomology measures this difference.

The identity rot(grad f) = 0 on R^3 corresponds to:

rot(grad f) = 0 is d^2 f = 0. Cause: Schwarz's theorem - mixed partial derivatives are equal, so all terms in the curl expansion cancel pairwise.

The Generalized Stokes Theorem

One formula. int_M d(omega) = int_{boundary M} omega. This contains Newton-Leibniz, Green, classical Stokes, and Gauss. All of integral calculus in six symbols. Cartan wrote this in 1945, and since then the subject has been seen differently: integration is not about regions - it is about the pairing between forms and manifolds.

The generalized Stokes theorem requires an oriented manifold with boundary. The orientation on M and the induced orientation on its boundary must be consistent. The induced orientation: outward normal first, then tangent vectors to the boundary.

Stokes theorem and Maxwell's Faraday law

Integral and differential forms of an equation

Faraday's law: oint_C E.dr = -d/dt iint_S B.dS. Left side = int_C E^flat. Right side: -d/dt of flux. Classical Stokes theorem: oint_C E.dr = iint_S curl E.dS. Equating with Faraday: iint_S curl E dS = -iint_S dB/dt dS for any surface S. Therefore curl E = -dB/dt (differential form). The generalized Stokes theorem converts between integral and differential forms of every conservation law.

The generalized Stokes theorem at dim omega = n-1, dim M = n is:

int_M d(omega) = int_{boundary M} omega is universal. n=1: Newton-Leibniz. n=2: Green or classical Stokes (depending on dimension of ambient space). n=3: Gauss.

Pullback: Forms on Manifolds

The pullback is what makes differential forms work on manifolds: a map F: N -> M pulls back any form on M to a form on N. Without pullback, one cannot even define what it means to integrate over a manifold described by charts. PyTorch backpropagation is structurally a pullback: the gradient of the loss is pulled back through layers. The chain rule is F*(d L) = d(F* L) in each layer.

Neural network backpropagation: each layer f_i: R^n -> R^m. The backward pass computes f_i*(dL) - the pullback of the loss gradient through f_i. The chain rule ensures f_i*(d(L circ f_{i+1} circ ...)) = d(f_i*(L circ ...)). Pullback commuting with d is exactly the chain rule.

Change of variables x = phi(u) with factor |det J_phi| in a multiple integral is:

phi*(dx^1 wedge ... wedge dx^n) = det(d phi) du^1 wedge ... wedge du^n. The Jacobian is the determinant of the differential of phi. Change of variables is exactly the pullback of the volume form.

Connections to other topics

Differential forms provide the universal language for integration, topology, and physics.

Theoretical physics — Related topic
Topology — Related topic
Machine learning — Related topic
Numerical analysis — Related topic

Итоги

k-form: antisymmetric k-covector field; 0-forms = functions, 1-forms = covectors, n-forms = volume elements
Exterior derivative d: Omega^k -> Omega^{k+1}, d^2 = 0 - follows from Schwarz's theorem on mixed partials
On R^3: d on 0-forms = gradient, d on 1-forms = curl, d on 2-forms = divergence
Generalized Stokes: int_M d(omega) = int_{boundary M} omega - Newton-Leibniz, Green, Stokes, Gauss as special cases
Pullback F*(omega) pulls forms back through a map; F*(d omega) = d(F* omega) - pullback commutes with d

Вопросы для размышления

d^2 = 0 encodes 'the boundary of a boundary is empty'. How does this algebraic identity encode topological information about a space?
The Cauchy-Riemann equations are the condition d(omega) = 0 for a 1-form in the complex plane. What does this say about analytic functions in terms of differential forms?
Pullback commutes with d: F*(d omega) = d(F* omega). This is the chain rule in disguise. In what sense is backpropagation in neural networks a sequence of pullbacks?

Связанные уроки

calc-26-divergence-theorem — Divergence theorem is the 3D special case of the generalized Stokes theorem
calc-28-manifold-int — Integration of forms on manifolds is the next step
calc-29-derham — de Rham cohomology is built on d^2 = 0