Tangent Vectors and Differential Forms

Tangent Spaces and Differential Forms

Tangent Vector Without Ambient Space

On a surface in ℝ³, a tangent vector is literally “an arrow lying on the surface.” But how do you define a tangent vector to an abstract manifold M not embedded in ℝⁿ?

An elegant solution: a tangent vector at point p is an equivalence class of smooth curves γ: (−ε, ε) → M with γ(0) = p (with the same speed in local coordinates). Or, equivalently, a differential operator v(f) = (f ∘ γ)'(0) — acting on functions (derivative along γ).

The second definition always works and reveals: “vector” = “a way to differentiate functions at a point.”

Tangent Space TₚM

TₚM is the vector space of all tangent vectors to M at point p. The dimension dim(TₚM) = n.

In a local chart φ = (x¹, ..., xⁿ): the basis of TₚM consists of partial derivatives ∂/∂xⁱ|_p, acting as (∂/∂xⁱ)(f) = ∂(f ∘ φ⁻¹)/∂xⁱ.

Any vector v ∈ TₚM decomposes as: v = vⁱ ∂/∂xⁱ (summation over i = 1,...,n).

Differential of a map F: M → N at point p: TₚF: TₚM → T_{F(p)}N — linear mapping. In coordinates: Jacobian matrix.

Vector Fields

A vector field X on M: smooth section of the tangent bundle TM → M. At each point p — a vector X(p) ∈ TₚM, smoothly depending on p.

In coordinates: X = Xⁱ(x) ∂/∂xⁱ, where Xⁱ are smooth functions.

Flow of a field X: system of ODEs dy/dt = X(y), y(0) = p. The solution Φ_t(p) is a one-parameter group of diffeomorphisms.

Lie brackets [X, Y]: the commutator of differential operators, again a vector field: X, Y = X(Y(f)) − Y(X(f)). In coordinates: [X, Y]ⁱ = Xʲ ∂Yⁱ/∂xʲ − Yʲ ∂Xⁱ/∂xʲ.

Geometric meaning: [X, Y] = 0 ↔ flows of X and Y “commute” (it does not matter in which order you move along X and Y).

Example: On ℝ², X = ∂/∂x, Y = ∂/∂y: [X, Y] = 0 (parallel translations commute). X = ∂/∂x, Y = x ∂/∂y: [X, Y] = ∂/∂y (moving along x, then along y, is not the same as the reverse order).

Differential Forms

A 1-form ω on M: at each point p — a linear functional ωₚ: TₚM → ℝ, smoothly depending on p.

In coordinates: ω = ωᵢ(x) dxⁱ, where dxⁱ are coordinate differentials (basis of the cotangent space).

Application: ω(X) = ωᵢ Xⁱ — a function on M.

Differential of a function: df = (∂f/∂xⁱ) dxⁱ — standard example of a 1-form.

k-form: antisymmetric multilinear functional on TₚM × ... × TₚM (k times). In coordinates: ω = ωᵢ₁...ᵢₖ dxⁱ¹ ∧ ... ∧ dxⁱₖ (∧ — wedge product, antisymmetric).

Wedge product: α ∧ β(X, Y) = α(X)β(Y) − α(Y)β(X) (for 1-forms). Antisymmetry: α ∧ β = −β ∧ α.

Exterior derivative d: k-form → (k+1)-form, d ∘ d = 0. In coordinates: d(ωᵢ dxⁱ) = (∂ωᵢ/∂xʲ) dxʲ ∧ dxⁱ.

de Rham Cohomology

Closed form: dω = 0. Exact form: ω = dη. From d² = 0: exact ⇒ closed. The converse is not always true.

Poincaré lemma: On a contractible domain (ℝⁿ, a ball) every closed form is exact. For spaces with “holes” this is no longer guaranteed.

Example: On ℝ² {0}: the form ω = (x dy − y dx)/(x² + y²) is closed (dω = 0). But ∮_{|r|=1} ω = 2π ≠ 0 → ω is not exact (there is no f : df = ω on all of ℝ² {0}).

de Rham groups: H^k(M) = {closed k-forms}/{exact k-forms}. Topological invariant: H^k(M) ≅ H^k_sing(M; ℝ) (de Rham’s theorem).

H¹(S¹) = ℝ (the 1-form dθ is not exact on S¹). H¹(ℝ²) = 0 (everything is contractible). H¹(T²) = ℝ² (two independent cycles on the torus).

Real Applications

Electromagnetism: F = dA (Faraday tensor = exterior derivative of the 4-potential). dF = 0 (Maxwell's equations without sources — closedness). Gauge invariance: A and A + dχ yield the same field F.

Thermodynamics: The 1-form δQ = T dS (heat) is not exact (no function Q: dQ = δQ). But dS is exact: along a cycle ∮ dS = 0. The second law of thermodynamics — closedness, not exactness of δQ/T = dS.

Differential Forms in Thermodynamics and Theoretical Physics

Differential forms are a language simultaneously describing field physics, thermodynamics, and mechanics without reference to a coordinate system. The one-form dU — the differential of internal energy — expresses the first law of thermodynamics: dU = T dS − p dV, where T dS and p dV are thermodynamic one-forms on the state space. The second law asserts that dS = δQ/T is an exact form (full differential) only for reversible processes, whereas the heat δQ of nonequilibrium processes is not an exact form — this mathematically expresses irreversibility. In electrodynamics, Maxwell's equations take the coordinate-invariant form: dF = 0 and d★F = J, where F = B + E∧dt is the two-form of the electromagnetic field, ★ is the Hodge operator. Such notation immediately generalizes to curved spacetime in General Relativity. In statistical mechanics, the Liouville measure (product of symplectic forms dq¹∧dp₁∧...∧dqⁿ∧dpₙ) is preserved along the Hamiltonian flow — the foundation of statistical physics and justification of ensembles. In modern geometry of partial differential equations, Cartan forms are used for invariant description of symmetry classes of equations and construction of conservation laws.

The Frobenius theorem in modern geometry generalizes to G-structures: flat connection (R = 0) on a bundle is integrable in the sense that parallel frames exist. This is used in computer vision for camera calibration via integrability of the connection on the SO(3)-bundle of orientations: the condition R = 0 means that the camera system can be coherently calibrated globally without accumulating errors. In processing tensor MRI images of the brain, bundles of orientations of neural fibers (diffusion MRI) use connection and torsion for tractography of neural pathways.

Assignment: (a) Compute [∂/∂x + y ∂/∂y, x ∂/∂y] on ℝ². (b) For the form ω = x dy − y dx on ℝ² {0}: compute dω. Is ω closed? Is it exact? (c) What is H²(S²)? Compute and interpret.