Nyimbi Odero

Series 01 · a numbered run · kept in public

Part 2: Vector Analysis in Curved Spaces

Chapter: Lesson 4: Multivariable Calculus II

Introduction to Curved Geometry

When we move from flat to curved spaces, our familiar calculus operations must be generalized carefully. Just as a person walking on a sphere can’t use flat maps without distortion, we need new tools to properly describe how vectors and fields behave on curved surfaces and in curved spaces.

Let’s organize our exploration using the mnemonic “CURVED”:

Covariant derivatives Understanding metrics Riemannian structure Vector transport Extended operators Differential forms

The Metric: Our Guide to Distance

The metric tensor gᵢⱼ is fundamental to understanding curved spaces. It tells us how to measure distances and angles:

  1. Line Element
ds2=gijdxidxjds^2 = g_{ij}dx^idx^j

This generalizes the Pythagorean theorem to curved spaces.

  1. Inner Product For vectors v and w:
⟨v,w⟩=gijviwj\langle \mathbf{v}, \mathbf{w} \rangle = g_{ij}v^iw^j

This tells us how to compute lengths and angles.

  1. Volume Element
dV=∣g∣dx1...dxndV = \sqrt{|g|}dx^1...dx^n

where |g| is the determinant of the metric.

Covariant Derivative: The True Directional Derivative

In curved space, ordinary derivatives don’t transform properly. We need the covariant derivative ∇μ:

  1. Definition For a vector field V:
∇μVν=∂μVν+ΓμλνVλ\nabla_\mu V^\nu = \partial_\mu V^\nu + \Gamma^\nu_{\mu\lambda}V^\lambda

where Γᵅβγ are the Christoffel symbols:

Γμνλ=>12gλσ(∂μgνσ+>∂νgμσ−∂σ>gμν)\Gamma^\lambda_{\mu\nu} = > \frac{1}{2}g^{\lambda\sigma}(\partial_\mu g_{\nu\sigma} + > \partial_\nu g_{\mu\sigma} - \partial_\sigma > g_{\mu\nu})

  1. Properties

    • Leibniz rule: ∇(fV) = (∇f)V + f∇V

    • Metric compatibility: ∇g = 0

    • Torsion-free: Γᵅβγ = Γᵅγβ

Divergence in Curved Space

The divergence generalizes to:

  1. Coordinate Expression
∇⋅V=1∣g∣∂i(∣g∣Vi)\nabla \cdot \mathbf{V} = \frac{1}{\sqrt{|g|}}\partial_i(\sqrt{|g|}V^i)
  1. Geometric Meaning Still measures flux per unit volume, but volume element is
∣g∣dnx\sqrt{|g|}d^nx
  1. Gauss’s Theorem
∫M∇⋅V,dV=∮∂MV⋅dA\int_M \nabla \cdot \mathbf{V},dV = \oint_{\partial M} \mathbf{V} \cdot d\mathbf{A}

remains valid with appropriate measure.

Curl in Curved Space

The curl operation becomes more subtle:

  1. Using the Metric
(curl V)i=ϵijk∇jVk(\text{curl }\mathbf{V})^i = \epsilon^{ijk}\nabla_j V_k

where εᵢⱼₖ is the Levi-Civita tensor.

  1. Stokes’ Theorem
∫S(∇×V)⋅dA=∮∂SV⋅dr\int_S (\nabla \times \mathbf{V}) \cdot d\mathbf{A} = \oint_{\partial S} \mathbf{V} \cdot d\mathbf{r}

remains valid with covariant derivatives.

Differential Forms: A Modern Approach

Differential forms provide a coordinate-independent way to handle vector calculus:

  1. k-Forms A k-form ω can be written:
ω=1k!ωi1...ikdxi1∧...∧dxik\omega = \frac{1}{k!}\omega_{i_1...i_k}dx^{i_1}\wedge...\wedge dx^{i_k}
  1. Exterior Derivative
dω=1k!∂jωi1...ikdxj∧dxi1∧...∧dxikd\omega = \frac{1}{k!}\partial_j\omega_{i_1...i_k}dx^j\wedge dx^{i_1}\wedge...\wedge dx^{i_k}

This unifies grad, curl, and div.

  1. Hodge Star
(⋆ω)∗i∗k+1...in=1k!∣g∣ϵi1...inωi1...ik(\star\omega)*{i*{k+1}...i_n} = \frac{1}{k!}\sqrt{|g|}\epsilon_{i_1...i_n}\omega^{i_1...i_k}

Allows us to define codifferential δ = ±d.

Physical Applications

  1. General Relativity Einstein’s equations:
Rμν−12Rgμν=8πGc4TμνR_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu} = \frac{8\pi G}{c^4}T_{\mu\nu}

involve covariant derivatives and curved space operations.

  1. Maxwell’s Equations In differential form notation:
dF=0dF = 0 d⋆F=⋆Jd\star F = \star J

where F is the electromagnetic field strength 2-form.

  1. Fluid Dynamics on Manifolds Navier-Stokes equations:
∇tvi+vj∇jvi=−1ρgij∂jp+ν∇j∇jvi\nabla_t v^i + v^j\nabla_j v^i = -\frac{1}{\rho}g^{ij}\partial_j p + \nu\nabla^j\nabla_j v^i

Advanced Topics

  1. Connection Forms The covariant derivative can be written:
∇XY=dY(X)+ω(X)Y\nabla_X Y = dY(X) + \omega(X)Y

where ω is the connection 1-form.

  1. Curvature Forms The Riemann tensor becomes:
R=dω+ω∧ωR = d\omega + \omega\wedge\omega

showing curvature as obstruction to parallel transport.

  1. Killing Vector Fields Vector fields X satisfying:
LXg=0\mathcal{L}_X g = 0

generate symmetries of the metric.

Computational Techniques

  1. Coordinate Calculations For a spherical metric:
ds2=dr2+r2(dθ2+sin⁡2θdϕ2)ds^2 = dr^2 + r^2(d\theta^2 + \sin^2\theta d\phi^2)

Calculate:

  • Christoffel symbols

  • Covariant derivatives

  • Divergence and curl

  1. Working with Forms For electromagnetic potential A:

    • Write F = dA

    • Calculate *F

    • Verify Maxwell’s equations

Exercises for Understanding

  1. Basic Computation On a sphere of radius R:

    • Calculate the divergence of V = (sin θ cos φ, sin θ sin φ, cos θ)

    • Interpret the result physically

  2. Physical Application For a Schwarzschild black hole:

    • Write down the metric

    • Calculate the Christoffel symbols

    • Find the geodesic equations

  3. Advanced Problem Using differential forms:

    • Show that d²ω = 0

    • Prove Stokes’ theorem locally

    • Calculate de Rham cohomology of a sphere

Looking Ahead

In our next section, we’ll explore integral theorems in curved spaces. These will be essential for understanding:

  • Global properties of manifolds

  • Topological invariants

  • Gauge theories

  • Quantum field theory in curved space

Remember: The mathematics of curved spaces isn’t just abstract theory — it’s how we describe the actual geometry of spacetime and the fields that live in it. Each generalization we’ve made is motivated by physical reality.

Next: Part 3 - Integral Theorems in Curved Space