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Vector Calculus in a Nutshell

"Gradient, divergence, and the Laplacian are the three tools that let differential geometry and analysis speak the same language."

Vector calculus provides the analytical tools without which neither the heat equation nor the Ricci flow could be formulated. Here the \(\mathbb{R}^n\) concepts are recalled briefly and immediately generalised to Riemannian manifolds.

1. Gradient

On \(\mathbb{R}^n\): for a smooth function \(f\), $$ \nabla f = \big(\partial_1 f, \dots, \partial_n f\big). $$

On a Riemannian manifold \((M, g)\): the gradient is the vector representing the differential, $$ g(\nabla f, X) = \mathrm{d}f(X) = X(f) \quad \forall X. $$ In coordinates: \((\nabla f)^i = g^{ij}\, \partial_j f\). The gradient is the vector field of steepest ascent of \(f\).

2. Divergence

On \(\mathbb{R}^n\): \(\operatorname{div} X = \sum_i \partial_i X^i\).

On \((M, g)\) in local coordinates: $$ \operatorname{div} X = \frac{1}{\sqrt{\det g}}\, \partial_i \big(\sqrt{\det g}\, X^i\big). $$ Intuitively \(\operatorname{div} X\) measures how much volume is "lost" or "gained" per unit time under the flow of \(X\) – the infinitesimal source strength of the vector field.

3. Laplacian

On \(\mathbb{R}^n\): $$ \Delta f = \operatorname{div}(\nabla f) = \sum_i \partial_i^2 f. $$

On \((M, g)\) – the Laplace–Beltrami operator: $$ \Delta_g f = \frac{1}{\sqrt{\det g}}\, \partial_i!\Big(\sqrt{\det g}\, g^{ij}\, \partial_j f\Big). $$ The Laplacian measures the deviation of a function from its local average. Functions with \(\Delta f \equiv 0\) are called harmonic.

4. Integral theorems

Three classical integral theorems are the foundation of every analytical geometry:

Divergence theorem (Gauss). For \(X\) on a compact domain \(\Omega \subset M\) with boundary \(\partial \Omega\): $$ \int_\Omega \operatorname{div} X\, \mathrm{d}V_g = \int_{\partial \Omega} g(X, \nu)\, \mathrm{d}A, $$ where \(\nu\) is the outward unit normal.

Green's identity. For \(f, h\) on \(\Omega\): $$ \int_\Omega (f\, \Delta h - h\, \Delta f)\, \mathrm{d}V_g = \int_{\partial \Omega} (f\, \partial_\nu h - h\, \partial_\nu f)\, \mathrm{d}A. $$

Stokes (general): for an \((n-1)\)-form \(\omega\) $$ \int_M \mathrm{d}\omega = \int_{\partial M} \omega. $$

These theorems allow partial derivatives to be controlled on average by boundary values – a mechanism that appears in every energy estimate of the Ricci flow.

5. Important identities

Identity Content
\(\operatorname{div}(fX) = f\operatorname{div} X + g(\nabla f, X)\) product rule
\(\Delta(fh) = f\Delta h + h\Delta f + 2 g(\nabla f, \nabla h)\) product rule for \(\Delta\)
$\int_M f\, \Delta f\, \mathrm{d}V = -\int_M \nabla f
\(\partial_t \int_M f\, \mathrm{d}V_{g(t)} = \int_M (\partial_t f - f\, R_g)\, \mathrm{d}V_{g(t)}\) Ricci-flow variant (with \(\partial_t g = -2\mathrm{Ric}\), hence \(\partial_t \log\sqrt{\det g} = -R\))

The last line is the elementary identity used in every variation of an integral under the Ricci flow – e.g. in Perelman's monotonicity proofs (Act 2, Article 05).

6. Why this is central in the Ricci flow

The Ricci flow is a parabolic differential equation \(\partial_t g = -2 \mathrm{Ric}\). Its linearisation contains \(\Delta\), and Perelman's energy/entropy functionals are evaluated in closed form by integration by parts. Without Gauss's divergence theorem and Green's identities there would be neither the \(\mathcal{F}\)- nor the \(\mathcal{W}\)-functional.

Cross-references

Sources

  • Lee, John M. (2018). Introduction to Riemannian Manifolds. Springer GTM 176, 2nd ed. Ch. 2.
  • Forster, Otto (2017). Analysis 3. Springer Spektrum, 8th ed.
  • Marsden, J. & Tromba, A. (2011). Vector Calculus. W. H. Freeman, 6th ed.
  • do Carmo, Manfredo P. (1992). Riemannian Geometry. Birkhäuser. App. A.