ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

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Divergence in polar coordinates

Example

For the Euclidean metric in polar coordinates, $div (X^{r} \partial_{r} + X^{θ} \partial_{θ}) = r^{- 1} \partial_{r} (r X^{r}) + \partial_{θ} X^{θ}$ .

Facts & Assumptions

Given: A polar chart with $r > 0$ , metric matrix $diag (1, r^{2})$ , and a smooth vector field $X$ .

[F1]

Coordinate formula for riemannian divergence: In coordinates, ${div}_{g} X = (\det G)^{- 1 / 2} \sum_{i = 1}^{n} \partial_{i} ((\det G)^{1 / 2} X^{i})$ .

Verification

technique · direct

1.1

The positive square root of the metric determinant is $r$ . Substituting into the divergence formula gives $div X = r^{- 1} {\partial_{r} (r X^{r}) + \partial_{θ} (r X^{θ})}$ . Since $\partial_{θ} r = 0$ , the second term is $\partial_{θ} X^{θ}$ , proving the formula.

F1given

2.1

The orthonormal polar frame is $e_{r} = \partial_{r}$ , $e_{θ} = r^{- 1} \partial_{θ}$ . Thus if $X = a e_{r} + b e_{θ}$ , its coordinate components are $X^{r} = a$ , $X^{θ} = b / r$ , and the formula becomes $r^{- 1} \partial_{r} (r a) + r^{- 1} \partial_{θ} b$ . For $X = r \partial_{r}$ , it gives $r^{- 1} \partial_{r} (r^{2}) = 2$ . For $X = \partial_{θ}$ it gives $0$ .

step 1.1∎

Source locator

Lee, p.423, definition of divergence and Exercise 16.31; Example 13.12, p.332, polar metric. The coordinate divergence theorem declared as F1 supplies the local coefficient formula used above.

Depends on

Coordinate formula for riemannian divergence

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John M. Lee, Introduction to Smooth Manifolds, second edition (standard reference, not scraped)