Curvature Tensor Calculator

Input Form

This calculator evaluates a 2D metric numerically at a single point.

Point and Display Controls

Report label

Coordinate 1 name

Coordinate 2 name

Coordinate 1 value

Coordinate 2 value

Precision

Zero tolerance

Metric Tensor Components

Use a symmetric 2×2 metric: g12 = g21.

g11

g12

g22

First Derivatives of Metric Components

∂g11/∂coord1

∂g11/∂coord2

∂g12/∂coord1

∂g12/∂coord2

∂g22/∂coord1

∂g22/∂coord2

Second Derivatives of Metric Components

Enter dxx, dxy, and dyy terms for each independent metric component.

∂²g11/∂coord1²

∂²g11/∂coord1∂coord2

∂²g11/∂coord2²

∂²g12/∂coord1²

∂²g12/∂coord1∂coord2

∂²g12/∂coord2²

∂²g22/∂coord1²

∂²g22/∂coord1∂coord2

∂²g22/∂coord2²

Example Data Table

The table below matches the built-in unit sphere example at θ = 1 and φ = 0. It should return K ≈ 1 and R ≈ 2.

Input	Value	Meaning
g11	1.000000	Meridional metric coefficient
g12	0.000000	Cross term
g22	0.708073	sin²(θ) at θ = 1
∂g22/∂θ	0.909297	sin(2θ) at θ = 1
∂²g22/∂θ²	-0.832294	2cos(2θ) at θ = 1
Expected K	1.000000	Unit sphere Gaussian curvature
Expected R	2.000000	Scalar curvature in 2D

Formula Used

This page evaluates local curvature from a symmetric 2D metric tensor g_ij and its first and second derivatives at one coordinate point.

Quantity	Formula
Inverse metric	g^ij = (g_ij)^-1
Christoffel symbols	Γⁱ_jk = 1/2 · g^im(∂_jg_km + ∂_kg_jm - ∂_mg_jk)
Riemann tensor	Rⁱ_jkl = ∂_kΓⁱ_jl - ∂_lΓⁱ_jk + Γⁱ_mkΓ^m_jl - Γⁱ_mlΓ^m_jk
Ricci tensor	Ric_jl = Rⁱ_jil
Scalar curvature	R = g^jlRic_jl
Gaussian curvature in 2D	K = R/2 = R₁₂₁₂ / det(g)

The calculator also computes the derivative of the inverse metric, Ricci norm, and the Kretschmann scalar for additional local geometry checks.

How to Use This Calculator

Enter coordinate labels and the point where the metric is evaluated.
Provide the symmetric metric coefficients g11, g12, and g22.
Enter all first derivatives for the independent metric components.
Enter second derivatives for g11, g12, and g22.
Choose precision and a zero tolerance for display cleanup.
Press Compute Curvature Tensor to generate results above the form.
Review scalar curvature, Gaussian curvature, Ricci tensor, and Riemann components.
Use the CSV and PDF buttons to export the current report.

FAQs

1. What does this calculator compute?

It computes a 2D curvature workflow numerically at one point: inverse metric, Christoffel symbols, mixed and lowered Riemann tensor components, Ricci tensor, scalar curvature, Gaussian curvature, Ricci norm, and the Kretschmann scalar.

2. Why is the calculator limited to two dimensions?

A two-dimensional version keeps the page practical, readable, and fast while still covering core curvature ideas. It also supports Gaussian curvature checks directly, which makes validation easier for teaching, research drafts, and local numeric testing.

3. Can I use it for non-Euclidean surfaces?

Yes. Any non-singular symmetric 2D metric can be evaluated numerically. Spherical, hyperbolic, polar, conformal, and custom local metrics are all acceptable as long as you provide consistent first and second derivatives at the chosen point.

4. What happens if det(g) is near zero?

The metric becomes nearly singular, so the inverse metric and curvature quantities become unstable. The calculator stops the computation when the determinant is too close to zero because Christoffel and curvature values would be unreliable.

5. Why are there two Gaussian curvature values?

One uses K = R/2, and the other uses the lowered Riemann component divided by det(g). Matching values are a helpful internal check that your derivatives and numeric precision are coherent.

6. Do I need symbolic formulas before using this tool?

No. You only need numeric values of the metric and its derivatives at one point. Those values may come from symbolic work, numerical differentiation, external software, or a known closed-form surface example.

7. What is the Kretschmann scalar here?

It is the full contraction R_ijklR^ijkl. This scalar measures total curvature intensity and is often useful when comparing local geometry strength across coordinate choices or across different surfaces.

8. Are exported files based on visible results?

Yes. The CSV and PDF exports are generated from the tables shown on the page after calculation. Recompute the calculator first if you change any input and want the downloads to match.