Calculator
Enter numeric values. For symbolic metrics, evaluate numerically at a chosen point first.
Example data
Example: polar coordinates (r, θ) in 2D Euclidean space.
| Input | Value | Notes |
|---|---|---|
| n | 2 | Two coordinates: r, θ |
| Mode | Build g = Jᵀ η J | Use Jacobian of (x,y) w.r.t (r,θ) |
| η | diag(1,1) | Euclidean base metric |
| J | [ [cosθ, -r sinθ], [sinθ, r cosθ] ] | Evaluate numerically at a point, e.g., r=2, θ=0.5 |
| Expected g | [ [1, 0], [0, r²] ] | At r=2 ⇒ g = diag(1, 4) |
Formula used
When you provide a Jacobian for a coordinate transformation, the metric is computed by:
From the metric, the calculator computes:
- Inverse metric gⁱʲ (matrix inverse).
- Determinant det(g) and volume element √|det(g)|.
- Optional Christoffel symbols: Γᵏᵢⱼ = ½ gᵏˡ(∂ᵢgⱼˡ + ∂ⱼgᵢˡ − ∂ˡgᵢⱼ).
How to use this calculator
- Select the dimension n for your coordinate system.
- Choose Build g = Jᵀ η J for coordinate transformations, or Direct for known metrics.
- If using Jacobian mode, pick a base signature (Euclidean, Minkowski, or custom).
- Fill the matrix fields with numeric values, then press Submit.
- Use Download CSV or Download PDF to export your results.
- For Christoffel symbols, enable the option and provide numeric ∂g blocks.
FAQs
1) What is a metric tensor?
A metric tensor gᵢⱼ defines distances and angles on a space. It converts coordinate changes into measurable lengths using ds² = gᵢⱼ duⁱ duʲ.
2) When should I use Jacobian mode?
Use Jacobian mode when you know a coordinate transformation from x to u. The calculator applies g = Jᵀ η J to produce the metric in the new coordinates.
3) Why must det(g) be nonzero?
A nonzero determinant means the metric is invertible. Invertibility is needed to raise indices, compute Γᵏᵢⱼ, and define consistent inner products.
4) What does √|det(g)| represent?
It is the volume density used in integrals on curved spaces. For example, the area element is √|det(g)| du¹ du² in two dimensions.
5) Can I enter symbolic expressions like cos(θ)?
This tool is numeric. Evaluate expressions at a point (choose θ, r, etc.), then enter the resulting numbers to compute g, gⁱʲ, and det(g).
6) How do I input derivatives for Christoffel symbols?
Enter one n×n matrix per derivative coordinate, separated by “|”. Within each matrix, separate rows by “;” and values by commas, like “0,0;0,0 | 0,0;0,0”.
7) What if my metric is not symmetric?
Most standard metrics are symmetric. If yours is not, the calculator still computes the inverse if possible, but interpretations like distances may not apply without additional structure.