Metric Tensor

Metric Tensor Calculations in Curvilinear Coordinates

1. Cartesian Coordinates (3D Euclidean Space)

In Cartesian coordinates (x, y, z), the line element ds2 is:

ds2 = dx2 + dy2 + dz2

Thus, the metric tensor gij is:

gij = 
[1  0  0]
[0  1  0]
[0  0  1]

2. Polar Coordinates (2D)

In polar coordinates (r, θ) with transformations:

x = r cos θ, y = r sin θ

The line element ds2 becomes:

ds2 = dr2 + r2 dθ2

Therefore, the metric tensor gij is:

gij = 
[1    0]
[0  r2]

3. Spherical Coordinates (3D)

In spherical coordinates (r, θ, φ) with transformations:

x = r sin θ cos φ
y = r sin θ sin φ
z = r cos θ

The line element ds2 becomes:

ds2 = dr2 + r2 dθ2 + r2 sin2 θ dφ2

The metric tensor gij is:

gij = 
[1         0                0]
[0      r2         0]
[0         0   r2 sin2 θ]

General Formula for Curvilinear Coordinates

To calculate the metric tensor gij in curvilinear coordinates (q1, q2, ..., qn), use:

gij = ∂xk / ∂qi * ∂xk / ∂qj

This approach can be applied to any curvilinear coordinate system.

Example Calculation in Polar Coordinates

1. Define transformations:

x = r cos θ, y = r sin θ

2. Calculate grr:

grr = (∂x/∂r)2 + (∂y/∂r)2 = 1

3. Calculate gθθ:

gθθ = (∂x/∂θ)2 + (∂y/∂θ)2 = r2