11.16.4 Tensor Coordinate Change Jacobian Factor

The Tensor Coordinate Change Jacobian Factor quantifies how tensor components transform under coordinate changes, essential in differential geometry and physics.

Tensor Coordinate Change Jacobian Factor is the matrix of partial derivatives relating one coordinate system to another that appears as the multiplicative element in every tensor transformation law, supplying the precise numerical factor by which each component of a tensor must be rescaled and recombined so that the tensor continues to represent the same underlying geometric object after the coordinate change.

Foundational Setting

Coordinates as Functions of One Another

When a manifold or space admits two coordinate systems, the coordinates ${\tilde{x}}^{i}$ of the new system can be expressed as functions of the coordinates $x^{j}$ of the old system. The Jacobian factor is built from the partial derivatives of this functional relationship.

The Two Jacobian Directions

There are two related but distinct Jacobian matrices relevant to tensor transformation, depending on the direction of differentiation:

J_{j}^{i} = \frac{\partial {\tilde{x}}^{i}}{\partial x^{j}}

K_{i}^{j} = \frac{\partial x^{j}}{\partial {\tilde{x}}^{i}}

These two matrices are inverses of one another whenever the coordinate change is invertible.

Role in Tensor Transformation Laws

Assignment to Contravariant Indices

Each upper, contravariant index of a tensor is multiplied by one factor of $J$ :

{\tilde{v}}^{i} = \sum_{j} J_{j}^{i} v^{j}

Assignment to Covariant Indices

Each lower, covariant index of a tensor is multiplied by one factor of $K$ , the matrix built from the reversed derivative direction:

{\tilde{ω}}_{i} = \sum_{j} K_{i}^{j} ω_{j}

Mixed Tensors and Combined Factors

For a tensor with both upper and lower indices, the Jacobian factor is applied once per index, with the appropriate matrix chosen according to whether that index is contravariant or covariant:

{\tilde{T}}_{i}^{j} = \sum_{k, l} J_{k}^{i} K_{j}^{l} T_{k}^{l}

The Determinant of the Jacobian Factor

Volume Scaling

The determinant of the Jacobian matrix measures how an infinitesimal volume element scales under the coordinate change:

d {\tilde{x}}^{1} \land \dots \land d {\tilde{x}}^{n} = det (J) d x^{1} \land \dots \land d x^{n}

Density and Tensor Weight

Quantities that pick up an extra factor of the Jacobian determinant during transformation, beyond the standard index-based factors, are called tensor densities. The power to which the determinant is raised is called the weight of the density, distinguishing true tensors, which have weight zero, from densities of nonzero weight.

Composition and Chain Rule Behavior

Sequential Coordinate Changes

When coordinates change through an intermediate system, the Jacobian factor for the overall transformation is the matrix product of the Jacobian factors of each step, a direct consequence of the multivariable chain rule:

J_{j}^{i} (total) = \sum_{k} J_{k}^{i} J_{j}^{k}

Local Validity

Because the Jacobian factor is built from derivatives evaluated at a point, it generally varies from point to point across a curved or nonlinearly parameterized space, so the tensor transformation law it governs is applied pointwise rather than globally with a single fixed matrix.

Summary of Key Traits

Defining Characteristics

The Jacobian factor consists of partial derivatives relating two coordinate systems.
Contravariant indices use one directional form of the Jacobian; covariant indices use its inverse form.
The determinant of the Jacobian factor governs volume and density scaling.
The factor composes multiplicatively under sequential coordinate changes and is generally position-dependent.