10.15.4 Tensor Coordinate Transformation Jacobian Relation

The Jacobian matrix governs how tensor components transform between coordinate systems in multilinear algebra.

Tensor Coordinate Transformation Jacobian Relation is the precise correspondence, within the coordinate transformation process, between the partial derivatives of the target chart's coordinates with respect to the source chart's coordinates and the multiplicative coefficients that carry tensor components from the source chart into the target chart, forming the mathematical link that makes the whole transformation process well defined.

Statement of the Relation

Defining the Jacobian From the Two Charts

Given the source chart coordinates $x^{i}$ and the target chart coordinates ${\bar{x}}^{j}$ , the Jacobian relation defines the transformation coefficients as the partial derivatives connecting them:

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

This is the relation that turns an abstract pair of coordinate charts into a concrete numerical array usable inside a transformation formula, and every subsequent step of the coordinate transformation process depends on having this relation correctly evaluated at the point in question.

The Reciprocal Relation

The Jacobian relation is paired with its reciprocal, the inverse Jacobian relation, obtained by holding the target coordinates fixed and differentiating the source coordinates with respect to them:

{(J^{- 1})}_{j}^{i} = \frac{\partial x^{i}}{\partial {\bar{x}}^{j}}

How the Relation Enters the Transformation

Insertion Into the Component Formula

The coordinate transformation process inserts the Jacobian relation directly into the tensor transformation law by matching each upper index of the tensor to a forward Jacobian factor and each lower index to an inverse Jacobian factor, so that a general mixed tensor picks up exactly as many Jacobian relations as it has indices:

{\bar{T}}_{l}^{k} = \sum_{i}^{n} \sum_{j}^{n} J_{i}^{k} {(J^{- 1})}_{l}^{j} T_{j}^{i}

Necessity of Evaluating at the Same Point

Both the forward and inverse Jacobian relations must be evaluated at the same point of the overlap between the source and target chart domains as the tensor component being transformed, since the derivatives making up the Jacobian relation are, in the general curvilinear case, different at every point, and using values taken from a different point would produce an inconsistent, meaningless result.

Consistency Enforced by the Relation

Chain Rule Compatibility

The Jacobian relation automatically satisfies the multivariable chain rule whenever a third chart is introduced, meaning that the Jacobian relation between the source and target charts, computed through an intermediate chart, agrees with the Jacobian relation computed directly between the source and target charts:

J_{i}^{k} = \sum_{m}^{n} {\overset{'}{J}}_{i}^{m} {\overset{''}{J}}_{m}^{k}

where the primed Jacobians denote the relations connecting the source chart to the intermediate chart and the intermediate chart to the target chart, respectively.

Product Identity With the Inverse

The Jacobian relation and its reciprocal must combine to the Kronecker delta whenever multiplied and summed over the shared index, which is the algebraic check confirming that the two relations genuinely describe forward and backward passage between the same pair of charts rather than two unrelated sets of derivatives.

Diagram of the Relation

Two Charts Linked by One Derivative Array

Failure Modes of the Relation

Non-Invertible Points

At any point where the Jacobian relation yields a matrix with a zero determinant, the reciprocal inverse Jacobian relation cannot be formed, and the coordinate transformation process cannot proceed for tensors carrying covariant indices at that point, since the required inverse derivatives simply do not exist there.

Discontinuities and Non-Smooth Transition Maps

If the transition map between the source and target charts fails to be smooth at some point, the Jacobian relation may be discontinuous or undefined there, and any tensor component transformed using values taken across such a discontinuity loses its meaning as a faithful representation of the same underlying tensor.