10.13.3 Tensor Jacobian Target Coordinate Role

The Tensor Jacobian links target coordinates to variable dependencies via differential transformations in multi-dimensional spaces.

Tensor Jacobian Target Coordinate Role is the function played by the new coordinate, appearing in the numerator of a Jacobian coordinate derivative entry, as the function being differentiated, marking that coordinate as the outcome of the change of coordinates rather than its starting point. It identifies which of the two coordinate systems involved in a Jacobian entry is being treated as the dependent function whose sensitivity to the source coordinate is being measured, a role that must be correctly recognized before the entry can be assigned to the correct column of the Jacobian matrix or contracted to produce the correct transformed index of a tensor.

Identifying the Target Coordinate

Position in the Derivative Expression

The target coordinate is the coordinate appearing in the numerator of a Jacobian entry, playing the role of the dependent function being differentiated with respect to the source coordinate.

\frac{\partial x^{j}'}{\partial x^{i}}

Here the primed coordinate in the numerator plays the target coordinate role, since it is this coordinate function whose rate of change is being measured.

Index Placement Reflecting the Target Role

The upper index of the Jacobian matrix entry is precisely the index of the target coordinate, so that recognizing an index as upper within the Jacobian notation immediately identifies the corresponding coordinate as playing the target role in that particular entry.

Function of the Target Coordinate Role

Determining the Column of the Jacobian Matrix

Because the upper index marks the target coordinate, and columns of the Jacobian matrix are conventionally organized by this upper index, the target coordinate role determines which column of the matrix a given derivative entry belongs to.

Producing the New Coordinate System in Tensor Contractions

When the Jacobian matrix is contracted against the components of a tensor, the target coordinate role corresponds to the new coordinate system, meaning the index playing this role is the free index labeling the transformed component being produced.

v^{j}' = \frac{\partial x^{j}'}{\partial x^{i}} v^{i}

Contrast With the Source Coordinate Role

Opposite Position in the Expression

While the target coordinate occupies the numerator of a Jacobian entry, the coordinate occupying the denominator plays the complementary source coordinate role, representing the coordinate system being transformed from rather than into.

Reversal Under the Inverse Jacobian

Taking the inverse Jacobian matrix exchanges these two roles entirely, so that the coordinate previously playing the target role in the forward Jacobian instead plays the source role in the inverse Jacobian, and vice versa.

\frac{\partial x^{i}}{\partial x^{j}'}

Consequences of the Target Coordinate Role

Necessity for Correct Labeling of Results

Misidentifying which coordinate in a Jacobian expression plays the target role would lead to mislabeling the free index of the resulting transformed component, producing an expression that does not correctly correspond to any single, well-defined coordinate in the new system.

Consistency With General Tensor Index Conventions

The target coordinate role parallels the general convention, already established for linear transformation matrices, that the upper index of a matrix factor is the one matching the free index of the newly produced quantity, extending that same convention naturally into the calculus-based Jacobian setting.

Schematic Representation

The diagram highlights the numerator of a Jacobian derivative entry as the location of the target coordinate, the function being differentiated representing the new coordinate system.