10.13.2 Tensor Jacobian Source Coordinate Role

The Tensor Jacobian Source Coordinate Role defines how coordinate transformations affect tensor derivatives in multivariable calculus.

Tensor Jacobian Source Coordinate Role is the function played by the old coordinate, appearing in the denominator of a Jacobian coordinate derivative entry, as the variable with respect to which the differentiation is taken, marking that coordinate as the starting point of the change of coordinates rather than its outcome. It identifies which of the two coordinate systems involved in a Jacobian entry is being treated as the independent variable of differentiation, a role that must be correctly recognized before the entry can be assigned to the correct row of the Jacobian matrix or contracted against the correct index of a tensor.

Identifying the Source Coordinate

Position in the Derivative Expression

The source coordinate is the coordinate appearing in the denominator of a Jacobian entry, playing the role of the independent variable against which the new coordinate function is differentiated.

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

Here the unprimed coordinate in the denominator plays the source coordinate role, since it is with respect to this coordinate that the derivative is taken.

Index Placement Reflecting the Source Role

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

Function of the Source Coordinate Role

Determining the Row of the Jacobian Matrix

Because the lower index marks the source coordinate, and rows of the Jacobian matrix are conventionally organized by this lower index, the source coordinate role determines which row of the matrix a given derivative entry belongs to.

Matching the Old Basis in Tensor Contractions

When the Jacobian matrix is contracted against the components of a tensor, the source coordinate role corresponds to the old coordinate system, meaning the index playing this role is the one that gets summed against the original, untransformed components of the tensor being converted to the new system.

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

Contrast With the Target Coordinate Role

Opposite Position in the Expression

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

Reversal Under the Inverse Jacobian

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

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

Consequences of the Source Coordinate Role

Necessity for Correct Contraction

Misidentifying which coordinate in a Jacobian expression plays the source role would lead to contracting the matrix against the wrong index of a tensor's components, producing a transformed component that does not correctly correspond to the intended new coordinate system.

Consistency With General Tensor Index Conventions

The source coordinate role parallels the general convention, already established for linear transformation matrices, that the lower index of a matrix factor is the one contracted against the original, untransformed quantity, extending that same convention naturally into the calculus-based Jacobian setting.

Schematic Representation

The diagram highlights the denominator of a Jacobian derivative entry as the location of the source coordinate, the variable of differentiation representing the old coordinate system.