6.13.5 Tensor Zero One Transformation Pattern

Tensor Zero One Transformation Pattern is the specific rule by which the components of a type zero-one tensor are recomputed under a change of coordinates, consisting of exactly one factor of the inverse Jacobian matrix applied to the original components and summed over the single dummy index shared between the old components and the transformation factor. This pattern is the simplest nontrivial instance of covariant transformation behavior, involving only one Jacobian factor rather than the products of several factors required for tensors of higher type, and it serves as the elementary building block from which every other covariant tensor's transformation pattern is assembled.

The Rule Itself

One Factor, One Contraction

The transformation pattern for a type zero-one tensor takes the old components, indexed by a dummy label, and contracts them against the partial derivative of each old coordinate with respect to the corresponding new coordinate, summing over the shared dummy index to produce each new component in turn.

Matrix Form of the Pattern

When the dimension of the space is finite, this pattern can be written as ordinary matrix multiplication, with the inverse Jacobian entries arranged into a square matrix and the old components arranged into a row, so that the new components form the row obtained by multiplying the old row against the inverse Jacobian matrix. This matrix picture is available precisely because the type zero-one transformation pattern involves only a single linear map applied once, with no further tensorial structure layered on top.

Contrast With the Contravariant Pattern

Inverse Versus Direct Jacobian

The type zero-one transformation pattern uses the inverse Jacobian, the derivative of old coordinates with respect to new coordinates, whereas the corresponding pattern for a type one-zero tensor uses the direct form of this same matrix, the derivative of new coordinates with respect to old. These two patterns are matrix inverses of one another whenever the coordinate change is invertible, which is precisely the condition required for the change of coordinates to be admissible in the first place.

Why the Inverse Choice Is Forced

The type zero-one pattern is required to use the inverse Jacobian precisely because a one-form, paired against a vector to yield an invariant scalar, must transform oppositely to the vector it is paired with, and the vector's own transformation uses the direct Jacobian. Reversing this assignment would break the invariance of the pairing under a change of coordinates, so the choice of inverse Jacobian for the type zero-one pattern is not a free convention but a structural necessity.

Behavior of the Pattern Under Composition and Special Cases

Composing Two Successive Transformations

Applying the type zero-one transformation pattern across two successive changes of coordinates, first to an intermediate system and then to a final system, reproduces exactly the same components obtained by applying the pattern once directly from the original system to the final one, since the inverse Jacobian matrices of two successive coordinate changes multiply together, in the reverse order corresponding to their composition, to give the inverse Jacobian of the combined change.

The Pattern for Linear Coordinate Changes

When the coordinate change is itself linear, the inverse Jacobian matrix is constant throughout the space rather than varying from point to point, and the type zero-one transformation pattern reduces to a single fixed matrix multiplication applied uniformly everywhere. For general, nonlinear coordinate changes, the inverse Jacobian matrix varies with position, so the same transformation pattern applies pointwise, with a potentially different matrix required at each distinct point of the space.