3.16.5 Tensor Covector Pullback Coordinate Expression

The tensor covector pullback expresses covector transformation under coordinate changes, linking tensor algebra to differential geometry.

Tensor Covector Pullback Coordinate Expression is the representation of the pullback of a covector in terms of components relative to chosen bases, obtained by writing the pullback map as a matrix acting on component tuples, with that matrix equal to the transpose of the matrix representing the original linear map, or, in the manifold setting, equal to the Jacobian matrix of partial derivatives of the coordinate functions.

Linear Algebra Coordinate Form

Matrix of the Original Map

Let f: V → W be a linear map represented, relative to bases of V and W, by a matrix F with entries F^i_j, so that a vector with components v^j maps to a vector with components:

w^{i} = \sum_{j} F_{j}^{i} v^{j}

Matrix of the Pullback

A covector ω on W, with components ω_i relative to the dual basis, pulls back to a covector on V with components obtained by the same matrix F, but contracted on the opposite index and applied on the left rather than the right:

(_{f} * ω_{j} = \sum_{i} ω_{i} F_{j}^{i}

Written as a matrix equation with ω as a row vector, the coordinate expression of the pullback is simply row-vector times matrix, which is equivalent to treating the pullback as multiplication by the transpose matrix F^T when ω is written as a column vector:

(^{*} ω = F^{T} ω

Manifold Coordinate Form

Jacobian as the Local Matrix

For a smooth map φ: M → N between manifolds with local coordinates x^i on M and y^a on N, the differential at a point is represented by the Jacobian matrix of partial derivatives:

J_{i}^{a} = \frac{\partial y^{a}}{\partial x^{i}}

Coordinate Formula for Pulling Back a Covector Field

A covector field ω = ω_a dy^a on N pulls back to the covector field on M obtained by substituting the Jacobian entries for each basis covector and summing over the repeated index:

φ^{*} ω = \sum_{i} (\sum_{a} ω_{a} \frac{\partial y^{a}}{\partial x^{i}}) d x^{i}

This formula is the working coordinate expression used in practice to compute the pullback of any 1-form once the coordinate representation y^a = y^a(x^1, ..., x^n) of the map φ is known explicitly.

Practical Computation Steps

Step 1: Express the Map in Coordinates

Write out each target coordinate y^a as an explicit function of the source coordinates x^1 through x^n.

Step 2: Differentiate

Compute every partial derivative ∂y^a / ∂x^i needed to fill in the Jacobian matrix at the relevant point or region.

Step 3: Substitute into Each Component of ω

Replace every occurrence of dy^a in the expression for ω with the corresponding linear combination of dx^i given by ∂y^a / ∂x^i, then collect terms by dx^i.

Step 4: Simplify

Combine like terms across all a for each fixed i to obtain the final coordinate components of the pulled-back covector field on M.

Special Cases

Coordinate Changes as Pullbacks

When φ is itself a change of coordinates on the same manifold, the coordinate expression of the pullback reduces to the classical transformation law for covector components under a change of basis, with the Jacobian playing the role of the transition matrix between the two coordinate charts.

Constant Linear Maps

When the underlying map is linear with constant matrix entries, all partial derivatives in the Jacobian are themselves constant, and the coordinate expression of the pullback reduces exactly to the linear algebra matrix formula F^T ω, showing that the manifold formula generalizes the linear algebra formula rather than replacing it.

Composability of Coordinate Expressions

Because the Jacobian of a composite map is the matrix product of the individual Jacobians, computing the coordinate expression of a pullback along a composite map can be done either by forming the composite Jacobian first or by pulling back successively through each individual map, with both approaches producing identical component formulas.