A tensor stays fixed while its component arrays change

At a point, a tangent vector can be viewed as a directional derivative. The coordinate basis is ∂μ, and its dual one-form basis is dxμ. The chain rule changes the derivative basis with an inverse Jacobian and the differential basis with the forward Jacobian. Requiring the full vector and one-form to remain the same then gives forward-Jacobian vector components and inverse-Jacobian covector components. These transformations concern components at the same event, not transporting a vector from one event to another.

A scalar differential df is a useful covector example. Its components are partial derivatives of f and act on a vector to return a directional derivative. Under a smooth coordinate transformation, the contraction is unchanged because the Jacobians cancel. A mixed tensor combines one factor for each index. Index placement therefore encodes which basis compensation is required. Contraction pairs one upper and one lower index; summing two upper component arrays as though both were Cartesian does not define an invariant without an additional metric.

The metric is a bilinear map on tangent vectors, so its two lower indices each acquire an inverse Jacobian. A coordinate shear of an ordinary Euclidean plane creates off-diagonal metric components even though the plane remains flat. The squared norm is computed with this transformed metric, not by summing the new coordinate components squared. Raising an index also requires the inverse metric. In Lorentzian spacetime the same bookkeeping applies, with the chosen signature supplying the time-direction sign.

The metric can change its component appearance drastically under a coordinate change, but its nondegenerate signature is preserved by an invertible transformation. Determinants transform with the square of the inverse Jacobian; a coordinate volume element transforms with the opposite Jacobian factor. These facts later combine into an invariant volume density. A tensor equation valid in one chart remains valid in any overlapping valid chart, whereas a statement that some connection coefficients vanish need not survive: connections have an additional inhomogeneous transformation term.

It is helpful to distinguish a passive coordinate change from an active diffeomorphism moving fields or points. Both can be written using related Jacobians, but their interpretation and evaluation points must be tracked. For our component examples the event and geometric tensor are held fixed while its coordinates change. A failed contraction check usually indicates a wrong inverse, a transposition error, or components evaluated at different points. It is not evidence that the scalar itself depends on the observer’s chosen coordinate notation.

Use the matrix lab to inspect the forward shear J=[[1,b],[0,1]]. Reset and set both controls to a=1,b=0, predicting determinant one. Reset and set a=1,b=1, again predicting determinant one but unequal singular values about 1.618 and 0.618. Current geometry is identity at Reset and interpolates under Play or Step; readouts always describe the selected final matrix. The lab transforms Euclidean geometry, not tensor indices automatically. The reciprocal covector and metric transformations must be calculated manually, and area preservation alone does not imply preservation of every ordinary component norm.