tensor transformation law
Suppose two people describe the same patch of space with different coordinate grids — one uses x coordinates, the other uses primed x' coordinates, perhaps polar versus Cartesian, perhaps two different moving frames. They will read off different component arrays for the same physical quantity. The tensor transformation law is the exact rule that says how one person's components convert into the other's. It is the precise statement of what it means for a quantity to be a tensor: not merely to have indices, but to have its indices change in this specific way.
The recipe follows the index positions. Each upper (contravariant) index transforms by a factor of the partial derivative of the new coordinate with respect to the old, partial x'^a / partial x^i, and each lower (covariant) index by the inverse factor, partial x^j / partial x'^b. So for a type (1,1) tensor the law reads T'^a_b = (partial x'^a / partial x^i)(partial x^j / partial x'^b) T^i_j, with i and j summed. Those partial-derivative matrices are the Jacobian of the coordinate change and its inverse. A type (p, q) tensor gets p Jacobian factors and q inverse-Jacobian factors. Because upper and lower factors are inverses of each other, contracting an up index with a down index makes those two factors cancel — which is exactly why a contraction produces a lower-rank tensor, and a full contraction produces a true scalar that all observers agree on.
This law is the gatekeeper of tensor calculus and the engine of general covariance in physics. If a candidate equation is written as equal-rank tensors on both sides with the same free indices, the transformation law guarantees that if it holds in one coordinate system it holds in every coordinate system — so a single derivation establishes a law of nature in all frames at once. The flip side is the standard cautionary tale: objects like the Christoffel symbols and the ordinary partial derivative of a tensor fail this law (they acquire stray non-tensorial terms), which is precisely why the covariant derivative had to be invented to repair the partial derivative back into a tensor.
A contravariant vector transforms as V'^a = (partial x'^a / partial x^i) V^i; a covariant vector as W'_b = (partial x^j / partial x'^b) W_j. Contract them: V'^a W'_a = (partial x'^a / partial x^i)(partial x^k / partial x'^a) V^i W_k. The two Jacobian factors collapse to delta^k_i, leaving V^i W_i — the same scalar in both frames.
Opposite transformation factors cancel on contraction — that is why a fully contracted tensor is a true scalar.
The law is the definition: 'is this a tensor?' means 'do its components obey this exact transformation rule?'. The ordinary partial derivative of a tensor fails it (extra second-derivative terms appear), which is the whole reason the covariant derivative exists.