Question

Explain this:
In coordinates x^i, the components T^{i1...ik}_{j1...jl} of a (k,l)-tensor transform under a change x → x' by
T'^{i1...ik}_{j1...jl}
= (∂x'^{i1}/∂x^{a1}) ... (∂x'^{ik}/∂x^{ak}) (∂x^{b1}/∂x'^{j1}) ... (∂x^{bl}/∂x'^{jl}) T^{a1...ak}_{b1...bl}.

Answer 1

Short answer: the formula is just the chain rule applied to each tensor index. Upper (contravariant) indices pick up factors of the Jacobian matrix ∂x'/∂x, lower (covariant) indices pick up factors of the inverse Jacobian ∂x/∂x'. The factors come from how the coordinate basis vectors ∂/∂x^i and the dual basis dx^i transform.

Derivation outline

1) Basis transformation (chain rule)
- For the coordinate basis vectors e_i := ∂/∂x^i,
∂/∂x^i = (∂x'^j/∂x^i) ∂/∂x'^j,
so e_i = (∂x'^j/∂x^i) e'_j.
- For the dual basis 1-forms dx^i,
dx'^j = (∂x'^j/∂x^i) dx^i,
hence dx^i = (∂x^i/∂x'^j) dx'^j.

2) Vectors and covectors
- A vector V = V^i e_i = V'^j e'_j. Substitute e_i =
(∂x'^j/∂x^i)e'_j and equate coefficients to get
V'^j = (∂x'^j/∂x^i) V^i.
So upper indices transform with ∂x'/∂x.
- A covector ω = ω_i dx^i = ω'_j dx'^j. Using dx^i =
(∂x^i/∂x'^j) dx'^j gives
ω'_j = (∂x^i/∂x'^j) ω_i.
So lower indices transform with ∂x/∂x' (the inverse Jacobian).

3) General (k,l)-tensor
Write T as components times tensor product of basis factors:
T = T^{i1...ik}_{j1...jl} e_{i1} ⊗ ... ⊗ e_{ik} ⊗ dx^{j1} ⊗ ... ⊗ dx^{jl}.
Replace each e_i and dx^j by their primed-basis expressions. Each upper index contributes a factor (∂x'^{ip}/∂x^{a_p}), each lower index contributes a factor (∂x^{b_q}/∂x'^{j_q}). Collecting all factors and renaming summed indices yields
T'^{i1...ik}_{j1...jl}
= (∂x'^{i1}/∂x^{a1}) ... (∂x'^{ik}/∂x^{ak})
(∂x^{b1}/∂x'^{j1}) ... (∂x^{bl}/∂x'^{jl})
T^{a1...ak}_{b1...bl},
with the usual Einstein summation over repeated indices.

Remarks
- The partial-derivative factors are the components of the Jacobian matrix and its inverse, evaluated at the point.
- Index position matters: upper = contravariant (transforms with ∂x'/∂x), lower = covariant (with ∂x/∂x').
- This transformation law guarantees tensorial quantities are coordinate-independent: the geometric multilinear map is the same, only components change under coordinate changes.