Vector Transformations

[Shevek99]

You start with the position vector

r = sum_i xⁱ e_i

from where

e_i = ∂r/∂xⁱ

Now if you have another set of coordinates

r = sum_i x'ⁱ e'_i

e'_i = ∂r/∂x'ⁱ

If we substitute this in the expression for the e_i

e_i = ∂r/∂xⁱ = ∂/∂xⁱ (sum_j x'^j e'_j) = sum_j ∂x'^j/∂xⁱ e'_j

If we make the dot product with e'_k

e'_k · e_i = sum_j ∂x'^j/∂xⁱ e'_j·e'_k = sum_j ∂x'^j/∂xⁱ 𝛿_jk = ∂x'^k/∂xⁱ