chain rule of Jacobian

$ \bm{J}_{\bm{g}∘\bm{f}}(a) = \bm{J}_{\bm{g}}(\bm{f}(a)) \bm{J}_{\bm{f}}(a)

where

f: ℝ^n→ℝ^m

g: ℝ^m → ℝ^l

あとで太字にする。

Proof

$ \text{LHS} = \left( \frac{∂(g∘f)_i}{∂x_j}(a) \right)_{i\,j}

$ = \left( \sum_{k∈1..m} \frac{∂g_i}{∂y_k}(f(a)) \cdot \frac{∂f_k}{∂x_j}(a) \right)_{i\,j}

$ = \left( \frac{∂g_i}{∂y_j}(f(a)) \right)_{i\,k} \left( \frac{∂f_i}{∂x_j}(a) \right)_{k\,j}

$ = \text{RHS}

ref.