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Jacobian of Composite Functions 📂Vector Analysis

Jacobian of Composite Functions

Theorem

Let’s assume we have two functions f:Rn→Rmf : \mathbb{R}^{n} \to \mathbb{R}^{m} and g:Rm→Rkg : \mathbb{R}^{m} \to \mathbb{R}^{k}. We denote the Jacobian of ff as J(f)J(f). Then, the following holds.

J(g∘f)=J(g)J(f) J(g \circ f) = J(g) J(f)

Explanation

Since the Jacobian is the most generalized derivative, the above theorem is a generalization of the chain rule.

Proof

By definition of the Jacobian,

J(g∘f)=[∂(g∘f)1∂x1⋯∂(g∘f)1∂xn⋮⋱⋮∂(g∘f)k∂x1⋯∂(g∘f)k∂xn]=[∂g1∂x1⋯∂g1∂xn⋮⋱⋮∂gk∂x1⋯∂gk∂xn] J(g \circ f) = \begin{bmatrix} \dfrac{\partial (g \circ f)_{1}}{\partial x_{1}} & \cdots & \dfrac{\partial (g \circ f)_{1}}{\partial x_{n}} \\ \vdots & \ddots & \vdots \\ \dfrac{\partial (g \circ f)_{k}}{\partial x_{1}} & \cdots & \dfrac{\partial (g \circ f)_{k}}{\partial x_{n}} \end{bmatrix} = \begin{bmatrix} \dfrac{\partial g_{1}}{\partial x_{1}} & \cdots & \dfrac{\partial g_{1}}{\partial x_{n}} \\ \vdots & \ddots & \vdots \\ \dfrac{\partial g_{k}}{\partial x_{1}} & \cdots & \dfrac{\partial g_{k}}{\partial x_{n}} \end{bmatrix}

Given gi=gi(f1(x),…,fm(x)))g_{i} = g_{i}(f_{1}(\mathbf{x}), \dots, f_{m}(\mathbf{x}))),

∂gi∂xj=∑ℓ=1m∂gi∂fℓ∂fℓ∂xj \dfrac{\partial g_{i}}{\partial x_{j}} = \sum \limits_{\ell=1}^{m} \dfrac{\partial g_{i}}{\partial f_{\ell}} \dfrac{\partial f_{\ell}}{\partial x_{j}}

Therefore,

J(g∘f)= [∑ℓ=1m∂g1∂fℓ∂fℓ∂x1⋯∑ℓ=1m∂g1∂fℓ∂fℓ∂xn⋮⋱⋮∑ℓ=1m∂gk∂fℓ∂fℓ∂x1⋯∑ℓ=1m∂gk∂fℓ∂fℓ∂xm]= [∂g1∂f1⋯∂g1∂fm⋮⋱⋮∂gk∂f1⋯∂gk∂fm][∂f1∂x1⋯∂f1∂xn⋮⋱⋮∂fm∂x1⋯∂fm∂xn]= J(g)J(f) \begin{align*} J(g \circ f) =&\ \begin{bmatrix} \sum \limits_{\ell=1}^{m} \dfrac{\partial g_{1}}{\partial f_{\ell}} \dfrac{\partial f_{\ell}}{\partial x_{1}} & \cdots & \sum \limits_{\ell=1}^{m} \dfrac{\partial g_{1}}{\partial f_{\ell}} \dfrac{\partial f_{\ell}}{\partial x_{n}} \\ \vdots & \ddots & \vdots \\ \sum \limits_{\ell=1}^{m} \dfrac{\partial g_{k}}{\partial f_{\ell}} \dfrac{\partial f_{\ell}}{\partial x_{1}} & \cdots & \sum \limits_{\ell=1}^{m} \dfrac{\partial g_{k}}{\partial f_{\ell}} \dfrac{\partial f_{\ell}}{\partial x_{m}} \end{bmatrix} \\ =&\ \begin{bmatrix} \dfrac{\partial g_{1}}{\partial f_{1}} & \cdots & \dfrac{\partial g_{1}}{\partial f_{m}} \\ \vdots & \ddots & \vdots \\ \dfrac{\partial g_{k}}{\partial f_{1}}& \cdots & \dfrac{\partial g_{k}}{\partial f_{m}} \end{bmatrix} \begin{bmatrix} \dfrac{\partial f_{1}}{\partial x_{1}} & \cdots & \dfrac{\partial f_{1}}{\partial x_{n}} \\ \vdots & \ddots & \vdots \\ \dfrac{\partial f_{m}}{\partial x_{1}} & \cdots & \dfrac{\partial f_{m}}{\partial x_{n}} \end{bmatrix} \\ =&\ J(g) J(f) \end{align*}

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