Jacobi Matrix

The Jacobi matrix is a special matrix consisting of partial derivatives that is vital in multivariate calculus. It is also sometimes referred to as the derivative matrix.

Definition §

Jacobi/Derivative Matrix ^definition

Let $f:{\mathbb{R}}^n\to {\mathbb{R}}^m$ be a function, expressed as $f(x_1,\dots ,x_n)=(f_1(x_1,\dots ,x_n),\dots ,f_m(x_1,\dots ,x_n))$. Then the Jacobi or derivative matrix of $f$ is a matrix of dimensions $m\times n$ given by:

$${\mathbf{J}}_f={\mathbf{D}}_f=\left[\begin{array}{cccc}\frac{\partial f_1}{\partial x_1}& \frac{\partial f_1}{\partial x_2}& \dots & \frac{\partial f_1}{\partial x_n}\\ \frac{\partial f_2}{\partial x_1}& \frac{\partial f_2}{\partial x_2}& \dots & \frac{\partial f_2}{\partial x_n}\\ ⋮& ⋮& \ddots & ⋮\\ \frac{\partial f_m}{\partial x_1}& \frac{\partial f_m}{\partial x_2}& \dots & \frac{\partial f_m}{\partial x_n}\end{array}\right]$$

The determinant of this matrix, $\det (\mathbf{J})$ is known as the Jacobian.

In some cases, the Jacobian is notated using:

$${\mathbf{J}}_f=\frac{\partial (f_1,f_2,\dots ,f_n)}{\partial (x_1,x_2,\dots ,x_n)}$$

For example, in the simple case of $f:{\mathbb{R}}^2\to \mathbb{R}$, i.e. $z=f(x,y)$, the Jacobi matrix degrades to the gradient vector:

$${\mathbf{J}}_f=\left[\begin{array}{cc}\frac{\partial f}{\partial x}& \frac{\partial f}{\partial y}\end{array}\right]=\nabla f$$

If $\vec{F}:{\mathbb{R}}^3\to {\mathbb{R}}^3$ is a 3D vector field, i.e. $\vec{F}(x,y,z)=(u(x,y,z),v(x,y,z),w(x,y,z))$ we have the Jacobi matrix:

$${\mathbf{J}}_F=\left[\begin{array}{ccc}\frac{\partial u}{\partial x}& \frac{\partial u}{\partial y}& \frac{\partial u}{\partial z}\\ \frac{\partial v}{\partial x}& \frac{\partial v}{\partial y}& \frac{\partial v}{\partial z}\\ \frac{\partial w}{\partial x}& \frac{\partial w}{\partial y}& \frac{\partial w}{\partial z}\end{array}\right]$$

Theorems §

#todo inverse of jacobi theorem (lecture 4)

T1: Jacobi Matrix of a composite function is the matrix product of composing Jacobi matrices

Let $h:{\mathbb{R}}^n\to {\mathbb{R}}^m$ be a composite function given by $(f\circ g)$ where:

• $g:{\mathbb{R}}^n\to {\mathbb{R}}^k$, where $g(u_1,\dots ,u_n)=(g_1(\dots ),\dots ,g_k(\dots ))$
• $f:{\mathbb{R}}^k\to {\mathbb{R}}^m$ where $f(v_1,\dots v_k)=(f_1(\dots ),\dots f_m(\dots ))$

That is, we have $h:{\mathbb{R}}^n\stackrel{g}{\to }{\mathbb{R}}^k\stackrel{f}{\to }{\mathbb{R}}^m$. Then:

$${\mathbf{J}}_h={\mathbf{J}}_f{|}_{(v_1,\dots ,v_k)\mapsto (g_1,\dots g_k)}{\mathbf{J}}_g$$

Or we can just directly compute using function composition:

$${\mathbf{J}}_h={\mathbf{J}}_{f\circ g}$$

Note

Examples §

1: 2D to 3D Jacobi Matrix

Let $f:{\mathbb{R}}^2\to {\mathbb{R}}^3$ given by:

$$f(x,y)=(x+y,x,y^2)$$

Find ${\mathbf{J}}_f$

Solution $f(u,v,w)$ where:

We can represent this function as

• $u=x+y$
• $v=x$
• $w=y^2$

Then the Jacobian is given by:

$${\mathbf{J}}_f=\left[\begin{array}{cc}\frac{\partial u}{\partial x}& \frac{\partial u}{\partial y}\\ \frac{\partial v}{\partial x}& \frac{\partial v}{\partial y}\\ \frac{\partial w}{\partial x}& \frac{\partial w}{\partial y}\end{array}\right]=\left[\begin{array}{cc}1& 1\\ 1& 0\\ 0& 2y\end{array}\right]$$

2: Function composition with Jacobi matrices

Let $f(u,v)=(u+v,u,v^2)$ and $g(x,y)=(x^2+1,y^2)$ be real functions. Find the Jacobi matrix of $f\circ g$

Solution $f:{\mathbb{R}}^2\to {\mathbb{R}}^3$ and $g:{\mathbb{R}}^2\to {\mathbb{R}}^2$. Then $f\circ g:{\mathbb{R}}^2\to {\mathbb{R}}^3$ By theorem 1, we know:

We know that

\begin{align} \mathbf{J}_{f \circ g} &= \mathbf{J}_{f}|_{(u,v) \mapsto (g_{1},g_{2})} \ \mathbf{J}_{g} = \begin{bmatrix} \frac{ \partial f_{1} }{ \partial u } & \frac{ \partial f_{1} }{ \partial v } \\ \frac{ \partial f_{2} }{ \partial u } & \frac{ \partial f_{2} }{ \partial v } \\ \frac{ \partial f_{3} }{ \partial u } & \frac{ \partial f_{3} }{ \partial v } \end{bmatrix} _{(u,v) \mapsto (g_{1},g_{2})} \begin{bmatrix} \frac{ \partial g_{1} }{ \partial x } & \frac{ \partial g_{1} }{ \partial y } \\ \frac{ \partial g_{2} }{ \partial x } & \frac{ \partial g_{2} }{ \partial y } \\

\end{bmatrix} \

&= \begin{bmatrix} 1 & 1 \ 1 & 0 \ 0 & 2v \end{bmatrix} _{(u,v) \mapsto (x^2 + 1, y^2)} \begin{bmatrix} 2x & 0 \ 0 & 2y \end{bmatrix}
= \begin{bmatrix} 1 & 1 \ 1 & 0 \ 0 & 2y^2 \end{bmatrix} \begin{bmatrix} 2x & 0 \ 0 & 2y \end{bmatrix} \ &= \begin{bmatrix} 2x & 2y \ 2x & 0 \ 0 & 4y^3 \end{bmatrix} \end{align}

3: More function composition

Compute ${\mathbf{D}}_{f\circ g}$ at the point $(x,y,z)=(0,1,2)$:

$$f(u,v,w)=u^2+v^2-wg(x,y,z)=(x^{2}y,y^2,e^{-xz})$$

Solution $h=f\circ g$. Then $h:{\mathbb{R}}^3\to \mathbb{R}$. Recall, by our theorem of Jacobi matrix of composite functions, our new composite function $h$ will have a Jacobi/Derivative matrix of size $3\times 1$.

Let

$${\mathbf{D}}_{f\circ g}={\mathbf{D}}_f{|}_{(u,v,w)\mapsto (x^{2}y,y^2,e^{-xz})}\cdot {\mathbf{D}}_g={\left[\begin{array}{ccc}\frac{\partial f}{\partial u}& \frac{\partial f}{\partial v}& \frac{\partial f}{\partial w}\end{array}\right]|}_{(u,v,w)\mapsto (x^{2}y,y^2,e^{-xz})}\cdot \left[\begin{array}{ccc}\frac{\partial g_1}{\partial x}& \frac{\partial g_1}{\partial y}& \frac{\partial g_1}{\partial z}\\ \frac{\partial g_2}{\partial x}& \frac{\partial g_2}{\partial y}& \frac{\partial g_2}{\partial z}\\ \frac{\partial g_3}{\partial x}& \frac{\partial g_3}{\partial y}& \frac{\partial g_3}{\partial z}\end{array}\right]$$

Solving it, we have:

$$\begin{array}{rl}& {\left[\begin{array}{ccc}2u& 2v& -1\end{array}\right]|}_{(u,v,w)\mapsto (x^{2}y,y^2,e^{-xz})}\cdot \left[\begin{array}{ccc}2xy& x^2& 0\\ 0& 2y& 0\\ -z{e}^{-xz}& 0& -x{e}^{-xz}\end{array}\right]\\ & =\left[\begin{array}{ccc}2x^{2}y& 2y^2& -1\end{array}\right]\cdot \left[\begin{array}{ccc}2xy& x^2& 0\\ 0& 2y& 0\\ -z{e}^{-xz}& 0& -x{e}^{-xz}\end{array}\right]\\ & =\left[\begin{array}{ccc}4x^3{y}^2+z{e}^{-2xz}& 2x^{4}y+4y^3& x{e}^{-2xz}\end{array}\right]\end{array}$$

Plugging in $(x,y,z)=(0,1,2)$, we get:

$$\left[\begin{array}{ccc}0+2& 0+4& 0\end{array}\right]=\left[\begin{array}{ccc}2& 4& 0\end{array}\right]$$

Applications §

• Integration By Substitution In Higher Dimensions requires the Jacobian when changing coordinate systems.
• Robotics: Using inverse kinematics to related https://medium.com/@prabakaran.r_43682/jacobian-transformation-real-life-applications-across-mathematics-robotics-and-engineering-9c1dd64bfd60#todo