jacobian

Jacobian matrix of symbolic function

Syntax

jacobian(f,v)

Description

jacobian(f,v) computes the Jacobian matrix of symbolic function f with respect to v. The (i,j) element of the result is $$\frac{\partial f\left(i\right)}{\partial \text{v}\left(j\right)}$$.

Examples

Jacobian of Vector Function

The Jacobian of a vector function is a matrix of the partial derivatives of that function.

Compute the Jacobian matrix of [x*y*z,y^2,x + z] with respect to [x,y,z].

syms x y z
jacobian([x*y*z,y^2,x + z],[x,y,z])

ans = 
$$\left(\begin{array}{ccc}y\hspace{0.17em}z& x\hspace{0.17em}z& x\hspace{0.17em}y\\ 0& 2\hspace{0.17em}y& 0\\ 1& 0& 1\end{array}\right)$$

Now, compute the Jacobian of [x*y*z,y^2,x + z] with respect to [x;y;z].

jacobian([x*y*z,y^2,x + z], [x;y;z])

ans = 
$$\left(\begin{array}{ccc}y\hspace{0.17em}z& x\hspace{0.17em}z& x\hspace{0.17em}y\\ 0& 2\hspace{0.17em}y& 0\\ 1& 0& 1\end{array}\right)$$

The Jacobian matrix is invariant to the orientation of the vector in the second input position.

Jacobian of Scalar Function

The Jacobian of a scalar function is the transpose of its gradient.

Compute the Jacobian of 2*x + 3*y + 4*z with respect to [x,y,z].

syms x y z
jacobian(2*x + 3*y + 4*z,[x,y,z])

ans = $$\left(\begin{array}{ccc}2& 3& 4\end{array}\right)$$

Now, compute the gradient of the same expression.

gradient(2*x + 3*y + 4*z,[x,y,z])

ans = 
$$\left(\begin{array}{c}2\\ 3\\ 4\end{array}\right)$$

Jacobian with Respect to Scalar

The Jacobian of a function with respect to a scalar is the first derivative of that function. For a vector function, the Jacobian with respect to a scalar is a vector of the first derivatives.

Compute the Jacobian of [x^2*y,x*sin(y)] with respect to x.

syms x y
jacobian([x^2*y,x*sin(y)],x)

ans = 
$$\left(\begin{array}{c}2\hspace{0.17em}x\hspace{0.17em}y\\ \sin \left(y\right)\end{array}\right)$$

Now, compute the derivatives.

diff([x^2*y,x*sin(y)],x)

ans = $$\left(\begin{array}{cc}2\hspace{0.17em}x\hspace{0.17em}y& \sin \left(y\right)\end{array}\right)$$

Jacobian of Coordinate Change

Specify polar coordinates $$r(t)$$, $$\varphi (t)$$, and $$\theta (t)$$ that are functions of time.

syms r(t) phi(t) theta(t)

Define the coordinate transformation form spherical coordinates to Cartesian coordinates.

R = [r*sin(phi)*cos(theta), r*sin(phi)*sin(theta), r*cos(phi)]

R(t) = $$\left(\begin{array}{ccc}\cos \left(\theta \left(t\right)\right)\hspace{0.17em}\sin \left(\varphi \left(t\right)\right)\hspace{0.17em}r\left(t\right)& \sin \left(\varphi \left(t\right)\right)\hspace{0.17em}\sin \left(\theta \left(t\right)\right)\hspace{0.17em}r\left(t\right)& \cos \left(\varphi \left(t\right)\right)\hspace{0.17em}r\left(t\right)\end{array}\right)$$

Find the Jacobian of the coordinate change from spherical coordinates to Cartesian coordinates.

jacobian(R,[r,phi,theta])

ans(t) = 
$$\left(\begin{array}{ccc}\cos \left(\theta \left(t\right)\right)\hspace{0.17em}\sin \left(\varphi \left(t\right)\right)& \cos \left(\varphi \left(t\right)\right)\hspace{0.17em}\cos \left(\theta \left(t\right)\right)\hspace{0.17em}r\left(t\right)& -\sin \left(\varphi \left(t\right)\right)\hspace{0.17em}\sin \left(\theta \left(t\right)\right)\hspace{0.17em}r\left(t\right)\\ \sin \left(\varphi \left(t\right)\right)\hspace{0.17em}\sin \left(\theta \left(t\right)\right)& \cos \left(\varphi \left(t\right)\right)\hspace{0.17em}\sin \left(\theta \left(t\right)\right)\hspace{0.17em}r\left(t\right)& \cos \left(\theta \left(t\right)\right)\hspace{0.17em}\sin \left(\varphi \left(t\right)\right)\hspace{0.17em}r\left(t\right)\\ \cos \left(\varphi \left(t\right)\right)& -\sin \left(\varphi \left(t\right)\right)\hspace{0.17em}r\left(t\right)& 0\end{array}\right)$$

Input Arguments

f — Scalar or vector function

symbolic expression | symbolic function | symbolic vector

Scalar or vector function, specified as a symbolic expression, function, or vector. If f is a scalar, then the Jacobian matrix of f is the transposed gradient of f.

v — Vector of variables or functions with respect to which you compute Jacobian

symbolic variable | symbolic function | symbolic vector

Vector of variables or functions with respect to which you compute Jacobian, specified as a symbolic variable, symbolic function, or vector of symbolic variables. If v is a scalar, then the result is equal to the transpose of diff(f,v). If v is an empty symbolic object, such as sym([]), then jacobian returns an empty symbolic object.

More About

Jacobian Matrix

The Jacobian matrix of the vector function f = (f₁(x₁,...,x_n),...,f_n(x₁,...,x_n)) is the matrix of the derivatives of f:

$$J\left(x_1,\dots x_n\right)=\left[\begin{array}{ccc}\frac{\partial f_1}{\partial x_1}& \cdots & \frac{\partial f_1}{\partial x_n}\\ ⋮& \ddots & ⋮\\ \frac{\partial f_n}{\partial x_1}& \cdots & \frac{\partial f_n}{\partial x_n}\end{array}\right]$$

Version History

Introduced before R2006a