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 =
[ y*z, x*z, x*y]
[   0, 2*y,   0]
[   1,   0,   1]

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 =
 
[ y*z, x*z, x*y]
[   0, 2*y,   0]
[   1,   0,   1]

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 =
[ 2, 3, 4]

Now, compute the gradient of the same expression.

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

ans =
 2
 3
 4

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 =
  2*x*y
 sin(y)

Now, compute the derivatives.

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

ans =
[ 2*x*y, sin(y)]