Note: This page has been translated by MathWorks. Please click here

To view all translated materals including this page, select Japan from the country navigator on the bottom of this page.

To view all translated materals including this page, select Japan from the country navigator on the bottom of this page.

Jacobian matrix

`jacobian(f,v)`

`jacobian(`

computes
the Jacobian matrix of `f`

,`v`

)`f`

with
respect to `v`

. The *( i,j)* element
of the result is $$\frac{\partial f\left(i\right)}{\partial \text{v}\left(j\right)}$$.

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])

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

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)]

`curl`

| `diff`

| `divergence`

| `gradient`

| `hessian`

| `laplacian`

| `potential`

| `vectorPotential`

Was this topic helpful?