# jacobian

Jacobian matrix

## Syntax

``jacobian(f,v)``

## Description

example

````jacobian(f,v)` computes the Jacobian matrix of `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 = [ 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)]```

## Input Arguments

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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`.

Vector of variables with respect to which you compute Jacobian, specified as a symbolic variable 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.

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### Jacobian Matrix

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

`$J\left({x}_{1},\dots {x}_{n}\right)=\left[\begin{array}{ccc}\frac{\partial {f}_{1}}{\partial {x}_{1}}& \cdots & \frac{\partial {f}_{1}}{\partial {x}_{n}}\\ ⋮& \ddots & ⋮\\ \frac{\partial {f}_{n}}{\partial {x}_{1}}& \cdots & \frac{\partial {f}_{n}}{\partial {x}_{n}}\end{array}\right]$`