What is the Jacobian matrix?

The Jacobian matrix is a matrix that composes the first-order partial derivatives of a multivariable function.

Suppose we want to differentiate two functions with respect to two variables, would that be possible? The answer is no, since using partial derivation or simple derivation, we only have one function with respect to one variable. In the case, where we have more than one function or one variable, we use the Jacobians matrix. A Jacobian matrix is also very useful for calculating the derivates of implicit or composite functions. 

For a function f: ℝ³ → ℝ, the derivative at J for a row vector is defined as:

Hence, the Jacobian matrix of the following function is written as follows:

Note: The determinant for the above Jacobian matrix is known as the Jacobian.

Here $f_1,f_2, f_3$ refer to the elements of the output vector, and $x_1,x_2, x_3$ refer to the elements of the input vector.

Example

Now that we know what a Jacobian matrix is, let's see how to compute the Jacobian matrix of a multivariable function.

Consider the function f: $R^2 → R^2$, with $(x,y)$ ↦ $(f_1(x,y), f_2(x,y))$, given by:

Find the Jacobian matrix at the point $(1,2)$.

Step 1: Calculate all the first-order partial derivatives of the function:

Step 2: Apply the formula of the Jacobian matrix. This function has two variables and two vector components, so the Jacobian matrix will be a $2×2$ square matrix:

Step 3: Put the $(1,2)$ given in the problem in the expression above:

Step 4: Calculate the expression:

Jacobian matrix determinant

In a Jacobian matrix, if $m = n$, with m rows and n columns, and the function f: ℝ3 → ℝ, is defined as:

$f (x, y) = (u (x, y), v (x, y))$

Then f is a function from ℝⁿ to itself and is a square A matrix that has an equal number of rows and columns.matrix. Hence, the Jacobian matrix is written as:

Then the determinant of a Jacobian matrix is:

Example

Compute the determinant for the following:

After applying the steps we learned in the first section to calculate the Jacobian matrix, we have the following:

Calculating the determinant:

Polar and spherical Cartesian transformation

Polar-Cartesian and spherical-Cartesian are the most important types of Jacobian matrices. These matrices are imperative, as they help in the conversion of one coordinate system into another.

Polar-Cartesian

The transformation from polar coordinates $(r,\theta)$ to Cartesian coordinates $(x, y)$, is written as:

• $x = r cos (θ)$

• $y = r sin (θ)$

We write the Jacobian matrix for the above coordinate change as follows:

Spherical-Cartesian transformation

The transformation from spherical coordinates to Cartesian coordinates $(x, y, z)$, is written as mentioned below:

The Jacobian matrix of $∂(x,y,z)∂(ρ,θ,ϕ)$ is calculated as:

Step 1: Compute the partial derivatives:

Step 2: Compute the Jacobian matrix:

The determinant simplifies to $ρ^2sin(ϕ)$, and since the volume of the prism is the product of its sides, we denote it by the following:

Conclusion

The Jacobian matrix is a representation of information about the local behavior of a function. It consists of first-order partial derivatives and helps us in converting one coordinate system into another.