Jacobian

Jacobian matrix is a matrix of partial derivatives. Jacobian is the determinant of the jacobian matrix. The matrix will contain all partial derivatives of a vector function. The main use of Jacobian is found in the transformation of coordinates. It deals with the concept of differentiation with coordinate transformation. In this article, let us discuss what is a jacobian matrix, determinants, and examples in detail.

What is Jacobian?

The term “Jacobian” often represents both the jacobian matrix and determinants, which is defined for the finite number of function with the same number of variables. Here, each row consists of the first partial derivative of the same function, with respect to the variables. The jacobian matrix can be of any form. It may be a square matrix (number of rows and columns are equal) or the rectangular matrix(the number of rows and columns are not equal).

Jacobian Matrix

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

\((\frac{\partial(f) }{\partial x_{1}}(P),\frac{\partial(f) }{\partial x_{2}}(P),….\frac{\partial(f) }{\partial x_{n}}(P) )\)

The jacobian matrix for the given matrix is given as:

\(\begin{bmatrix} \frac{\partial f_{1}}{\partial x_{1}} & \frac{\partial f_{1}}{\partial x_{2}} & \cdots &\frac{\partial f_{1}}{\partial x_{m}} \\ \frac{\partial f_{2}}{\partial x_{1}}& \frac{\partial f_{2}}{\partial x_{2}} & \cdots & \frac{\partial f_{2}}{\partial x_{m}}\\ \frac{\partial f_{3}}{\partial x_{1}}& \frac{\partial f_{3}}{\partial x_{2}} &\cdots & \frac{\partial f_{3}}{\partial x_{m}} \end{bmatrix}\)

The determinant for the above jacobian matrix is called a jacobian.

Jacobian Determinant

In a jacobian matrix, if m = n = 2, and the function f: ℝ3 → ℝ, is defined as:

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

Hence, the jacobian matrix is written as:

\(J = \begin{bmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y}\\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} \end{bmatrix}\)

Therefore, the determinant of a jacobian matrix is

\(det(J)= \begin{vmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y}\\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} \end{vmatrix}\)

 

\(det(J)= \left | \frac{\partial u}{\partial x}\frac{\partial v}{\partial y} – \frac{\partial u}{\partial y}\frac{\partial v }{\partial x}\right |\)

Polar and Spherical Cartesian Transformation

For a normal cartesian to polar transformation, the equation can be written as:

x = r cos θ

y = r sin θ

The jacobian determinant is written as:

\(J (r,\theta ) = \begin{vmatrix} \frac{\partial x}{\partial r} & \frac{\partial x}{\partial \theta } \\ \frac{\partial y}{\partial r}& \frac{\partial y}{\partial \theta } \end{vmatrix}\)

Using these partial differentiation on the polar equations we get,

\(J (r,\theta ) = \begin{vmatrix} cos \theta & -r sin\theta \\ sin \theta & r cos\theta \end{vmatrix}\)

J (r, θ ) = r (sin2 θ + cos2θ) = r (1)

J (r, θ) = 1

Jacobian Example

Question: Let x (u, v) = u2 – v2 , y (u, v) = 2 uv. Find the jacobian J (u, v).

Solution:

Given: x (u, v) = u2 – v2

y (u, v) = 2 uv

We know that,

\(J (u, v ) = \begin{bmatrix} x_{u} & x_{v} \\ y_{u} & y_{v} \end{bmatrix}\) \(J (u, v ) = \begin{bmatrix} 2u & -2v \\ 2v & 2u \end{bmatrix}\)

J (u, v) = 4u2 + 4v2

Therefore, J (u, v) is 4u2 + 4v2

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