Introduction to the Jacobian Matrix
Calculus involves many fundamental concepts that allows us to solve complex problems such as the transformation between coordinates. For this transformation, the Jacobian is used. It is a method or formula that uses the concept of matrix and determinant to find the behaviour of a function with the change in its variables. Let us understand more about the Jacobian matrix and determinant formula.
Understanding of the Jacobian
The term Jacobian refers to the both jacobian matrix and its determinant. A jacobian matrix contains all of the partial derivatives of the vector functions defined on the same number of variables such as u(x,y) and v(x,y). A Jacobian can be in any matrix form such as it can be a square, column or row matrix.
In calculus the Jacobian is used to transform between coordinates. For example, it transforms a function f(u,v) to f(x,y) by using the jacobian matrix and determinant. In simple words, a Jacobian is nothing but a determinant of all partial derivatives of a vector function.
Jacobian Matrix Formula
The Jacobian matrix collects all first order derivatives of a function depending on more than one variable. In general the Jacobian of a function is a combination of the Jacobian matrix and its determinant. Let’s see how the matrix of jacobians looks like.
For a function f, the Jacobian matrix is represented as;
It is simply just a representation of all partial derivatives of f in a matrix. There are m rows and n columns in a jacobian matrix. If $f(x,y) = (u(x,y),v(x,y))$ and $m = n = 2$, then the Jacobian matrix of order 2 is,
And the determinant is represented as;
Since the Jacobian is used to transform a function between different coordinates. Therefore, it can also transform a function from cartesian to polar coordinates. So for,
$x=r\cos \theta$
$y=r\sin \theta$
The Jacobian from cartesian to polar coordinates will be,
Now we have to calculate all partial derivatives one by one.
$\frac{\partial x}{\partial r}=\frac{\partial}}{\partial r}(r\cos \theta)=\cos \theta$
$\frac{\partial x}{\partial \theta}=-r\sin \theta$
$\frac{\partial y}{\partial r}=\sin\theta$
$\frac{\partial y}{\partial \theta}=r\cos \theta$
Now using the above derivatives in the jacobian matrix, we get
How to calculate the Jacobian of a function?
In calculus, the Jacobian is used to solve the nonlinear system of a vector function. It makes the calculation easy with the use of first order derivatives. We can calculate Jacobian by using the following steps.
- Identify the variables of the vector function e.g., (u(x,y),v(x,y)).
- Calculate all partial derivatives one-by-one.
- Now write the partial derivatives in the matrix form.
- Find the determinant of the matrix. If the order of the Jacobian matrix is more than 2, then use the expansion method of matrices to solve it.
- Simplify to get a solution.
Let’s understand how to find jacobian in the following examples.
Jacobian Example
Compute Jacobian $J(u, v)$ for, $x(u,v)=u^2-v62,y(u,v)=2uv$.
Given that,
$x(u,v)=u^2-v^2$
And,
$y(u,v)=2uv$
To calculate Jacobian, we have to transform from x(u, v), y(u, v) to J(u, v).
Using the Jacobian formula,
Finding the determinant, we get
$J(u,v)=4u^2+4v^2$
Related Topics
Like Jacobian, partial derivative and power rule of differentiation are fundamental concepts in differential calculus. It is because the Jacobian cannot be calculated without following derivative rules. These rules are defined as,
Partial derivative:
If a function depends on more than one variable, the partial derivative is used to calculate its derivative. It is represented as;
$\frac{d}{dx}[f(x,y)]=\frac{df}{dx} + \frac{df}{dy}$
Above formula says that we can get partial derivative by adding the derivative of f(x,y) with respect to x and y.
Power Rule:
The power rule is a rule to calculate derivatives of an algebraic expression with some power. The formula for power rule for xn is;
$f’(x) = \frac{d}{dx}[x^n]=nx^{n-1}\frac{d}{dx}(x)$
Where, n is a real number.
Conclusion
In calculus, the Jacobian is an important mathematical concept. It allows us to study how small changes in one set of variables affect another set of variables in a given function. The Jacobian plays a key role in many applications such as optimization problems, linear transformations, and change of variables in integrals. It is an essential concept for students and practitioners of mathematics, physics, engineering, and many other fields.