To get an idea about Jacobians, one requires knowledge of matrices, determinants, functions and partial differentiation.
I) Matrices and Determinants :
A matrix is a rectangular arrangement of numbers in rows and columns. When , the matrix is known as a square matrix. Determinants are defined for square matrices. The simplest square matrix is
One can expand a determinant using cofactors.
Note that Jacobians are determinants whose elements are partial derivatives.
II) Functions of Several Variables :
This is often the case in real world applications. For example, the temperature at a given point may be a function of space and time . So depends on variables.
III) Partial Differentiation :
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Let be a function of . Then,by definition of differentiation,
By removing the limit, we can approximate the left hand side to right hand side.
Now, instead of , if we have a vector function , then
The thing is the Jacobian. This means, while dealing with derivatives of functions of several variables, to get an approximation, one needs to differentiate each function w.r.t. each independent variable. This is what Jacobians are for.
Jacobians are termed as functional determinants. Let be functions of independent variables . Then the determinant
is the Jacobian, which is sometimes denoted by
Similarly, for functions of ,
While changing the coordinate system from one to another, Jacobians are useful. Thus,
A corollary to this is,
Jacobians of Implicit Functions
Let be implicit functions of such that following equations are satisfied :
Then, for ,
Useful to solve problems :
Partial Derivatives of Implicit Functions
Let be implicit functions of such that
Similarly, one can get and other derivatives.
For functional dependence of and ,
For dependence of 3 functions, one has to equate the Jacobian to .