# What exacty is the role played by Jacobian or Wronskian?

In many of our derivations or in differential equations we come across the terms Jacobian or Wronskian. For example, to check the linear independence of solutions of differential equations, we ensure that the Wronskian is non zero. What role do these play? What do they actually account for?

The Jacobian is an $$m\times n$$ matrix and it consists of first-order derivatives of all the variables of a given function $$f$$. The Jacobian matrix is an $$m\times n$$ matrix that gives the best linear approximation of $$f$$ near the point $$x\in \mathbb{R}^n$$. If we have a square matrix, then $$f:\mathbb{R}^n\to\mathbb{R}^n$$ and the Jacobian tells us that $$f$$ is invertible if the Jacobian at a point is non-zero.
On the other hand, the Wronskian is used to show that a set of solutions are linearly independent, as long as the Wronskian does not vanish. For functions $$f_1,...,f_n$$ the Wronskian is the determinant of an $$n\times n$$ matrix defined on an interval $$x\in I$$. Unlike the Jacobian, it includes higher derivatives than the first derivatives (here, we must have $$(n-1)^{\text{th}}$$ derivatives).