A Note on Jacobians

The matrix representation for the Backward Euler solution introduced a partial derivative term in the integration algorithm. These partial derivatives are otherwise referred to as Jacobians.

How does a partial derivative differ from its “impartial” counterpart? A partial derivative is one in which the dependent variable is differentiated with respect to only one of the other variables. Partial derivatives bring in the concept of dimensionality in a real world problem. Heat transfer problems in Mechanical Engineering are a good intuitive example.

Mathematically, let us assume a function



Here, y is a function of x and y. A normal derivative is usually with respect to x (assuming x is the independent variable), while the partial derivative is with respect to y. In the above case, the partial derivative, or the Jacobian of f with respect to y is



The Jacobian of a function that has multiple dependent variables is thus a matrix of partial derivatives. For example, if f is a function defined by



then the partial derivatives are



Following the above logic, the Jacobian matrix for a series of ODE’s can be represented as follows: