Heat and Thermodynamics

1.1 Partial Differentiation

Here we shall discuss differentiation of a function containg more than one independent variable. To start with, we shall consider the case of a function u which contains two independent variables x and y. The derivative of u, when x varies and y remains constant, is called the partial derivative of u with respect to x. Similarly, the partial derivative of u with respect to y is the differential coefficient of u with respect to y, when x remains constant.

 

Notation: Partial derivatives of u with respect to x and y are denoted by ∂u/∂x and ∂u/∂y. Similar notations are used for higher derivatives. The formal definition of ∂u/∂x, when u = f (x, y), is

 

 

Similarly,

 

 

Example 1: Find the partial derivative of ...