So far, while studying calculus, we have dealt with functions of single variable, i.e. . are few examples. Irrespective of their complexity, the variable always depended on the value of independent variable . We also defined the derivatives and integrals of and studied few applications of them.
More often than not, we encounter situations, where a function needs more than 1 independent variable for its definition. Such functions are known as functions of several variables. e.g. a function of 2 variables is
Thus, without knowing values of both and simultaneously, we cannot get a unique value of .
One can define a function of as many variables as one wants. (Of course, it should make some sense.) In many of the problems in mechanical engineering, the functions are of at the most 4 independent variables; viz. 3 space variables, and a time variable .
The partial differentiation involves obtaining the derivatives of functions of several variables.
- Definition and Rules
Let be a function of 2 independent variables and . To differentiate partially w.r.t , we treat as a constant and follow the usual process of differentiation. Thus,
Thus, the definition is similar to that of ordinary differentiation. The condition of existence of the limit is necessary.
Note that we use the letter for partial derivatives and the letter for ordinary derivatives.
The rules of for differentiation of addition, subtraction, multiplication, division are same as ordinary differentiation.
- Derivatives of Higher Order
Having obtained the first order derivatives and , we now define the second order derivatives, i.e.
For a function of 2 variables, four 2nd order derivatives are possible. These are sometimes written as
If the function and its derivatives are continuous, then we have
One can define derivatives of order by following the same procedure.
- Types of Problems (Crucial from exam point of view)
I) Based on the definition and the commutative property of partial differentiation
II) Based on the concept of composite functions (Mostly involve the relations between cartesian and polar coordinates)
- Homogeneous Functions (Already encountered in M II , 1st unit)
When the sum of indices of the variables in a function is same for all terms, the function is said to be homogeneous of degree equal to the sum.
is an example. (Degree )
Note that each term must be explicitly of the form . Thus, is NOT a homogeneous function.
- Euler’s Theorem (by Leonhard Euler)
For a homogeneous function of degree ,
As a consequence of this,
Similarly, if is a homogeneous function of 3 independent variables of degree , then
- Total Derivatives
Consider a function . If it so happens that and themselves are functions of another variable , then the total derivative of w.r.t. is defined as
Thus, if we are given a function , we would differentiate it w.r.t. , thus getting . Instead, if is expressed as and and , then obtaining the total derivative of will be equivalent to getting
We will discuss the applications of partial differentiation in the next unit.