Module 6

Partial differentiation

6.1 Definition :

Let be a function of two independent variable x and y. Suppose now x changes to, y remaining constant. Then x will change to .

The limit of    as , if it existed is called the partial derivative of z with respect to x and it is denoted by    or    or  .

Similarly we can find 

If

find 

Sol:

Differentiate z wrt x with y as constant

Diff w.r.t y as x as constant

6.2 Some standard functions:

6.3 Some standard rules:

6.4 Sums on Standard Formula

Sol:

Sol:

Sol:

Type II

6.5 Partial Derivatives of first order of a function of a function.

Sol:

2.     Find the value of  n , so that  . satifies the equation

Sol

Further diff wrt

Adding (i) & (ii) we get.

3.    

Sol:

Type III

6.6 Partial Derivatives of second order of a function of a function.

Q1.

Sol:

Again diff wrt x

Detailed Explained of the above step

Again differentiating  x  wrt  y

Sol:

Now

Using the formula

Eq (i) becomes.

Sol: Since    treating  y constant

Treating x  constant .................(2)

 Diff again partially wrt x, we get

Now diff (1) partially writ y we get.

4.    

Sol: Differentiating x  partially wrt y get,  .

Differentiating this paritially wrt x we get.

Diff this again partially wrt y we get

Differentiating    probably

wrt  y

From (i) & (ii)

6.7 Variable treated as constant

 

As we want   we

must express  u, v  as function

6.8 Composite function

 It is given by

Sol:

     =

6.9 First order Linear PDE

A first-order PDE for an unknown function  is said to be linear if it can be expressed in the form

The PDE is said to be quasilinear if it can be expressed in the form

A PDE which is neither linear nor quasi-linear is said to be nonlinear.

6.10 First order Nonlinear PDEs

The general first-order nonlinear PDE for an unknown function  is given by

Here  is a function of , , and .

The term "nonlinear" refers to the fact that  is a nonlinear function of  and . For instance, the equation involves a quadratic expression in  and .