Unit 3

Partial Differentiation

3.1 Partial derivatives

Partial Differentiation

If

Prove that

Partial differentiation of function of function

If z = f(u) and . Then z becomes a function of x & y. In this case, z becomes a function of x & y.

i.e.

Then

,

Similarly

If

Then z becomes a function of x, y & z.

…………….

1. If where

Prove that

2.     If V = show that

3.     If show that

4.     If then prove that

3.2 Eulers theorem

A polynomial in x & y is said to be Homogeneous expression in x & y of degree n. If the degree of each term in the expression is same & equal to n.

e.g.

is a homogeneous function of degree 3.

To find the degree of homogeneous expression f(x, y).

1. Consider
2. Put . Then if we get .

Then the degree of is n.

Ex.

Consider

Put

.

Thus degree of f(x, y) is

Note that

If be a homogeneous function of degree n then z can be written as

Differentiation of Implicit function

Suppose that we cannot find y explicitly as a function of x. But only implicitly through the relation f(x, y) = 0.

Then we find

Since

diff. P. w.r.t. x we get

i.e.

Similarly,

It f (x, y, z) = 0 then z is called implicit function of x, y. Then in this case we get

Ex.

Find if

Ex. Find . If , &

Ex. If , where

Find

Ex. If

Then find

Eulers Theorem on Homogeneous functions:

Statement:

If be a homogeneous function of degree n in x & y then,

Deductions from Euler’s theorem

1. If be a homogeneous function of degree n in x & y then,

.

2.     If be a homogeneous functions of degree n in x & y and also then,

And

Where

Ex.

If , find the value of

Ex.

If then find the value of

Ex. If then prove

That

Ex. If the prove that

Ex. If  then show

That

3.3 Composite function

a)     Let and , then z becomes a function of , In this case, z is called a composite function of .

i.e.

b)    Let possess continuous partial derivatives and let possess continuous partial derivatives, then z is called a composite function of x and y.

i.e.

&

Continuing in this way, …..

Ex. If Then prove that

Ex. If then prove that

Where is the function of x, y, z.

Ex. If where ,

then show that,

i)      

Ii)   

Notations of partial derivatives of the variable to be treated as a constant

Let

and

i.e.

Then means the partial derivative of u w.r.t. x treating y const.

To find from given reactions we first express x in terms of u & v.

i.e. & then diff. x w.r.t. u treating v constant.

To find express v as a function of y and u i.e. then diff. v w.r.t. y treating u as a const.

Ex. If , then find the value of

.

Ex. If , then prove that

3.4 Method of Lagrange’s multipliers.

Let be a function of x, y, z which to be discussed for stationary value.

Let be a relation in x, y, z

for stationary values we have,

i.e.    … (1)

Also from we have

    … (2)

Let ‘’ be undetermined multiplier then multiplying equation (2) by and adding in equation (1) we get,

     … (3)

     … (4)

      … (5)

Solving equation (3), (4) (5) & we get values of x, y, z and .

1. Decampere a positive number ‘a’ in to three parts, so their product is maximum

Solution:

Let x, y, z be the three parts of ‘a’ then we get.

    … (1)

Here we have to maximize the product

i.e.

By Lagrange’s undetermined multiplier, we get,

       … (2)

       … (3)

        … (4)

i.e.

        … (2)’

        … (3)’

        … (4)

And

From (1)

Thus .

Hence their maximum product is  .

2.     Find the point on plane nearest to the point (1, 1, 1) using Lagrange’s method of multipliers.

Solution:

Let be the point on sphere which is nearest to the point . Then shortest distance.

Let

Under the condition    … (1)

By method of Lagrange’s undetermined multipliers we have

       … (2)

       … (3)

i.e. &

       … (4)

From (2) we get

From (3) we get

From (4) we get

Equation (1) becomes

i.e.

y = 2

If where x + y + z = 1.

Prove that the stationary value of u is given by,

Reference Book:

1. G.B. Thomas and R.L. Finney, Calculus and Analytic geometry, 9th Edition, Pearson, Reprint, 2002.

2. Erwin kreyszig, Advanced Engineering Mathematics, 9th Edition, John Wiley & Sons,2006.

3. Veerarajan T., Engineering Mathematics for the first year, Tata McGraw-Hill, New Delhi,2008.

4. Ramana B.V., Higher Engineering Mathematics, Tata McGraw Hill New Delhi, 11th Reprint, 2010.

5. D. Poole, Linear Algebra: A Modern Introduction, 2nd Edition, Brooks/Cole, 2005.

6. B.S. Grewal, Higher Engineering Mathematics, Khanna Publishers, 36th Edition, 2010