Unit 3

Partial Differentiation

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.

…………….

- If where

Prove that

2. If V = show that

3. If show that

4. If then prove that

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).

- Consider
- 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

- 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

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

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 .

- 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