3.1 Introduction to functions of several variables | unit 3 partial differentiation

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Unit –3

Partial Differentiation

3.1 Introduction to functions of several variables

Partial Differentiation

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 function of x & y.

i.e.

Then

Similarly

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

…………….

If where

Prove that

2. If V = show that

3. If show that

4. If then prove that

3.2 Partial Derivatives

Partial Differentiation of composite function

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

i.e.

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

i.e.

Continuing in this way, …..

Ex. If Then prove that

Ex. If then prove that

Where is function of x, y, z.

Ex. If where ,

then show that,

ii)

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

Let

and

i.e.

Then means 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

3.3 Homogeneous Function/ Equation:-

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 Eulers 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

3.4 Partial derivative of Composite Function, Total Derivative, Change of Independent variables

Jacobians, Errors and Approximations, maxima and minima

Jacobians

If u and v be continuous and differentiable functions of two other independent variables x and y such as

, then we define the determine

as Jacobian of u, v with respect to x, y

Similarly ,

JJ’ = 1

Actually Jacobins are functional determines

Ex.

Calculate
If
If

4. find

5. If and , find

7. If

8. If , ,

JJ1 = 1

If ,

JJ1=1

Jacobian of composite function (chain rule)

Then

Ex.

Where

2. If and

Find

3. If

Find

Jacobian of Implicit function

Let u1, u2 be implicit functions of x1, x2 connected by f1, f2 such there

Then

Similarly,

Ex.

Find

Partial derivative of implicit functions

Consider four variables u, v, x, y related by implicit function.

Then

Ex.

If and

Find

If and

Find

Reference Books:

1. Advanced Engineering Mathematics by Erwin Kreyszig (Wiley Eastern Ltd.)

2. Advanced Engineering Mathematics by M. D. Greenberg (Pearson Education)

3. Advanced Engineering Mathematics by Peter V. O’Neil (Thomson Learning)

4. Thomas’ Calculus by George B. Thomas, (Addison-Wesley, Pearson)

5. Applied Mathematics (Vol. I & Vol. II) by P.N.Wartikar and J.N.Wartikar Vidyarthi Griha Prakashan, Pune.

6. Linear Algebra –An Introduction, Ron Larson, David C. Falvo (Cenage Learning, Indian edition)

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