UNIT-3

PARTIAL DIFFERENTIATION 2

Q 1: Let x(u,v) = u2 – v2, y(u,v)=uv.find the jacobain j(u,v).

Solution:

Given that x(u,v) =u2-v2 and y(u,v)= 2uv

We know that,

J(u,v)=

Therefore, J(u,v) = 4u2+4v2

Q 2: Find if u= 2xy, v= x2 – y2 and x = r.cos , y = r.sin

Solution:

=

=

Q 3: Expandin power of up to second  terms.

Given function

Here  

Now,  

and

The Taylor’s expansion in power is

Q 4: Expand the  in power of x and y up to third term?

Given function 

Now,

By Maclaurins expansion

+…

Q 5: Let z = f(x, y)

A5)

Now for stationary point dz = 0

&

This gives the set of values of x and y for which maxima or minim occurs

Now find

We called it as r, s, t resp.

Thus function has maximum or minimum

ifrt – s2>0

i.e.

further if

; function is minimum at (x, y) &

; Function is maximum at (x, y)

Q6: Decampere a positive number ‘a’ in to three parts, so their product is maximum

A6)

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  .

Q 7: Find the point on plane nearest to the point (1, 1, 1) using Lagrange’s method of multipliers.

A7) 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.