Unit-5

Applications of Multivariable Differential Calculus

Question-1: If x = r sin , y = r sin , z = r cos, then show that

sin  also find 

Sol.  We know that,

                                       =

                                       =

                                       = ( on solving the determinant)

                                      = 

Now using first propert of Jacobians, we get

Question-2: If u = x + y + z ,uv = y + z , uvw = z ,  find

Sol.  Here we have,

              x = u – uv = u(1-v)

             y = uv – uvw = uv( 1- w)

And      z = uvw

So that,

                              =

Apply

                             =

Now we get,

=  u²v(1-w) + u²vw

                          = u²v

Question-3: If u = xyz , v = x² + y² + z²  and w = x + y + z, then find   J = 

Sol. Here u ,v and w are explicitly given , so that first we calculate

= yz(2y-2z) – zx(2x – 2z) + xy (2x – 2y)

= 2[yz(y-z)-zx(x-z)+xy(x-y)]

= 2[x²y - x²z - xy² + xz² + y²z - yz²]

= 2[x²(y-z)  - x(y² - z²) + yz (y – z)]

= 2(y – z)(z – x)(y – x)

= -2(x – y)(y – z)(z – x)

By the property,

JJ’ =  1

Question-4: If u = 2axy, v = then prove that-

Sol. Here we have,

                                     u = 2axy, v =

Then

    Here - 

    So that

Now,

Hence-

Hence proved.

Question-5: If x + y + z = u,  y + z = uv , z = uvw, then prove that-

Sol. Suppose

And

Hence,

Hence proved.

Question-6: Show that and are functionally dependent.

Sol. Here we have-

and

Now we will find out the Jacobian to check the functional dependence.

                        =

Here Jacobian is zero, so we can conclude that these functions are functionally dependent.

Question-7: Prove that u, v, w are functionally dependent, where-

Sol. Here we have 

Now we will find out the Jacobian of the given functions-

                                                          =

=

= 0

Therefore, u, v, w are functionally dependent.

Question-8: Find the percentage error in the area of an ellipse where error of ly is made in measuring it’s major and minor axes.

Solution:

Let A be the area of an ellipse and ‘a’ & ‘b’ are its semi major and minor axes

Taking log on both sides.

Differentiating we get,

Percentage error ion the area of an ellipse = 2%

Question-9: Find out the maxima and minima of the function

Sol.

Given

…(i)

Partially differentiating (i) with respect to x we get

….(ii)

Partially differentiating (i) with respect to y we get

….(iii)

Now, form the equations

Using (ii) and (iii) we get

using above two equations

Squaring both side we get

Or 

This show that

Also we get

Thus we get the pair of value as

Now, we calculate

      Putting above values in

      At point (0,0)  we get

      So, the point (0,0) is a saddle point.

      At point we get

        So the point is the minimum point where

          In case 

         So the point is the maximum point where

Question-10: Determine the maxima and minima of when 1

Sol. Suppose    ……….. (1)

And =0  …………… (2)

Differentiate partially equation (1) and (2) w.r.t. x and y respectively, we get-

And                            

Now using Lagrange’s equations, we get-

Which gives-

or    ……… (3)

Which gives-

…… (4)

Which gives-

……. (5)

Multiply these equations by x, y, z respectively and adding, we get-

                                     (

Hence we get-

                             f + 0   then 

Put in (3) , (4) and (5), we get-

x(1 – fa) = 0   ,   y(1 – fb) = 0 and z(1 – fc) = 0

We get-

These gives the maximum and minimum values of f.

Question-11: Divide 24 into three parts such that the continued product of the first, square of second and cube of third may be maximum.

Let first number be x, second be y and third be z.

According to the question

Let the  given function  be  f

And the relation

By Lagrange’s Method

….(i)

Partially differentiating (i) with respect to x,y and z and  equate  them  to zero

….(ii)

….(iii)

….(iv)

From (ii),(iii)  and (iv) we get

On solving

Putting it in given relation we get

Or 

Or

Thus the first number is 4 second is 8 and third is 12

Unit-5

Applications of Multivariable Differential Calculus

Question-1: If x = r sin , y = r sin , z = r cos, then show that

sin  also find 

Sol.  We know that,

                                       =

                                       =

                                       = ( on solving the determinant)

                                      = 

Now using first propert of Jacobians, we get

Question-2: If u = x + y + z ,uv = y + z , uvw = z ,  find

Sol.  Here we have,

              x = u – uv = u(1-v)

             y = uv – uvw = uv( 1- w)

And      z = uvw

So that,

                              =

Apply

                             =

Now we get,

=  u²v(1-w) + u²vw

                          = u²v

Question-3: If u = xyz , v = x² + y² + z²  and w = x + y + z, then find   J = 

Sol. Here u ,v and w are explicitly given , so that first we calculate

= yz(2y-2z) – zx(2x – 2z) + xy (2x – 2y)

= 2[yz(y-z)-zx(x-z)+xy(x-y)]

= 2[x²y - x²z - xy² + xz² + y²z - yz²]

= 2[x²(y-z)  - x(y² - z²) + yz (y – z)]

= 2(y – z)(z – x)(y – x)

= -2(x – y)(y – z)(z – x)

By the property,

JJ’ =  1

Question-4: If u = 2axy, v = then prove that-

Sol. Here we have,

                                     u = 2axy, v =

Then

    Here - 

    So that

Now,

Hence-

Hence proved.

Question-5: If x + y + z = u,  y + z = uv , z = uvw, then prove that-

Sol. Suppose

And

Hence,

Hence proved.

Question-6: Show that and are functionally dependent.

Sol. Here we have-

and

Now we will find out the Jacobian to check the functional dependence.

                        =

Here Jacobian is zero, so we can conclude that these functions are functionally dependent.

Question-7: Prove that u, v, w are functionally dependent, where-

Sol. Here we have 

Now we will find out the Jacobian of the given functions-

                                                          =

=

= 0

Therefore, u, v, w are functionally dependent.

Question-8: Find the percentage error in the area of an ellipse where error of ly is made in measuring it’s major and minor axes.

Solution:

Let A be the area of an ellipse and ‘a’ & ‘b’ are its semi major and minor axes

Taking log on both sides.

Differentiating we get,

Percentage error ion the area of an ellipse = 2%

Question-9: Find out the maxima and minima of the function

Sol.

Given

…(i)

Partially differentiating (i) with respect to x we get

….(ii)

Partially differentiating (i) with respect to y we get

….(iii)

Now, form the equations

Using (ii) and (iii) we get

using above two equations

Squaring both side we get

Or 

This show that

Also we get

Thus we get the pair of value as

Now, we calculate

      Putting above values in

      At point (0,0)  we get

      So, the point (0,0) is a saddle point.

      At point we get

        So the point is the minimum point where

          In case 

         So the point is the maximum point where

Question-10: Determine the maxima and minima of when 1

Sol. Suppose    ……….. (1)

And =0  …………… (2)

Differentiate partially equation (1) and (2) w.r.t. x and y respectively, we get-

And                            

Now using Lagrange’s equations, we get-

Which gives-

or    ……… (3)

Which gives-

…… (4)

Which gives-

……. (5)

Multiply these equations by x, y, z respectively and adding, we get-

                                     (

Hence we get-

                             f + 0   then 

Put in (3) , (4) and (5), we get-

x(1 – fa) = 0   ,   y(1 – fb) = 0 and z(1 – fc) = 0

We get-

These gives the maximum and minimum values of f.

Question-11: Divide 24 into three parts such that the continued product of the first, square of second and cube of third may be maximum.

Let first number be x, second be y and third be z.

According to the question

Let the  given function  be  f

And the relation

By Lagrange’s Method

….(i)

Partially differentiating (i) with respect to x,y and z and  equate  them  to zero

….(ii)

….(iii)

….(iv)

From (ii),(iii)  and (iv) we get

On solving

Putting it in given relation we get

Or 

Or

Thus the first number is 4 second is 8 and third is 12