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.

1.   Calculate
2. If
3. If

ST

4.      find

5.     If and , find

6.    

7.     If  

8.     If , ,

JJ1 = 1

If ,

JJ1=1

Jacobian of composite function (chain rule)

Then

Ex.

1. If

Where

2.     If   and

Find

3.     If

Find

Jacobian of Implicit function

Ex.

If

If

Find

Partial derivative of implicit functions

Ex.

If and

Find

If and

Find

Find

If

Find

Let f(x, y) be a continuous function of x & y. Let be the increments x & y resp. Then the value of f(x, y) will be

Expanding above expression by taylor’s theorem & since are very small Hence their product and higher powers are negligible and hence neglected we get,

i.e.

Similarly,

If f be a function of variables x, y, z, t, ………. Then we have

Note that

1. dx, dy, dz, ……. may be taken as actual errors or interments in x, y, z ……… resp. while df is the approximate error in f.
2. ……….. denotes relative errors in x, y, z,………….. resp.
3. ………….
4. Denotes percentage errors in, x, y, z, …………. resp.

Ex.

Q1) Find the percentage error in the area of an ellipse where error of ly is made in measuring it’s major and minor axes.

S1)

Q2) The density of a body is calculated from its’s weight W in air and win water if error are made in , find the error in .

S2)

Q3) Find the percentage error in computing the parallel resistance r of three resistances r1, r2, r3 from the formula.

Where

r1, r2, r3 are each in error of 1 & 2y.

Q4) Find the approximate value of

S4)

1. ; function is minimum at (x, y) &
2. ; Function is maximum at (x, y)

Note that

1. If ; then function will not have either maxima or minima such point is called saddle point.
2. If ; then more details are required to justly maxima or minima

Ex. Discuss the stationary values of

Ex. Find the values of x and y for which x2 + y2 + 6x = 12 has a minimum values and find its minimum value.

Divide 120 into three parts so that the sum of their product. Taken two at a times shall be maximum.

Using Lagrange’s method divide 24 into three parts. Such that continued product of the first, square of second, cube of third may be maximum.

Find the maximum and minimum value of x2 + y2 when 3x2 + 4xy + 6y2 = 140

is satisfied.

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 .

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  .

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,