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
