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.
Then z becomes a function of x, y & z.
- If where
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.
is a homogeneous function of degree 3.
To find the degree of homogeneous expression f(x, y).
- Put . Then if we get .
Then the degree of is n.
Thus degree of f(x, y) is
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
diff. P. w.r.t. x we get
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. If , where
Eulers Theorem on Homogeneous functions:
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,
If , find the value of
If then find the value of
Ex. If then prove
Ex. If the prove that
Ex. If then show
a) Let and , then z becomes a function of , In this case, z is called a composite function of .
b) Let possess continuous partial derivatives and let possess continuous partial derivatives, then z is called a composite function of x and y.
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,
Notations of partial derivatives of the variable to be treated as a constant
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
Let ‘’ be undetermined multiplier then multiplying equation (2) by and adding in equation (1) we get,
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
Let x, y, z be the three parts of ‘a’ then we get.
Here we have to maximize the product
By Lagrange’s undetermined multiplier, we get,
Hence their maximum product is .
2. Find the point on plane nearest to the point (1, 1, 1) using Lagrange’s method of multipliers.
Let be the point on sphere which is nearest to the point . Then shortest distance.
Under the condition … (1)
By method of Lagrange’s undetermined multipliers we have
From (2) we get
From (3) we get
From (4) we get
Equation (1) becomes
y = 2
If where x + y + z = 1.
Prove that the stationary value of u is given by,
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