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State and Prove Euler’S Theorem for Three Variables. - Applied Mathematics 1

Sum

State and prove Euler’s Theorem for three variables.

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Solution

Statement: If u=f(x, y, z) is a homogeneous function of degree n, then -`x(delu)/(delx)+y(delu)/(dely)+z(delu)/(delz)=n u`

Let, u=f(x, y, z) is a homogeneous function of degree n.
Putting X = x t, Y = y t, Z = z t.
f(X,Y,Z) = tn f(x,y,z) ………. (1)
Diff LHS w.r.t t,

`(delf)/(delt)=(delf)/(delx).(delx)/(delt)+(delfdely)/(delydelt)+(delfdelz)/(delzdelt)`

`(delf)/(delt)=x(delf)/(delx).+y(delf)/(dely)+z(delf)/(delz)`…… (2)

Diff RHS w.r.t. t,

`(delf)/(delt)=nt^(n-1)f(x,y,z)`

Now put t = 1, we get `(delf)/(delt)=nf(x,y,z)`……… (3)

From equation 2 and 3, we get

`x(delf)/(delx).+y(delf)/(dely)+z(delf)/(delz)=nf(x,y,z)`

`x(delf)/(delx).+y(delf)/(dely)+z(delf)/(delz)=n u`

Hence proved

Concept: Euler’s Theorem on Homogeneous functions with two and three independent variables (with proof)
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