Question

State and Prove Euler’s Theorem for three variables.

 

Answer 1

Euler’s theorem: 

Statement: If ‘u’ is a homogenous function of three variables x, y, z of degree ‘n’ then Euler’s theorem
States that 

`x del_u/del_x+ydel_u/del_y+z del_u/del_z=n u`

Proof:
Let u = f (x, y, z) be the homogenous function of degree ‘n’. 

Let X = xt, Y = yt, Z = zt 

∴` (delx)/(delt)=x; (dely)/(delt)=y;(delz)/(delt)=z`        …(1) 

  At t = 1,                                              …(2)

X = x, Y = y, Z = z 

∴ `(delf)/(delx)=(delf)/(delx);(delf)/(dely)=(delf)/(dely);(delf)/(delz)=(delf)/(delz)`          …(3)

Now, f (X, Y, Z) = `t^n` f (x, y, z)               …(4

∴f → X, Y, Z → x, y, z, t 

Differentiating (4) partially` w.r.t. 't', (delf)/(delx).(delx)/(delt)+(delf)/(del).(dely)/(delt)+(delf)/(delz).(delz)/(delt)=nt^(n-1)f(x,yz)`

∴ `(delf)/(delx).x+(delf)/(dely).y+(delf)/(delz).z=n(1)^(n-1)f(x,y,z)`                            (From 1,2 & 3) 

∴ `x(delu)/(delx)+y(delu)/(dely)+z(delu)/(delz)=n u`