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Question

If F(u)= f(x,y,z) be a homogeneous function of degree n in x,y,z, then xux+yuy+zuz=

A
F(u)F(u)
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B
n F (u)
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C
nu
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D
None of these
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Solution

The correct option is B n F (u)
A function f is called homogeneous of degree n, then it will satisfy the equation-
f(tx,ty,tz)=tnf(x,y,z)
f(x,y,z)=F(u)
Let,
p=tx
q=ty
r=tz
Therefore,
ddt(p,q,r)=ntn1f(x,y,z)
fpdpdt+fqdqdt+frdrdt=ntn1F(u)(F(u)=f(x,y,z))
xfp+yfq+zfr=ntn1F(u)
Substituting t=1, we get
xfx+yfy+zyz=nF(u)
Thus xfx+yfy+zyz=nF(u)
Hence the correct answer is nF(u).

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