Question

If F(u)= f(x,y,z) be a homogeneous function of degree n in x,y,z, then $x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}+z\frac{\partial u}{\partial z}$=

• A. $\frac{F\left(u\right)}{F\left(u\right)}$
• B. n F (u)
• C. nu
• D. None of these

Answer 1

Answer: (B)

A function $f$ is called homogeneous of degree $n$, then it will satisfy the equation-

$f(tx,ty,tz)=t^{n}f(x,y,z)$

$f(x,y,z)=F\left(u\right)$

Let,

$p=tx$

$q=ty$

$r=tz$

Therefore,

$\frac{d}{dt}(p,q,r)=n{t}^{n-1}f(x,y,z)$

$\frac{\partial f}{\partial p}\frac{dp}{dt}+\frac{\partial f}{\partial q}\frac{dq}{dt}+\frac{\partial f}{\partial r}\frac{dr}{dt}=n{t}^{n-1}F\left(u\right)(\because F\left(u\right)=f(x,y,z))$

$\Rightarrow x\frac{\partial f}{\partial p}+y\frac{\partial f}{\partial q}+z\frac{\partial f}{\partial r}=n{t}^{n-1}F\left(u\right)$

Substituting $t=1$, we get

$x\frac{\partial f}{\partial x}+y\frac{\partial f}{\partial y}+z\frac{\partial y}{\partial z}=nF\left(u\right)$

Thus $x\frac{\partial f}{\partial x}+y\frac{\partial f}{\partial y}+z\frac{\partial y}{\partial z}=nF\left(u\right)$

Hence the correct answer is $nF\left(u\right)$.