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# multivariable calculus Watch

1. Hi, I have problem solving this question. pls help!!

Qns: Suppose f(x,y,z) is a function which satisfies,

f(tx, ty, tz) = t^5*f(x, y, z), for all x,y,z,t ∈ R

Show that

x(∂f/∂x) + y(∂f/∂y) + z(∂f/∂z) =Cf,
for some constant C.

Ans: Let x,y,z be functions of t,

f( tx(t), ty(t), tz(t) ) = t^5*f( x(t), y(t), z(t) ), for all x,y,z,t R

differentiate both sides wrt. t,

(∂f/∂x){x(t)+t(dx/dt)} + (∂f/∂y){y(t)+t(dy/dt)} + (∂f/∂z){z(t)+t(dz/dt)}
=5t^4 * f( x(t), y(t), z(t) ) + t^5 * (
(∂f/∂x)(dx/dt) + (∂f/∂y)(dy/dt) + (∂f/∂z)(dz/dt) )

after manipulating the eqn, I got

x(t)*
(∂f/∂x) + y(t)*(∂f/∂y) + z(t)*(∂f/∂z) = 5t^4 * f( x(t), y(t), z(t) ) + (t^5 - t) * ( (∂f/∂x)(dx/dt) + (∂f/∂y)(dy/dt) + (∂f/∂z)(dz/dt) )

and I'm stuck here now. any hints on how to go about solving this?

thanks!
2. (Original post by nigepang)
Hi, I have problem solving this question. pls help!!

Qns: Suppose f(x,y,z) is a function which satisfies,

f(tx, ty, tz) = t^5*f(x, y, z), for all x,y,z,t ∈ R

Show that

x(∂f/∂x) + y(∂f/∂y) + z(∂f/∂z) =Cf,
for some constant C.

Ans: Let x,y,z be functions of t,

f( tx(t), ty(t), tz(t) ) = t^5*f( x(t), y(t), z(t) ), for all x,y,z,t R

differentiate both sides wrt. t,

(∂f/∂x){x(t)+t(dx/dt)} + (∂f/∂y){y(t)+t(dy/dt)} + (∂f/∂z){z(t)+t(dz/dt)}
=5t^4 * f( x(t), y(t), z(t) ) + t^5 * (
(∂f/∂x)(dx/dt) + (∂f/∂y)(dy/dt) + (∂f/∂z)(dz/dt) )

after manipulating the eqn, I got

x(t)*
(∂f/∂x) + y(t)*(∂f/∂y) + z(t)*(∂f/∂z) = 5t^4 * f( x(t), y(t), z(t) ) + (t^5 - t) * ( (∂f/∂x)(dx/dt) + (∂f/∂y)(dy/dt) + (∂f/∂z)(dz/dt) )

and I'm stuck here now. any hints on how to go about solving this?

thanks!
I can't read that. Why not use LaTeX?

Maybe this will help. http://mathworld.wolfram.com/EulersH...onTheorem.html
3. (Original post by nigepang)
[SIZE=3][FONT=arial]Hi, I have problem solving this question. pls help!!

Qns: Suppose f(x,y,z) is a function which satisfies,

f(tx, ty, tz) = t^5*f(x, y, z), for all x,y,z,t ∈ R

Show that

x(∂f/∂x) + y(∂f/∂y) + z(∂f/∂z) =Cf,
for some constant C.

Ans: Let x,y,z be functions of t,
Let tx=u, ty=v and tz=w
then
f(u,v,w)=t^5*f(x,y,z)

Differentiate both side with respect t
then use t=1 substitution

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