Understanding the Concept of Gradient for a Zero Function

[HAMJOOP]

Suppose F(x,y,z) = 0
grad (F) = 0 ?

e.g. F = x + y + z

grad (F) = <1,1,1> =/= <0,0,0> ??

I don't know why I get an opposite result

 

[SteamKing]

What opposite result are you talking about?

The derivative of a constant is always zero, regardless of the value of the constant.

Hint: F(x,y,z) = x + y + z is not a constant function.

 

[lurflurf]

grad(x+y+z)=<1,1,1> =/= <0,0,0> =grad(0)
x+y+z=/=0

There is no contradiction, why would you expect one?

 

[VantagePoint72]

Because if $F(x,y,z)=x+y+z$ then it's not a zero function? And it doesn't have a critical point at (0,0,0) either. Maybe I'm not clear what exactly what you're asking. $F(x,y,z)=x+y+z$ equals zero at zero but the value of a function at one point doesn't tell you anything about the value of its derivative. Derivatives depend on the value of a function in the neighbourhood of the point.

 

[HallsofIvy]

Hamjoop, could you please come back and explain more about what you are asking/thinking? A "zero function", to me, is exactly what it says: F(x,y,z)= 0 for all x, y, and z. And that is certainly not true for F(x,y,z)= x+ y+ z. What is your idea of a "zero function"?