# Gradient of a zero function

1. Jan 15, 2013

### HAMJOOP

Suppose F(x,y,z) = 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

2. Jan 16, 2013

### SteamKing

Staff Emeritus
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.

3. Jan 16, 2013

### lurflurf

x+y+z=/=0

There is no contradiction, why would you expect one?

4. Jan 16, 2013

### 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.

5. Jan 16, 2013

### 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"?