Understanding the Concept of Gradient for a Zero Function

In summary, the conversation is about the function F(x,y,z) = x + y + z and its derivative grad(F). The individual is confused about the results they are getting and clarifies that F(x,y,z) = x + y + z is not a constant function and therefore does not have a critical point at (0,0,0). They also mention that the value of a function at one point does not determine the value of its derivative, as derivatives depend on the function's values in the neighborhood of the point. The expert explains that a "zero function" is a function where F(x,y,z) = 0 for all x, y, and z, which is not the case for F(x,y,z)
  • #1
HAMJOOP
32
0
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
 
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  • #2
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
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?
 
  • #4
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
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"?
 

FAQ: Understanding the Concept of Gradient for a Zero Function

What is a gradient of a zero function?

The gradient of a zero function is equal to zero at every point. This means that the function has no slope or direction of change at any point, resulting in a flat or horizontal line.

How is the gradient of a zero function calculated?

The gradient of a zero function is calculated by taking the derivative of the function with respect to its independent variable. Since the derivative of a constant value is always zero, the gradient of a zero function will also be zero.

What does the gradient of a zero function tell us about the function?

The gradient of a zero function tells us that the function has no change in value as the independent variable changes. This means that the function has a constant value, resulting in a flat or horizontal line.

Can a zero function have a non-zero gradient?

No, a zero function by definition has a gradient of zero at every point. This means that the function has no slope or direction of change, and therefore cannot have a non-zero gradient.

What is the significance of the gradient of a zero function in real-life applications?

The gradient of a zero function often represents a state of equilibrium or balance in real-life applications. For example, in physics, the gradient of a zero function can represent a state of zero acceleration, resulting in a constant velocity. In economics, it can represent a point of no change in supply and demand, resulting in a market equilibrium.

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