How can gradient be zero if its a normal vector?

In summary, the conversation discusses the physical interpretation of the gradient, which is a vector that is normal to equipotential surfaces. It is noted that this definition may not apply to all surfaces, as some may not be equipotential. The conversation then explores the possibility of a surface having a normal vector of 0, which may contradict the assumption that the gradient is a normal vector. It is also mentioned that equipotential surfaces are defined by a constant value of the potential function, but there may be cases where this does not result in a unique surface.
  • #1
aaryan0077
69
0
Physical interpretation of gradient says that its a vector normal to equipotential (or level) surface [tex]\phi(x,y,z) = 0[/tex]
but what about other surfaces, say the surface which are not equipotential?
This is my first question.

ok, now
as [tex]grad \phi[/tex] is a vector normal to surface it can't be 0. Because that would mean that surface have no normal vector, or say a normal vector of indeterminate direction (as 0 vector is of indeterminate direction). how can it be possible that a surface has no normal vector, more specifically a 0 vector as its normal vector?
But I have seen many examples in which [tex]grad \phi[/tex] is 0.
So doesn't that contradicts the assumption that [tex]grad \phi[/tex] is a normal vector?
 
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  • #2
Equipotential surfaces are defined by the eqution:
[tex]\phi(x,y,z) = C[/tex] (C is constant)
But consider the contant potential field ,say [tex]\phi(x,y,z) = 2[/tex],can you find a unique equipotential surface for it?
 

1. How can the gradient be zero if it's a normal vector?

The gradient can be zero if it's a normal vector because the gradient is a vector that represents the direction of maximum change in a function. When the gradient is zero, it means that there is no change in the function in any direction, which can occur at a local maximum or minimum point of the function.

2. Does a zero gradient always indicate a maximum or minimum point?

No, a zero gradient does not always indicate a maximum or minimum point. It can also occur at a saddle point, where the function has both increasing and decreasing directions. Additionally, the gradient can be zero at a flat point or a point of inflection.

3. Can the gradient be zero at multiple points in a function?

Yes, the gradient can be zero at multiple points in a function. This can occur when the function has multiple maximum or minimum points, saddle points, or flat points.

4. How is the gradient related to the normal vector?

The gradient and the normal vector are related because both are perpendicular to the level curves or surfaces of a function. The gradient is perpendicular to the level curves in two dimensions, while the normal vector is perpendicular to the level surfaces in three dimensions.

5. Is the gradient always perpendicular to the level curves or surfaces?

Yes, the gradient is always perpendicular to the level curves or surfaces. This is because the gradient represents the direction of maximum change in the function, and the level curves or surfaces represent the points where the function has a constant value.

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