How can gradient be zero if its a normal vector?

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SUMMARY

The discussion centers on the concept of the gradient as a normal vector to equipotential surfaces defined by the equation \(\phi(x,y,z) = C\). It is established that if the gradient \(\nabla \phi\) equals zero, it implies the absence of a normal vector, leading to an indeterminate direction. The participants explore the implications of constant potential fields, such as \(\phi(x,y,z) = 2\), questioning the existence of unique equipotential surfaces in such scenarios. The conversation highlights the necessity of understanding the conditions under which the gradient can be zero without contradicting its role as a normal vector.

PREREQUISITES
  • Understanding of vector calculus, specifically gradients.
  • Familiarity with the concept of equipotential surfaces.
  • Knowledge of potential fields and their mathematical representations.
  • Basic comprehension of normal vectors in three-dimensional space.
NEXT STEPS
  • Study the mathematical properties of gradients in vector calculus.
  • Explore the implications of constant potential fields on equipotential surfaces.
  • Learn about the geometric interpretation of normal vectors in three-dimensional surfaces.
  • Investigate examples of surfaces where the gradient is zero and their physical significance.
USEFUL FOR

Students and professionals in physics, mathematics, and engineering who are studying vector calculus, particularly those interested in the properties of gradients and equipotential surfaces.

aaryan0077
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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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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?
 

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