# Proof on why del is normal to surface?

• unscientific
In summary: Same thing I said before, again. Something like ##\frac{ d \phi }{ds}=0## is neither right nor wrong until you say how s is related to phi. But yes, you can make it work by doing that.
unscientific

## Homework Statement

Simple proof on why ∇∅ is normal to surface of ∅(x,y,z) = constant

## The Attempt at a Solution

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unscientific said:

## Homework Statement

Simple proof on why ∇∅ is normal to surface of ∅(x,y,z) = constant

## The Attempt at a Solution

The 'attempt at a proof' is a good 'relevant equation' maybe with a little fixing. Now what's the attempt at a proof?

Dick said:
The 'attempt at a proof' is a good 'relevant equation' maybe with a little fixing. Now what's the attempt at a proof?

Sorry, I attached the wrong picture!

unscientific said:
Sorry, I attached the wrong picture!

Ok that's more like it. Now if you want to make the string of equations function as a proof you need to add some words explaining why it's a proof. What kind of a curve are you taking ##\vec s## to be? And why does the last line show grad(phi) is normal to the surface?

Let s(t) be a curve lying in the surface $\Phi(x,y,z)= const$.

Show, using the chain rule, that, along that curve, $$\frac{d\Phi}{dt}= \frac{ds}{dt}\cdot\nabla \Phi= 0$$.

Dick said:
Ok that's more like it. Now if you want to make the string of equations function as a proof you need to add some words explaining why it's a proof. What kind of a curve are you taking ##\vec s## to be? And why does the last line show grad(phi) is normal to the surface?

s is the distance along the curve, t is the unit tangent vector to the curve...

does this proof make sense?

unscientific said:
s is the distance along the curve, t is the unit tangent vector to the curve...

does this proof make sense?

Same thing I said before. A proper proof involves at least a little narrative in words as to what things are, like you just did, and how the final equation justifies the conclusion that grad(phi) is normal to the level surface of phi. Supply those and it will work fine.

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Dick said:
Same thing I said before. A proper proof involves at least a little narrative in words as to what things are, like you just did, and how the final equation justifies the conclusion that grad(phi) is normal to the level surface of phi. Supply those and it will work fine.

Thanks! I will do that in future. The math is correct, right?

unscientific said:
Thanks! I will do that in future. The math is correct, right?

Same thing I said before, again. Something like ##\frac{ d \phi }{ds}=0## is neither right nor wrong until you say how s is related to phi. But yes, you can make it work by doing that. The math is correct in that sense.

## 1. What does it mean for a del to be normal to a surface?

A del being normal to a surface means that it is perpendicular to the surface at a specific point. This is important in understanding the orientation and behavior of the surface in relation to the del.

## 2. Why is it important to prove that del is normal to a surface?

Proving that del is normal to a surface is important in many fields of science, such as physics and mathematics, as it allows for a better understanding of the behavior and properties of the surface. It also helps in solving various equations and problems related to the surface.

## 3. How is the proof of del being normal to a surface carried out?

The proof of del being normal to a surface is usually carried out using mathematical equations and principles, such as the dot product and cross product. It involves finding the direction and magnitude of the del and comparing it to the direction and orientation of the surface at a specific point.

## 4. Can del be normal to a surface at every point?

No, del can only be normal to a surface at a specific point where it intersects the surface. At other points on the surface, del may have a different orientation and therefore, will not be normal to the surface.

## 5. What are some real-life applications of understanding del being normal to a surface?

Understanding del being normal to a surface has various applications in science and engineering. It is used in the study of fluid dynamics, electromagnetism, and mechanics, among others. It also helps in practical applications such as designing structures and calculating forces acting on them.

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