# Proof on why del is normal to surface?

1. Feb 1, 2013

### unscientific

1. The problem statement, all variables and given/known data

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

2. Relevant equations

3. The attempt at a solution

Last edited: Feb 1, 2013
2. Feb 1, 2013

### Dick

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

3. Feb 1, 2013

### unscientific

Sorry, I attached the wrong picture!

4. Feb 1, 2013

### Dick

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?

5. Feb 1, 2013

### HallsofIvy

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

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

6. Feb 1, 2013

### unscientific

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

does this proof make sense?

7. Feb 1, 2013

### Dick

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.

Last edited: Feb 2, 2013
8. Feb 2, 2013

### unscientific

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

9. Feb 2, 2013

### Dick

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.