Proof on why del is normal to surface?

[unscientific]

Homework Statement

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

Homework Equations

The Attempt at a Solution

 

[Dick]

Homework Statement

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

Homework Equations

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?

 

[unscientific]

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!

 

[Dick]

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?

 

[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.

 

[unscientific]

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?

 

[Dick]

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.

 

[unscientific]

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?

 

[Dick]

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.