- #1

- 388

- 18

f( x , y , z , ...) = C

then the normal direction to it is simply the (unit vector of the) divergence of the function. How has this been proven?

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- I
- Thread starter swampwiz
- Start date

- #1

- 388

- 18

f( x , y , z , ...) = C

then the normal direction to it is simply the (unit vector of the) divergence of the function. How has this been proven?

- #2

- #3

- 8,676

- 2,734

Take two nearby points ##A## and ##B## on the surface ##f(x,y,z,...) = C##, let ##\Delta \vec{r}## be the displacement from ##A## to ##B##. Then the change in ##f## will be approximately given by ##f(B) - f(A) = \nabla f \cdot \Delta \vec{r}##. Since ##f## is constant on the surface, ##f(B) - f(A) = 0##. So we have:

##\nabla f \cdot \Delta \vec{r} = 0##

which implies that ##\nabla f## is orthogonal to ##\Delta \vec{r}##.

- #4

lavinia

Science Advisor

Gold Member

- 3,245

- 627

Take two nearby points ##A## and ##B## on the surface ##f(x,y,z,...) = C##, let ##\Delta \vec{r}## be the displacement from ##A## to ##B##. Then the change in ##f## will be approximately given by ##f(B) - f(A) = \nabla f \cdot \Delta \vec{r}##. Since ##f## is constant on the surface, ##f(B) - f(A) = 0##. So we have:

##\nabla f \cdot \Delta \vec{r} = 0##

which implies that ##\nabla f## is orthogonal to ##\Delta \vec{r}##.

Equivalently, if c(t) is a curve on the surface ##f(x,y,z,...)=C## then ##f(c(t) = C## so ##df/dt = 0##. By the Chain rule ##0=df/dt = ∇f⋅dc/dt##. Since ##dc/dt## is tangent to the surface for all such curves ##c(t)##, ##∇f## is normal to the surface.

Last edited:

- #5

- #6

- 388

- 18

Take two nearby points ##A## and ##B## on the surface ##f(x,y,z,...) = C##, let ##\Delta \vec{r}## be the displacement from ##A## to ##B##. Then the change in ##f## will be approximately given by ##f(B) - f(A) = \nabla f \cdot \Delta \vec{r}##. Since ##f## is constant on the surface, ##f(B) - f(A) = 0##. So we have:

##\nabla f \cdot \Delta \vec{r} = 0##

which implies that ##\nabla f## is orthogonal to ##\Delta \vec{r}##.

This works for me. I presume that making this rigorous requires some tedious δ-ε analysis.

- #7

- 16,977

- 6,774

Why? All you need is in #4. It is just the chain rule, which I assume you have already done your epsilon delta construction for.This works for me. I presume that making this rigorous requires some tedious δ-ε analysis.

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