Why is dF expressed as ∇F · dr in scalar function analysis?

[aaaa202]

My teacher often writes for a scalar function F:

dF = \nablaF \bullet dr

Why is it you are allowed to do this. Shouldn't you use the pythagorean theorem?

 

[I like Serena]

My teacher often writes for a scalar function F:

dF = \nablaF \bullet dr

Why is it you are allowed to do this. Shouldn't you use the pythagorean theorem?

Hi aaaa202!

It's an identity that follows from the definition of the derivative of a function of multiple coordinates.
Consider that the total derivative is:
$$dF = {\partial F \over \partial x}dx + {\partial F \over \partial y}dy + {\partial F \over \partial z}dz$$
And the right hand side is:
$$\nabla F \cdot d\mathbf{r} = \begin{bmatrix}{\partial F \over \partial x}\\{\partial F \over \partial y}\\{\partial F \over \partial x}\end{bmatrix} \cdot \begin{bmatrix}dx\\dy\\dz\end{bmatrix} = {\partial F \over \partial x}dx + {\partial F \over \partial y}dy + {\partial F \over \partial z}dz$$

I guess you could say that he pythagorean theorem is not involved because any application of it cancels left and right in the identity.

 

[aaaa202]

hmm yes I kinda got this already, but I'm just unsure how to interpret the term:

dF = dF/dx * dx + dF/dy * dy + dF/dz *dz

What does this infinitesimal bit of represent? Surely its not a vector, since the result is a scalar. Surely it can't be an infinitesimal part of its length either, since dF =√(dFx^2 + dFy^2 + dFz^2)
So what does it mean this dF?

 

[I like Serena]

hmm yes I kinda got this already, but I'm just unsure how to interpret the term:

dF = dF/dx * dx + dF/dy * dy + dF/dz *dz

What does this infinitesimal bit of represent? Surely its not a vector, since the result is a scalar. Surely it can't be an infinitesimal part of its length either, since dF =√(dFx^2 + dFy^2 + dFz^2)
So what does it mean this dF?

You can interpret infinitesimals as small delta's.

dF is the change in F(x,y,z) if you change the coordinates by (dx,dy,dz) to F(x+dx,y+dy,z+dz).
You can write this as: dF=F(x+dx,y+dy,z+dz)-F(x,y,z).
This is the (scalar) difference in F between 2 points in space.
As such dF =√(dFx^2 + dFy^2 + dFz^2) does not apply, since it is not a vector.

 

[aaaa202]

Okay yes, but I just don't find the idea of adding up the changes of the function in respectfully the x-, y- and x- direction very interesting.
So far I've seen it used, but rather length has been used (or maybe I'm wrong) - for instance in continuity considerations you require that f(x,y) is well defined inside the cirfumference of an infinitesimal circle.