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Help me understand this:by aaaa202
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#1
Feb2612, 06:09 AM

P: 1,005

My teacher often writes for a scalar function F:
dF = [itex]\nabla[/itex]F [itex]\bullet[/itex] dr Why is it you are allowed to do this. Shouldn't you use the pythagorean theorem? 


#2
Feb2612, 08:13 AM

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P: 6,187

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. 


#3
Feb2712, 04:20 AM

P: 1,005

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? 


#4
Feb2712, 09:44 AM

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P: 6,187

Help me understand this:
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. 


#5
Feb2712, 11:41 AM

P: 1,005

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. 


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