Register to reply

Help me understand this:

by aaaa202
Tags: None
Share this thread:
aaaa202
#1
Feb26-12, 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?
Phys.Org News Partner Science news on Phys.org
Physical constant is constant even in strong gravitational fields
Montreal VR headset team turns to crowdfunding for Totem
Researchers study vital 'on/off switches' that control when bacteria turn deadly
I like Serena
#2
Feb26-12, 08:13 AM
HW Helper
I like Serena's Avatar
P: 6,189
Quote Quote by aaaa202 View Post
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?
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
#3
Feb27-12, 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?

I like Serena
#4
Feb27-12, 09:44 AM
HW Helper
I like Serena's Avatar
P: 6,189
Help me understand this:

Quote Quote by aaaa202 View Post
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
#5
Feb27-12, 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.


Register to reply

Related Discussions
I don't understand war. Social Sciences 45
How to understand something you don't know/cant get Academic Guidance 5
Help me to understand. Introductory Physics Homework 1
Don't understand Introductory Physics Homework 2
Please help me understand this General Math 16