10 Jul '16 14:28>5 edits
In light of recent topics of infinitesimals and such I was hoping to get some clarification on modeling using differentials.
For example: lets look at the derivation of velocity in polar coordinates for curvilinear motion ( r,Theta coordinates).
Position: r = r* u_r
Velocity: d/dt ( r ) = d/dt (r)* u_r + r*d/dt ( u_r )
The model has to determine the rate of change in direction:
d/dt ( u_r )
They do this by saying a small change in angle Theta results in a change u_r in the direction of u_Theta
Then define a new vector u_r' that follows the relationship:
u_r' = u_r + δu_r
Then using the small angle approximation
δu_r = δTheta*u_Theta
What i'm trying to wrap my head around:
How does u_r' have the same magnitude as u_r = 1 if δu_r is some small quantity orthogonal ( the direction of u_Theta ) to u_r ?
Philosophically is this correct because of the nature of infinitesimals...? because it doesn't seem to hold absolutely true in mathematics?
For example: lets look at the derivation of velocity in polar coordinates for curvilinear motion ( r,Theta coordinates).
Position: r = r* u_r
Velocity: d/dt ( r ) = d/dt (r)* u_r + r*d/dt ( u_r )
The model has to determine the rate of change in direction:
d/dt ( u_r )
They do this by saying a small change in angle Theta results in a change u_r in the direction of u_Theta
Then define a new vector u_r' that follows the relationship:
u_r' = u_r + δu_r
Then using the small angle approximation
δu_r = δTheta*u_Theta
What i'm trying to wrap my head around:
How does u_r' have the same magnitude as u_r = 1 if δu_r is some small quantity orthogonal ( the direction of u_Theta ) to u_r ?
Philosophically is this correct because of the nature of infinitesimals...? because it doesn't seem to hold absolutely true in mathematics?