Limits in case of 'vector functions'

[Ashu2912]

Hi! I am an amateur to the world of calculus...
I have a doubt with the limits in case of functions which are vectors...
Actually, I require it's application in physics...
Suppose we have a vector r (all vectors in bold face), which represents the general position vector of a point on any arbitrary curve. Then is dr defined just as in normal functions, as lim $$\Delta$$r
$$\Delta$$r$$\rightarrow$$0 ?
If this is the case will dr have any direction in space, depending upon the direction of $$\Delta$$r, or will it be along r?

 

[HallsofIvy]

For r(t), with t a parameter, dr(t)/dt will be a tangent vector to the curve. Since t is a real variable, dr will have direction tangent to the curve.

If the curve happened to be a circle, for example, dr will be perpendicular to r.

 

[Ashu2912]

Actually, the problem I'm facing is as follows:
In a 3D space, there is an electric field E, which is a vector field and is a function of the position vector r. We have to calculate the work done i.e. E.Displacement in going from A (position vector a) to B (position vector b). In the book, they have divided the line joining A and B into elements with length dl. However, shouldn't dr
which is $$\Delta$$r as $$\Delta$$ -> 0, as by vector addition dl is actually dr, the difference between r and r + dr?

 

[HallsofIvy]

No. In one case, dl is tangent to the given curve. In the other, because r is the position vector r is a "position vector" (the vector from the origin to the given point), dr points from the given point directly away from the origin.

 

[Ashu2912]

You mean dr should point towards the point from the origin, along r ? Also, if dy/dx can't be expressed as
lim $$\Delta$$x -> 0; $$\Delta$$ y / lim $$\Delta$$x -> 0; $$\Delta$$x,
as lim $$\Delta$$x -> 0; $$\Delta$$ x = 0, what do we infer if we have a 'dl' element?