Voltage difference, line integral

[kougou]

Homework Statement

I don't understand the follow formula of the integral :

Integral of ( E dot dL) from B to A
What direction is the dr vector? Is it the direction of the line integral?

Say I want to derive the formula for electric potential due to a point in Space. E has a direction vector of ar
assume dL is pointing from infinity to A (-ar), which is opposite of the Eletric Field. And the formula ends up with a negative sign, which is not consistent with the formal formula.
Why?

 

[Simon Bridge]

The integral is you described, "Integral of ( E dot dL) from B to A", is $$\int_B^A \vec{E}\cdot\vec{dL}$$... that what you meant?

What direction is the dr vector? Is it the direction of the line integral?

dr points wherever you define it to point.

The integral is along a line, which, in general, will curve in most coordinate systems.
But does the potential difference in an electric field depend on the path taken? What does the rest of the theory say?

assume dL is pointing from infinity to A (-ar), which is opposite of the Electric Field. And the formula ends up with a negative sign, which is not consistent with the formal formula.

I think you may have misplaced a minus sign.