Confused about the different types of line integrals

[asdf1]

what's the difference between
$$\int_{a}^{b} f(r(t)) dt$$
and
$$\int_{a}^{b} F(r(t)) (dr/dt)dt$$?
(where F is a vector function)

because when I'm calculating those two types of questions, the first question just uses dt to integrate the line integral but in the 2nd question, i have to differentiate dr?

 

[d_leet]

what's the difference between
$$\int_{a}^{b} f(r(t)) dt$$
and
$$\int_{a}^{b} F(r(t)) (dr/dt)dt$$?
(where F is a vector function)

because when I'm calculating those two types of questions, the first question just uses dt to integrate the line integral but in the 2nd question, i have to differentiate dr?

What is r though? Doesn't r have a fairly standard definition ( if we're talking about the vector) as r = (x, y, z) = (x(t), y(t), z(t)) the dr is (dx, dy, dz) Maybe I;ve got it wrong but its late so I apologize if this isn't correct, but they at least SEEM to be more or less the same thing.

 

[Benny]

From what I know, I would say that the second integral is supposed to be evaluated with respect to t. A more conventional way of writing the second integral is:

$$\int\limits_a^c {F\left( {r\left( t \right)} \right)} \bullet r'\left( t \right)dt$$

Assuming that F and r are vector functions of t. That representation should clear up some confusion. Also, the (dr/dt) or r'(t) is essentially just some vector. So when it is 'dotted' with F(r(t)), which is another vector, your integrand is just a scalar function.

 

[asdf1]

i think that the 1st "f" is only a scalar function...