Confused about the different types of line integrals

  • Thread starter asdf1
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  • #1
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what's the difference between
[tex]\int_{a}^{b} f(r(t)) dt[/tex]
and
[tex]\int_{a}^{b} F(r(t)) (dr/dt)dt[/tex]?
(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???
 

Answers and Replies

  • #2
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asdf1 said:
what's the difference between
[tex]\int_{a}^{b} f(r(t)) dt[/tex]
and
[tex]\int_{a}^{b} F(r(t)) (dr/dt)dt[/tex]?
(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.
 
  • #3
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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:

[tex]
\int\limits_a^c {F\left( {r\left( t \right)} \right)} \bullet r'\left( t \right)dt
[/tex]

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.
 
  • #4
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i think that the 1st "f" is only a scalar function...
 

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