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Section 3 Key Concepts

  • The line integral \(\int_{C} \vec{F} \cdot d\vec{r}\) of a vector field \(\vec{F}\) over a curve \(C\) is defined as follows:

    • Break up the curve \(C\) into small pieces.
    • For the initial point \(t_i\) of each small piece of curve, take the derivative \(\vec{r}\,'(t_i)\text{.}\) This gives the tangent vector to \(C\) at the point \(\vec{r}(t_i)\) in the direction of the orientation of \(C\text{.}\)
    • The dot product \(\vec{F}(\vec{r}(t_i)) \cdot\vec{r}\,'(t_i)\) is the dot product of the vector in the vector field \(\vec{F}\) at the point \(\vec{r}(t_i)\) together with the tangent vector \(\vec{r}\,'(t_i)\) at \(\vec{r}(t_i)\text{.}\) It is positive if they point in the same direction, negative if in the opposite direction, and zero if perpendicular.
    • We can construct the Riemann sum of the dot products \(\vec{F}(\vec{r}(t_i)) \cdot\vec{r}\,'(t_i)\) over all the pieces of the curve.
    • The integral \(\int_{C} \vec{F} \cdot d\vec{r}\) is defined to be the limit of this Riemann sum as the lengths of the pieces of curve go to zero.
  • By the way it is defined, the line integral \(\int_{C} \vec{F} \cdot d\vec{r}\) is a scalar measurement of the flow of the vector field \(\vec{F}\) along the curve \(C\text{:}\) that is, the integral is positive if \(\vec{F}\) generally flows in the direction as \(C\text{,}\) negative if \(\vec{F}\) generally flows in the opposite direction to \(C\text{,}\) and zero if perpendicular or there is the same flow with as against.
  • If \(C\) is parameterized by \(\vec{r}(t)\) for \(a\leq t\leq b\text{,}\) then to calculate the integral \(\int_{C} \vec{F} \cdot d\vec{r}\) we use the formula
    \begin{equation*} \int_{C} \vec{F} \cdot d\vec{r} =\int_{a}^{b} \vec{F}(\vec{r}(t)) \cdot \vec{r}\,'(t) ~dt. \end{equation*}
    Note that if \(C\) is not already parameterized, this means we need to find a parameterization for \(C\text{,}\) including the bounds.
  • Since \(\int_{C} \vec{F} \cdot d\vec{r}\) measures flow along \(C\text{,}\) we can often estimate its value from a graph of the vector field \(\vec{F}\text{.}\)
  • If \(C\) is a closed curve, we often call the integral \(\int_{C} \vec{F} \cdot d\vec{r}\) the circulation of \(\vec{F}\) around \(C\text{,}\) and to emphasize that \(C\) is closed, we use the alternate integral notation \(\oint_{C} \vec{F} \cdot d\vec{r}\text{.}\)
  • Since geometrically a line integral of a vector field calculates flow along the curve, it does not depend on the chosen parameterization for the curve. However, it does depend on both orientation, since opposite direction will give the opposite sign, and on the curve itself: that is, if \(C_1\) and \(C_2\) are two different curves between the same two points, we expect the integrals over these curves to be different.
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