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2025-07-27 15:26:00 -04:00
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quarto/integral_vector_calculus/line_integrals.qmd
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# Line and Surface Integrals
# Line and surface integrals
{{< include ../_common_code.qmd >}}
@@ -340,7 +340,7 @@ W = integrate(F(r(t)) ⋅ T(r(t)), (t, 0, 2PI))
There are technical assumptions about curves and regions that are necessary for some statements to be made:
* Let $C$ be a [Jordan](https://en.wikipedia.org/wiki/Jordan_curve_theorem) curve - a non-self-intersecting continuous loop in the plane. Such a curve divides the plane into two regions, one bounded and one unbounded. The normal to a Jordan curve is assumed to be in the direction of the unbounded part.
* Let $C$ be a [Jordan](https://en.wikipedia.org/wiki/Jordan_curve_theorem) curve---a non-self-intersecting continuous loop in the plane. Such a curve divides the plane into two regions, one bounded and one unbounded. The normal to a Jordan curve is assumed to be in the direction of the unbounded part.
* Further, we will assume that our curves are *piecewise smooth*. That is comprised of finitely many smooth pieces, continuously connected.
* The region enclosed by a closed curve has an *interior*, $D$, which we assume is an *open* set (one for which every point in $D$ has some "ball" about it entirely within $D$ as well.)
* The region $D$ is *connected* meaning between any two points there is a continuous path in $D$ between the two points.
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:::{.callout-note}
## Note
For a Jordan curve, the positive orientation of the curve is such that the normal direction (proportional to $\hat{T}'$) points away from the bounded interior. For a non-closed path, the choice of parameterization will determine the normal and the integral for flow across a curve is dependent - up to its sign - on this choice.
For a Jordan curve, the positive orientation of the curve is such that the normal direction (proportional to $\hat{T}'$) points away from the bounded interior. For a non-closed path, the choice of parameterization will determine the normal and the integral for flow across a curve is dependent---up to its sign---on this choice.
:::