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Section 3 Key Concepts

  • The Fundamental Theorem of Calculus for Line Integrals: Let \(C\) be a smooth curve parameterized by \(\vec{r} (t)\) with \(a\leq t\leq b\) and suppose \(f\) is a differentiable function of two or three variables whose gradient vector is continuous on \(C\text{.}\) Then

    \begin{equation*} \int_{C} \nabla f \cdot d\vec{r} =f(\vec{r} (b)) -f(\vec{r} (a)). \end{equation*}

    In particular, if you can find a potential function of a vector field, to evaluate a line integral of the vector field you just evaluate the potential function at the endpoints and take the difference.

  • A region \(U\) is simply connected if every simple closed curve completely contained in \(C\) can be shrunk to a point in \(U\) without breaking and without leaving \(U\text{.}\) For a region in \(2\) space, this is equivalent to containing no holes.

  • The Curl Test: Let \(\vec{F}\) be a \(C^1\) vector field on a simply connected set \(U\text{.}\) Then \(\vec{F}\) is a conservative vector field if and only if \({\rm curl}\, \vec{F} =\vec{0}\) (or if the scalar curl of \(\vec{F}\) is \(0\) if \(\vec{F}\) is in \(2\) space). Note that \(U\) being simply connected is a necessary condition.

  • A vector field \(\vec{F}\) is called path-independent if \(\int_{C_{1}} \vec{F} \cdot d\vec{r} =\int_{C_{2}} \vec{F} \cdot d\vec{r}\) for any two paths \(C_{1}\) and \(C_{2}\) between any two points. It is path-dependent if there exist curves \(C_1\) and \(C_2\) between the same two points with \(\int_{C_{1}} \vec{F} \cdot d\vec{r} \neq \int_{C_{2}} \vec{F} \cdot d\vec{r}\text{.}\)

  • It is often easier to visualize line integrals over closed curves, so an equivalent definition for path independence is: a vector field \(\vec{F}\) is path-independent if and only if \(\int_{C}\vec{F} \cdot d\vec{r} =0\) for any closed curve \(C\text{.}\)

  • Any gradient vector field is path-independent, and conversely a path-independent field is always a gradient vector field. This gives a geometric way to decide whether or not a vector field is a gradient vector field: test for path dependence or independence.

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