Conservative Vector Fields: Learn It 1

  • Identify simple and closed curves in various regions
  • Find potential functions for conservative fields
  • Apply the Fundamental Theorem for Line Integrals
  • Test a vector field to determine whether it is conservative

Curves and Regions

Before we can work with the theorems in this section, we need to establish some geometric vocabulary about curves and regions. The definitions here will set us up for generalizing the Fundamental Theorem of Calculus to line integrals.

Let’s start with two important types of curves. A closed curve begins and ends at the same point. A simple curve doesn’t cross itself. When a curve is both closed and simple, we call it a simple closed curve (Figure 1).

closed and simple curves

A curve [latex]C[/latex] is closed if it has a parameterization [latex]\mathbf{r}(t)[/latex], [latex]a\leq t\leq b[/latex], where [latex]\mathbf{r}(a)=\mathbf{r}(b)[/latex] and the parameterization traverses the curve exactly once.

[latex]\\[/latex]

A curve [latex]C[/latex] is simple if it doesn’t cross itself. More precisely, [latex]C[/latex] is simple if it has a parameterization [latex]\mathbf{r}(t)[/latex], [latex]a\leq t\leq b[/latex], where [latex]\mathbf{r}[/latex] is one-to-one over [latex](a, b)[/latex]. Note that we can have [latex]\mathbf{r}(a)=\mathbf{r}(b)[/latex], which would make the simple curve also closed.

An image showing eight curves and their types. The first curve is neither simple nor closed; it has two endpoints and crosses itself twice. The second curve is simple but not closed; it does not cross itself and has two endpoints. The third curve is closed but is not simple; it crosses itself a few times. The fourth is a simple closed curve; it does not cross itself and has no endpoints. The fifth is a simple, not closed curve; it does not cross itself, but it has endpoints. The sixth is a simple, closed curve; it does not cross itself and has no endpoints. The seventh is closed but not a simple curve; it crosses itself but has no endpoints. The last is not simple and not closed; it crosses itself and has endpoints.
Figure 1. Types of curves that are simple or not simple and closed or not closed.
Is the curve with parameterization [latex]{\bf{r}}(t)=\left\langle\cos{t},\frac{\sin{(2t)}}2\right\rangle, \ 0\leq{t}\leq2\pi[/latex] a simple closed curve?

Now we need to define two types of regions. Many theorems in this module relate an integral over a region to an integral over its boundary. For these theorems to work, we need the region to be either connected or simply connected.

A connected region is one where you can draw a path between any two points without leaving the region. A simply connected region goes further—it’s a connected region with no holes in it.

connected and simply connected regions

A region [latex]D[/latex] is connected if, for any two points [latex]P_1[/latex] and [latex]P_2[/latex] in [latex]D[/latex], there exists a path from [latex]P_1[/latex] to [latex]P_2[/latex] that stays entirely inside [latex]D[/latex].

[latex]\\[/latex]

A region [latex]D[/latex] is simply connected if [latex]D[/latex] is connected and any simple closed curve [latex]C[/latex] inside [latex]D[/latex] can be continuously shrunk to a point while staying entirely inside [latex]D[/latex]. In two dimensions, this means the region is connected and has no holes.

All simply connected regions are connected, but not all connected regions are simply connected (Figure 3).

A diagram showing simply connected, connected, and not connected regions. The simply connected regions have no holes. The connected regions may have holes, but a path can still be found between any two points in the region. The not connected region has some points that cannot be connected by a path in the region. Here, this is illustrated by showing two circular shapes that are defined as part of region D1 but are separated by white space.
Figure 3. Not all connected regions are simply connected. (a) Simply connected regions have no holes. (b) Connected regions that are not simply connected may have holes but you can still find a path in the region between any two points. (c) A region that is not connected has some points that cannot be connected by a path in the region.
Is the region in the below image connected? Is the region simply connected?<img src="/apps/archive/20220422.171947/resources/4fa7f7d2717cf20dc660c0bca880c7c7de267432" data-media-type="image/jpeg" alt="A shaded circle with an open space in the shape of a circle inside it but very close to the boundary." id="9">