Answer
See proof
Work Step by Step
Step 1: Since we are integrating a vector field over a smooth curve, we are dealing with a line integral. Let's say that we have a vector field \(\vec{F}\), and we want to determine its line integral over a smooth curve \(\vec{C}\): \[ \int_{\vec{C}} \vec{F} \cdot d\vec{r} \] We want to discuss the possible ways to determine if the vector field \(\vec{F}\) is a conservative vector field. Is there a way to tell that the line integral of a conservative vector field is \(0\) without evaluating the integral explicitly?
Step 2: The first method of integrating a conservative vector field along a smooth curve that we are going to discuss is simply treating the vector field \(\vec{F}\) as if it weren't conservative. Meaning that we will do a basic line integral of a vector field by parameterizing the curve \(\vec{C}\) over which we want to integrate. Let \(\vec{C} = \vec{r}(t)\), where \(t\) is the parameter in some range \(t_0 \leq t \leq t_1\). We also want to describe the vector field \(\vec{F}\) in terms of its value at the parameter \(t\): \(\vec{F}(t)\). We can now turn the line integral over \(\vec{C}\) in (1) into a definite integral with respect to \(t\) by applying \(d\vec{r}(t) = \vec{r}'(t) dt\): \[ \int_{\vec{C}} \vec{F} \cdot d\vec{r} = \int_{t_0}^{t_1} \vec{F}(t) \cdot \vec{r}'(t) dt \] Now all that is left in order to solve (1) is to find \(\vec{F}(t)\) and \(\vec{r}'(t)\) for the specific problem given.
Step 3: The second method that we will show takes into account that the vector field \(\vec{F}\) is conservative. Conservative vector fields can be expressed in terms of their potential function \(\phi\): \[ \vec{F} = \nabla \phi \] From (2), follows the conclusion, called The Fundamental Theorem of Line Integrals, that can help us evaluate (1) between the points \(\vec{P}\) and \(\vec{Q}\) that are connected by a smooth curve \(\vec{C}\): \[ \int_{\vec{C}} \vec{F} \cdot d\vec{r} = \phi(\vec{Q}) - \phi(\vec{P}) \] Here, we have turned the problem of determining the line integral into a problem of finding the potential function \(\phi\) of the vector field \(\vec{F}\) at the starting \(\vec{P}\) and ending \(\vec{Q}\) points of the curve \(\vec{C}\), which can sometimes be easier than doing a proper line integral.
Step 4: Finally, we will note that we can evaluate the line integral (1) with ease if the path of integration \(\vec{C}\) is a closed smooth curve. A closed smooth curve is a smooth curve that starts and ends at the same point. If a conservative vector field \(\vec{F}\) is defined with its derivatives over some region where the closed smooth curve \(\vec{C}\) lies, then the line integral in (1) is equal to \(0\). This result can be simply obtained by looking at (3) and applying that the path of integration has the same starting and ending point: \(\vec{C} = \vec{P} = \vec{Q}\).
Step 5: Let's sum up the methods above: 1. We can do a proper line integral by parameterizing the curve and the given vector field \(\vec{F}\). 2. We can use the fact that the vector field \(\vec{F}\) is conservative, and determine the line integral by finding its potential function \(\phi\). 3. We can use the fact that the line integral of a conservative vector field is \(0\) if the path of integration is a closed smooth curve that lies in the same region where \(\vec{F}\) and its derivatives are defined.
