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A line integral is a type of integral where the function to be integrated is evaluated along a curve. In the context of vector fields, a line integral can be thought of as finding the total effect of the field along a certain path or curve. This can represent various physical concepts, such as the work done by a force field along a trajectory. The line integral of a vector field \( \mathbf{F} \) along a curve \( C \) is denoted as \( \int_{C} \mathbf{F} \cdot d \mathbf{r} \). Here, \( d \mathbf{r} \) represents a differential element of the curve.Line integrals are particularly significant in fields like physics and engineering because they allow us to sum up a vector field's influence over a certain path. To compute it, one needs to parameterize the curve and understand its direction. The orientation of the curve is crucial since line integrals are sensitive to direction; reversing the direction may change its sign. In our problem, the parameterization of curve \( C \) is given by \( \mathbf{r}(t) = \sin \left( \frac{\pi t}{2} \right) \mathbf{i} + \ln t \mathbf{j} \), providing a clear path from point \( t=1 \) to \( t=2 \).

A potential function is usually associated with a conservative vector field. If a vector field \( \mathbf{F} \) is conservative, then there exists a scalar potential function \( f(x, y) \) such that \( \mathbf{F} = abla f \). This means that the vector field can be represented as the gradient of a scalar field. The existence of a potential function implies that the line integral of the vector field between two points is path-independent.To find a potential function for a given vector field, we typically need to perform integration. For the vector field \( \mathbf{F}(x, y) = (e^y + ye^x)\mathbf{i} + (xe^y + e^x)\mathbf{j} \), the potential function \( f(x, y) \) is found by integrating the components of \( \mathbf{F} \). Specifically, integrating \( e^y + ye^x \) with respect to \( x \) yields \( ye^x + g(y) \). Ensuring that this potential also satisfies the \( j \)-component, we find \( g(y) = e^y \), leading to the potential function \( f(x, y) = ye^x + e^y \). This function reflects how the vector field accumulates values through space and provides insight into the behavior across different points of the field.

Path independence is a defining characteristic of conservative vector fields. It means that the value of the line integral \( \int_{C} \mathbf{F} \cdot d \mathbf{r} \) depends only on the starting and ending points of the path, not on the specific path taken between them. This property is a direct consequence of the existence of a potential function.In our example, the path independence allows us to compute the line integral by simply evaluating the potential function at the endpoints of the given curve. The points on the path, \( \mathbf{r}(1) = \mathbf{i} \) and \( \mathbf{r}(2) = \ln(2)\mathbf{j} \), are used to find the values of the potential function, which we then subtract to find the line integral's value. This simplification stems from the fact that in conservative fields, different paths resulting in the same endpoints yield the same integral value, thus confirming the utility of potential functions to bypass the integration of each trajectory. In conclusion, path independence not only showcases a neat mathematical property but also makes solving such problems much more efficient.