91Ó°ÊÓ

Understanding whether a vector field is conservative can be determined by examining its curl. The curl of a vector field, denoted as \( abla \times \mathbf{F} \), measures the tendency of the field to rotate around points. For a vector field to be conservative, this curl must be zero in all regions where the field is defined.
To calculate the curl in two dimensions, we use the formula: \( abla \times \mathbf{F} = \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \). Here, \( \mathbf{F} = P \mathbf{i} + Q \mathbf{j} \) is the given vector field. If the field is conservative, it essentially means that there is no rotational movement within the field, allowing it to be expressed as the gradient of some scalar potential function.
In practice, calculating the curl is a matter of finding these partial derivatives and seeing if they cancel each other out. This is performed in steps to ensure accuracy, as any discrepancies can point to a mislabeled field or an incorrect calculation. By concluding that the curl is zero, such as in the provided solution, we confirm that the vector field, indeed, is conservative.

When a vector field is conservative, like \( \mathbf{F} \) in our exercise, it means there exists a potential function \( f(x, y) \) such that \( abla f = \mathbf{F} \). This potential function essentially serves as an alternative description of the vector field, where the direction and magnitude of the field can be derived from the potential itself.
To find the potential function, start by integrating the component \( P \) of the vector field with respect to \( x \). This gives a partial potential function in terms of \( x \) and an unknown function \( g(y) \), which accounts for any terms solely dependent on \( y \).
Next, differentiate this integration result with respect to \( y \) and set it equal to the \( Q \) component. This comparison helps in finding \( g(y) \). By solving for \( g(y) \), the complete expression for \( f(x, y) \) is obtained.
In the exercise, once the potential function is created, it simplifies the calculation of line integrals across given paths by turning a path integral into an evaluation of the potential at boundary points.

Once a potential function is known for a conservative vector field, computing a line integral becomes much easier. A line integral in the context of a vector field is typically written as \( \int_{C} \mathbf{F} \cdot d \mathbf{r} \). It represents the accumulated effect of the vector field along a curve \( C \).
However, for conservative fields, this calculation can be reduced significantly. Instead of having to carry out complex integrals over curves, we only need to evaluate the potential function at the endpoints of the curve.

• First, the curve \( C \) is parameterized. In the exercise, this is expressed as \( x = t^3 - 1, \ y = t^6 - t \) with bounds on \( t \) from 0 to 1.
• The endpoints are determined by substituting these bounds into the parameterization equations to yield starting point \((x_0, y_0)\) and ending point \((x_1, y_1)\).
• Finally, the value of the line integral is computed as \( f(x_1, y_1) - f(x_0, y_0) \), which utilizes the found potential function \( f(x, y) \).

This approach is much more efficient and particularly in this problem, highlights a key advantage of identifying a vector field as conservative.