91Ó°ÊÓ

A potential function, often denoted by \( f \), is a scalar function whose gradient yields a vector field \( \mathbf{F} \). Think of it as a mathematical way to "summarize" a vector field using a single function. The mathematical relationship is given by the equation \( abla f = \mathbf{F} \). When a potential function exists for a vector field, that vector field is conservative.

In the step-by-step solution provided for the exercise, finding the potential function involves calculating \( f(x, y, z) \). This is done by integrating components of the vector field \( \mathbf{F} \) with respect to their corresponding variables while determining the unknown parts through subsequent differentiation and comparing it with the original vector components. By following these integration steps carefully, we obtain the potential function \( f(x, y, z) = \arctan(xy) + \arcsin(yz) + \ln|z| + C \), where \( C \) is a constant of integration.

A conservative field is a special type of vector field where the line integral of the vector field between any two points is path-independent. This means, regardless of the path taken between two points, the integral will yield the same result. A key characteristic of a conservative field is that it can be expressed as the gradient of a scalar potential function.

In mathematical terms, if a vector field \( \mathbf{F} \) is conservative, there exists a function \( f \) such that \( \mathbf{F} = abla f \). Additionally, the curl of the vector field is zero, which is the condition \( abla \times \mathbf{F} = \mathbf{0} \). This property is crucial for verifying whether a vector field is conservative, as observed in the original exercise where checks were made through components of the curl. Once these conditions are verified, one can proceed to determine the potential function. In this exercise, after confirming \( abla \times \mathbf{F} = \mathbf{0} \), the field \( \mathbf{F} \) was shown to be conservative, enabling us to find the potential function \( f \).

The curl of a vector field is an expression that measures the rotation of the field around a point. It helps to understand how a field "swirls" around a particular location. For a vector field \( \mathbf{F} = M\mathbf{i} + N\mathbf{j} + P\mathbf{k} \), the curl is given by the cross-product \( abla \times \mathbf{F} \).

The components of the curl are determined as:

• \( \frac{\partial P}{\partial y} - \frac{\partial N}{\partial z} \) for the \( \mathbf{i} \) component,
• \( \frac{\partial M}{\partial z} - \frac{\partial P}{\partial x} \) for the \( \mathbf{j} \) component,
• \( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \) for the \( \mathbf{k} \) component.

To determine if a field is conservative, the curl must be evaluated, and all of these components must equal zero. In the exercise, evaluating \( abla \times \mathbf{F} = \mathbf{0} \) confirmed that the vector field was conservative. Thus, understanding the curl is a pivotal step in recognizing the nature of a vector field and ensuring the possibility of finding a potential function.