91Ó°ÊÓ

A gradient field is a type of vector field that can be represented as the gradient of a scalar function, often denoted as \( abla f \). This means that the vector field \( \mathbf{F} \) is derived from a potential function \( f(x, y) \). Each component of the vector field corresponds to the partial derivative of \( f \) with respect to a variable. In simpler terms:

• The \( i \)-component of the vector field is the partial derivative of the function with respect to \( x \).
• The \( j \)-component is the partial derivative with respect to \( y \).

For example, if given \( \mathbf{F}(x, y) = (1 + x y) e^{x y} \mathbf{i} + x^2 e^{x y} \mathbf{j} \), we understand that:

• \( \frac{\partial f}{\partial x} = (1 + x y) e^{x y} \)
• \( \frac{\partial f}{\partial y} = x^2 e^{x y} \)

By integrating these components, we can reconstruct the function \( f \) such that the gradient matches the original field. This process is crucial when we seek to use properties of conservative fields.

Path independence is a fascinating property of line integrals associated with gradient fields. When a vector field is the gradient of some function, the integral of the field along any path only depends on the initial and final points, not on the specific path taken. Thus, we say the field is path-independent.This is extremely useful in calculations involving line integrals. Instead of computing the integral over a complex path, we simply evaluate the potential function at the endpoints:

• If \( \mathbf{F} = abla f \), compute \( f \) at the start and end of the path.
• The difference \( f(x_2, y_2) - f(x_1, y_1) \) gives us the value of the integral.

For instance, in the exercise, path independence allows us to calculate \( \int_C \mathbf{F} \cdot d \mathbf{r} \) over curve \( C \) by evaluating \( f \) at the endpoints \((1, 0)\) and \((0, 2)\), simplifying our work significantly.

Exact differential forms are concepts tied closely to gradient fields and path independence. If a vector field \( \mathbf{F} \) is the gradient of a potential function \( f \), then \( \mathbf{F} \cdot d\mathbf{r} \) is an exact differential form. This means there exists a potential function whose differential matches the vector field:

• An exact form satisfies \( abla f = \mathbf{F} \).
• The condition \( \frac{\partial^2 f}{\partial x \partial y} = \frac{\partial^2 f}{\partial y \partial x} \) holds true, ensuring the function is well-defined.

Consequently, if \( F \) is exact, it implies path independence and confirms that the line integral is determined solely by the values of the potential function at the endpoints of a path. Recognizing exact forms is key in confirming that the vector field is conservative, as demonstrated in the solved problem where verifying \( f(x, y) \) yields \( abla f = \mathbf{F} \). This validation reassures the accuracy of using path endpoint evaluations for integral solutions.