91Ó°ÊÓ

A gradient field is a special type of vector field that can be expressed as the gradient of a scalar function. In mathematics, a vector field \( \vec{F} \) is said to be a gradient field if there exists a scalar function \( f \) such that \( \vec{F} = \vec{abla} f \), where \( \vec{abla} \) denotes the gradient operator. This means that the components of the vector field are the partial derivatives of \( f \) with respect to each variable. When dealing with gradient fields in three dimensions, you typically identify that the vector field components \( F_1, F_2, \) and \( F_3 \) correspond to the partial derivatives \( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \) and \( \frac{\partial f}{\partial z} \), respectively. Therefore, finding \( f \) involves integration.
In the given exercise, the vector field \( \vec{F} \left( \begin{array}{l} x \ y \ z \end{array} \right) = \left[ \begin{array}{c} F_{1}(x, y) \ F_{2}(x, y) \ 0 \end{array} \right] \) suggests that our scalar function \( f \) depends only on \( x \) and \( y \), because \( F_3 = 0 \). This hints that \( f \) is a function of two variables, even though it's defined in \( \mathbb{R}^{3} \). The requirement \( D_2 F_1 = D_1 F_2 \) corresponds to Clairaut's theorem, which ensures the interchangeability of mixed partial derivatives, is crucial for this vector field \( \vec{F} \) to be a gradient.

Partial derivatives play a fundamental role in multivariable calculus, particularly when working with functions of multiple variables. They are used for evaluating the change of a function with respect to one variable, while all other variables are held constant. This is essential in understanding how functions behave in higher-dimensional spaces.
For a function \( f(x, y, z) \), the partial derivatives \( \frac{\partial f}{\partial x} \), \( \frac{\partial f}{\partial y} \), and \( \frac{\partial f}{\partial z} \) measure how \( f \) changes as each of these variables change separately. In the context of gradient fields, partial derivatives are integrated back to retrieve the scalar function \( f \) from its gradient. For this exercise, the conditions \( F_1 = \frac{\partial f}{\partial x} \) and \( F_2 = \frac{\partial f}{\partial y} \) are pivotal. By integrating \( F_1 \) with respect to \( x \) and \( F_2 \) with respect to \( y \), and ensuring their mixed partial derivatives are equal, we ensure consistency and correct reconstruction of the function \( f \).
Understanding partial derivatives ties directly into Clairaut's Theorem, which gives a deeper insight into their symmetry properties. This symmetry helps establish the conditions under which a vector field can be expressed as a gradient.

Clairaut's Theorem, a pivotal result in calculus, concerns the equality of mixed partial derivatives. It states that if all second partial derivatives of a function exist and are continuous in a neighborhood, the mixed partial derivatives can be interchanged, i.e., \( \frac{\partial^2 f}{\partial y \partial x} = \frac{\partial^2 f}{\partial x \partial y} \).
This theorem is crucial in establishing whether a given vector field \( \vec{F} \) can be a gradient of a scalar function. The criterion \( D_2 F_1 = D_1 F_2 \) in the exercise directly uses Clairaut's Theorem to determine if \( \vec{F} \) is indeed a gradient field. Essentially, when checking whether a vector field is gradient, Clairaut's Theorem provides the symmetry condition required for the vector components' partial derivatives. This symmetry ensures that a potential function \( f \) exists such that its gradient recovers the components of \( \vec{F} \). In the context of the given problem, verifying \( D_2 F_1 = D_1 F_2 \) ensures that there's a scalar function \( f \) satisfying \( \vec{F} = \vec{abla} f \). Clairaut's insight into the equality of mixed partials thus becomes a powerful tool in verifying the consistency of a potential function against a vector field.