91Ó°ÊÓ

When dealing with conservative vector fields, a potential function is a key concept. A vector field is said to be conservative if there exists a scalar function, often called the potential function, from which the vector field can be derived as a gradient. This means that the vector field represents the rate of change of the potential function in space.
In the given exercise, the vector field \( \mathbf{F}(x, y) = 3y^2 \mathbf{i} + 6xy \mathbf{j} \) has been verified to be conservative. This verification means there is a potential function \( f(x, y) \) such that the vector field \( \mathbf{F} \) is the gradient of \( f(x, y) \).

• The potential function was found by integrating the components of the vector field.
• Integration requires us to handle each component separately.
• The resulting expression \( f(x, y) = 3xy^2 + C \) combines these integrations where \( C \) is an arbitrary constant.

This reveals how the vector field stems from a potential that governs its behavior, demonstrating the integral connection between conservative vector fields and their potential functions.

Partial derivatives play a crucial role in understanding vector fields, particularly when examining whether a vector field is conservative. A partial derivative measures how a function changes as one of its variables changes, while the other variables are held constant.
In our example, to determine whether \( \mathbf{F}(x, y) \) is a conservative vector field, we need to find the partial derivatives of its components \( P(x, y) \) and \( Q(x, y) \):

• \( \frac{\partial P}{\partial y} = \frac{\partial}{\partial y}(3y^2) = 6y \)
• \( \frac{\partial Q}{\partial x} = \frac{\partial}{\partial x}(6xy) = 6y \)

These computations are essential because, in a 2D space, a vector field is conservative if \( \frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x} \), as seen in this exercise. The equality of these partial derivatives confirms that the vector field has the necessary property of being conservative.

The curl of a vector field is a vector operator that describes the rotation or the "twisting" force of a field around a point. In two dimensions, the concept simplifies, focusing primarily on the differences in partial derivatives of the vector field's components.
This exercise leverages the fact that in a 2D vector field, for it to be conservative, the curl must be zero. In mathematical terms, this means:

• \( \frac{\partial Q}{\partial x} = \frac{\partial P}{\partial y} \)

These equations ensure that the vector field shows no rotation and is entirely determined by a scalar potential function.
In \( \mathbf{F}(x, y) = 3y^2 \mathbf{i} + 6xy \mathbf{j} \), both partial derivatives of the components matched (\( 6y \)), confirming that the curl is indeed zero. This zero curl is the hallmark of a conservative field, suggesting that it is possible to find a potential function that exactly describes \( \mathbf{F} \). This function dictates the path-independent nature of the field, which is a fundamental property of conservative vector fields.