91Ó°ÊÓ

A potential function is a scalar field associated with a vector field, where the vector field is derived as the gradient of the scalar field. For a vector field \( \mathbf{F} \) to have a potential function \( f \), \( \mathbf{F} \) must be conservative. This means there must exist a function \( f(x, y, z) \) such that its gradient \( abla f = \mathbf{F} \). If you can find such a function, then the vector field is conservative and path-independent.
For instance, in our given vector field \( \mathbf{F}(x, y, z) = 2x \sin z \mathbf{i} + 2y \cos z \mathbf{j} + (x^2 \cos z - y^2 \sin z) \mathbf{k} \), we identified \[ f(x, y, z) = x^2 \sin z + y^2 \cos z \] as the potential function. This function satisfies \( abla f = \mathbf{F} \), verifying the vector field as conservative.

Path independence is a fundamental property of conservative vector fields. It implies that the line integral of the vector field along a path only depends on the endpoints and not the specific path taken. This is a very useful property in physics and engineering because it simplifies the computation of work and energy.
In mathematical terms, for a vector field \( \mathbf{F} \) that is path-independent, the line integral \( \int_{c} \mathbf{F} \cdot d\mathbf{r} \) between two points \( A \) and \( B \) is given by \( f(B) - f(A) \), where \( f \) is the potential function of \( \mathbf{F} \).

• If the vector field is conservative, as we established with the potential function, then it is also path-independent.
• This correlates to our example where the vector field \( \mathbf{F} \) satisfies the conditions of having a potential function, reaffirming its path-independence.

The curl of a vector field \( \mathbf{F} \) is a vector that describes the rotation or "twisting" effect of \( \mathbf{F} \) at a point. In three-dimensional space, the curl is given by:
\[ abla \times \mathbf{F} \ = \left( \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z} \right) \mathbf{i} + \left( \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x} \right) \mathbf{j} + \left( \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \right) \mathbf{k} \]
For a vector field to be conservative, its curl must be zero everywhere in its domain. A zero curl indicates no rotational tendency, confirming two things:

• The vector field is conservative.
• There exists a potential function \( f \) such that \( abla f = \mathbf{F} \).

In our example, calculating the curl of \( \mathbf{F} \) returned zero: \[ abla \times \mathbf{F} = \mathbf{0} \], proving that the field is conservative. This allows us to find a potential function and confirms the field's path independence.