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A potential function is a scalar function whose gradient yields a given vector field. In simpler terms, if you have a conservative vector field, you can "uncover" a potential function that essentially "produces" that field by taking its gradient.
A vector field is considered conservative if there exists a potential function, denoted here as \( f(x, y, z) \). For a vector field \(\mathbf{F}\) expressed in terms of its components \( (F_1, F_2, F_3) \), finding this potential function involves these steps:

• Identify components of \(\mathbf{F}\)
• Integrate each component with respect to its respective variable
• Combine results, ensuring it forms one consistent function including undetermined functions of the other variables

In practice, computing this involves operations such as partial integration and acknowledging that constants can vary, but disappear when the gradient is recalculated. Once potential function \( f \) is found, it means that movement from one point to another in the field is path-independent and depends only on the endpoints in the potential.

The curl of a vector field helps in determining whether a field is conservative. Mathematically, curl measures the rotational effect at any point within the field.
The curl of a vector field \( \mathbf{F} = (F_1, F_2, F_3) \) is calculated using the formula:
\[abla \times \mathbf{F} = \left( \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z}, \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x}, \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \right)\]If the curl is zero throughout its entire domain, the field is conservative, implying the presence of a potential function. The vector field \(\mathbf{F}(x, y, z) = (x + yz, y + xz, z + xy)\) was checked as follows:

• \( \frac{\partial F_3}{\partial y} = x \) and \( \frac{\partial F_2}{\partial z} = x \)
• \( \frac{\partial F_1}{\partial z} = y \) and \( \frac{\partial F_3}{\partial x} = y \)
• \( \frac{\partial F_2}{\partial x} = z \) and \( \frac{\partial F_1}{\partial y} = z \)

Subsequently, the calculation \( abla \times \mathbf{F} = (0, 0, 0) \) verified that the field is indeed conservative. This essential zero result signifies no rotational component, further suggesting the field can be described entirely by a potential function.

Integration of vector components is key in determining the potential function of a conservative vector field. Each component of the vector field \( (F_1, F_2, F_3) \) needs to be integrated separately.
For the vector field \( \mathbf{F}(x, y, z) = (x + yz, y + xz, z + xy) \), the integration is performed as follows:

• Integrate \( F_1 = x + yz \) with respect to \( x \) results in: \[ f(x, y, z) = \int (x + yz) \, dx = \frac{x^2}{2} + xyz + g(y, z) \]
• Integrate \( F_2 = y + xz \) with respect to \( y \) results in: \[ f(x, y, z) = \int (y + xz) \, dy = \frac{y^2}{2} + xyz + h(x, z) \]
• Integrate \( F_3 = z + xy \) with respect to \( z \) results in: \[ f(x, y, z) = \int (z + xy) \, dz = \frac{z^2}{2} + xyz + p(x, y) \]

Each integration brings an arbitrary function depending on the ignored variables during the integration process. These are combined and simplified to produce the consistent potential function: \[ f(x, y, z) = \frac{x^2}{2} + \frac{y^2}{2} + \frac{z^2}{2} + xyz + C \]where \( C \) is a constant, underlining the independence from the path in the field and emphasizing the role of ending conditions. This process highlights how each component contributes to forming the final, complete potential function for the vector field.