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The concept of a potential function, often denoted by the Greek letter \( \varphi \), is crucial in vector calculus, especially when dealing with gradient fields. Essentially, a potential function is a scalar field whose gradient results in a given vector field. If you have a gradient field \( \mathbf{F} \), the function \( \varphi \) can be thought of as the source of that field.
To put it simply, if you know \( \varphi \), you can find \( \mathbf{F} \) by computing the gradient of \( \varphi \). This is an incredibly powerful tool because it allows us to evaluate line integrals easily by using potential differences, a concept linked to conservative fields.

• The gradient of \( \varphi(x, y, z) = xy + xz + yz \) is calculated to obtain the vector field \( \mathbf{F} = \langle y+z, x+z, x+y \rangle \).
• The function \( \varphi(x, y, z) \) is the potential function for the given line integral problem, meaning that any line integral over a path C can potentially be simplified using \( \varphi \).

These properties make potential functions highly relevant in computations involving vector fields and integrals.

A gradient field is a vector field represented by the gradient of a scalar potential function \( \varphi \). If we have a function \( \varphi(x, y, z) \), its gradient \( abla \varphi \) is a vector field. This vector is directed such that it points in the direction of the greatest rate of increase of \( \varphi \).
Calculating the gradient vector involves partial derivatives:

• Take partial derivatives of \( \varphi \) with respect to each variable: \( \partial \varphi / \partial x \), \( \partial \varphi / \partial y \), and \( \partial \varphi / \partial z \).
• These derivatives together form the gradient field: \( \mathbf{F} = abla \varphi = \langle y+z, x+z, x+y \rangle \).

Gradient fields have a special property - they are conservative fields. This means that the line integral over any closed loop is zero. It also means that the line integral between two points is path-independent, making calculations more straightforward when using potential functions.

The fundamental theorem for line integrals connects the concepts of potential functions, gradient fields, and line integrals. It gives us a way to evaluate line integrals easily if we know the potential function.
Here's how the theorem functions:

• If \( \mathbf{F} \) is a gradient field of a potential function \( \varphi \), then for a smooth curve C going from point A to point B, the line integral of \( \mathbf{F} \) is \( \varphi(B) - \varphi(A) \).
• This method transforms the potentially complex task of calculating a line integral into finding the values of \( \varphi \) at two points.

In the problem, the endpoints of the curve C are used to directly apply this theorem, \( \int_{C} \mathbf{F} \cdot d\mathbf{r} = \varphi(B) - \varphi(A) = 320 - 0 = 320 \). This approach highlights the elegance and simplicity offered by the fundamental theorem, making complex integrals manageable by leveraging path independence.