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A gradient vector field is essentially a representation of how a scalar function changes at any given point in space. To find the gradient of a function, we take the derivative, or rate of change, with respect to each variable in the function. Here, the function is given as \( f(x, y) = (x + y)^2 \). To find its gradient, we denote it as \( abla f \) which is composed of partial derivatives with respect to \( x \) and \( y \).

Let's calculate these partial derivatives:

• With respect to \( x \), the derivative is \( 2(x + y) \).
• With respect to \( y \), the derivative is also \( 2(x + y) \).

Therefore, the gradient vector field \( abla f \) is \( (2(x + y), 2(x + y)) \). This vector field tells us how the value of \( f \) changes as we move in the plane. It always points in the direction of greatest increase of the function, and its magnitude reflects how steep that increase is.

A vector field is termed conservative if it is the gradient of some scalar function. This implies certain properties that are useful, especially in calculating line integrals.

• The most notable property is that the work done by a conservative vector field over a closed path is zero.
• This occurs because, with conservative fields, the line integral is path-independent and solely dependent on the start and end points.

In the context of our example, since the vector field \( abla f = (2(x+y), 2(x+y)) \) arises from the gradient of the function \( f(x, y) \), it's conservative. So, regardless of the path taken, as long as it forms a closed loop, the work done by this vector field is zero.

The Fundamental Theorem of Line Integrals bridges the relationship between the integration of a conservative vector field over a curve and the evaluation of its scalar potential function. This theorem particularly simplifies calculations by stating that the line integral of a vector field that is the gradient of a function depends only on the endpoints of the path and not on the specific path taken—this is known as path independence.

Mathematically, if \( \mathbf{F} = abla f \), and there’s a curve \( C \) with endpoints \( A \) and \( B \), the line integral of \( \mathbf{F} \) along \( C \) is given by:\[ \int_C \mathbf{F} \cdot d\mathbf{r} = f(B) - f(A) \]In our case, since the path is a circle returning to its start point, \( A = B \). Thus, the line integral or the work done around a closed path is \( f(B) - f(A) = 0 \). This straightforward property leads directly to the conclusion that the work done by a conservative field over a closed path is zero, perfectly applying to our gradient vector field example.