Final answer:
In the expression \(\\abla \cdot \mathbf{F}\), you compute the divergence, a scalar quantity indicating the rate at which quantity flows out of or into a point in space. It is calculated by taking the dot product of the del operator with the components of the vector field. The correct answer is B.
Step-by-step explanation:
In the expression \(\\abla \cdot \mathbf{F}\), what you compute is the divergence of the vector field \(\mathbf{F}\). The divergence is a scalar quantity that represents the rate at which "quantity" is emanating from a point. If we imagine the vector field as arrows representing the flow of a fluid or gas, the divergence at a point will tell you whether there is more of the substance flowing out of the point than into it (a positive divergence), or whether there is more flowing in than out (a negative divergence).
To calculate the divergence of a vector field in three-dimensional space, we take the dot product of the del operator (also called the gradient operator) and the vector field in question. Mathematically, if the vector field \(\mathbf{F}\) has components \(F_x, F_y, F_z\), the divergence is given by:
\(div \mathbf{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}\)
The concept of divergence is widely used in physics, particularly in the field of electromagnetism and fluid dynamics. For instance, in electromagnetism, Gauss's law states that the divergence of the electric field is directly proportional to the electric charge density. In fluid dynamics, the divergence theorem, also known as Gauss's divergence theorem, relates the flow (flux) of a vector field through a surface to the divergence of the vector field within the volume enclosed by that surface.