derive an expression for divergence
Answers
The divergence of different vector fields. The divergence of vectors from point (x,y) equals the sum of the partial derivative-with-respect-to-x of the x-component and the partial derivative-with-respect-to-y of the y-component at that point: {\displaystyle \nabla \!\cdot (\mathbf {V} (x,y))={\frac {\partial \ {\mathbf {V} _{x}(x,y)}}{\partial {x}}}+{\frac {\partial \ {\mathbf {V} _{y}(x,y)}}{\partial {y}}}}
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point.
As an example, consider air as it is heated or cooled. The velocity of the air at each point defines a vector field. While air is heated in a region, it expands in all directions, and thus the velocity field points outward from that region. The divergence of the velocity field in that region would thus have a positive value. While the air is cooled and thus contracting, the divergence of the velocity has a negative value.