The idea of divergence and curl

Vector fields

We can think of a vector-valued function F : R2 → R2 as representing fluid flow in two dimensions, so that F(x,y) gives the velocity of a fluid at the point (x,y). In this case, we may call F(x,y) the velocity field of the fluid. More generally, we refer to a function like F(x,y) as a two-dimensional vector field. You can read more about how we can visualize the fluid flow by plotting the velocity F(x,y) as vector positioned at the point (x,y).

We can do the same thing for a three-dimensional fluid flow with velocity represented by a function F : R3 → R3. In this case, F(x,y,z) is the velocity of the fluid at the point (x,y,z), and we can visualize it as the vector F(x,y,z) positioned a the point (x,y,z). We refer to F(x,y,z) as a three-dimensional vector field.

Divergence

The divergence of a vector field is relatively easy to understand intuitively. Imagine that the vector field F below gives the velocity of some fluid flow. It appears that the fluid is exploding outward from the origin.

This expansion of fluid flowing with velocity field F is captured by the divergence of F, which we denote div F. The divergence of the above vector field is positive since the flow is expanding.

In contrast, the below vector field represents fluid flowing so that it compresses as it moves toward the origin. Since this compression of fluid is the opposite of expansion, the divergence of this vector field is negative.

The divergence is defined for both two-dimensional vector fields F(x,y) and three-dimensional vector fields F(x,y,z). A three-dimensional vector field F showing expansion of fluid flow is shown in the below CVT. Again, because of the expansion, we can conclude that div F > 0.

Vector fields

We can think of a vector-valued function F : R2 → R2 as representing fluid flow in two dimensions, so that F(x,y) gives the velocity of a fluid at the point (x,y). In this case, we may call F(x,y) the velocity field of the fluid. More generally, we refer to a function like F(x,y) as a two-dimensional vector field. You can read more about how we can visualize the fluid flow by plotting the velocity F(x,y) as vector positioned at the point (x,y).

We can do the same thing for a three-dimensional fluid flow with velocity represented by a function F : R3 → R3. In this case, F(x,y,z) is the velocity of the fluid at the point (x,y,z), and we can visualize it as the vector F(x,y,z) positioned a the point (x,y,z). We refer to F(x,y,z) as a three-dimensional vector field.

Divergence

The divergence of a vector field is relatively easy to understand intuitively. Imagine that the vector field F below gives the velocity of some fluid flow. It appears that the fluid is exploding outward from the origin.

This expansion of fluid flowing with velocity field F is captured by the divergence of F, which we denote div F. The divergence of the above vector field is positive since the flow is expanding.

In contrast, the below vector field represents fluid flowing so that it compresses as it moves toward the origin. Since this compression of fluid is the opposite of expansion, the divergence of this vector field is negative.

The divergence is defined for both two-dimensional vector fields F(x,y) and three-dimensional vector fields F(x,y,z). A three-dimensional vector field F showing expansion of fluid flow is shown in the below CVT. Again, because of the expansion, we can conclude that div F > 0.