What do div, curl, and grad refer to?

Div, Curl, and Grad, or Divergence, Curl, and Gradient are basic operations in vector calculus and are used extensively in fluid dynamics and electromagnetism. The terms div, curl, and grad are still used in a number of texts, but the more modern notion is to use the "del" or "nabla" symbol, which looks like an upside-down triangle.

The divergence describes how the "flow" of the field is expanding at a given point. In electromagnetism, div B = 0 indicates that the magnetic field is free from divergences and hence there are no magnetic monopoles in classical E&M.

Curl describes the rotation of the flow described by the vector field, i.e. the tendency for a point particle in the field to rotate about some point.

Finally, grad or gradient describes how a given scalar function is changing in each direction. For example, the force on a particle (a vector) is given as F = -grad V where V(x,y,z) is the (scalar) potential.

The properties of div, curl, and grad give rise to many important identities and theorems in calculus, such as Gauss' theorem, Stokes theorem, and so forth. While they seem complex and confusing, they dramatically simplify the analysis of many physical systems, including the flow of blood as visualized in the graphic.