Curl and divergencesteemCreated with Sketch.

In the context of vector calculus, "curl" and "divergence" are two important differential operators used to describe the behavior of vector fields.

Curl

The curl of a vector field measures the rotation or the rotational tendency at a point in the field. It is a vector operation that describes the infinitesimal rotation of a 3-dimensional vector field. The curl of a vector field (\mathbf{F} = (F_x, F_y, F_z)) is defined as:

[ \text{curl} , \mathbf{F} = \nabla \times \mathbf{F} = \left( \frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z} , \frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x} , \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \right) ]

where (\nabla \times \mathbf{F}) denotes the curl of (\mathbf{F}).

Divergence

The divergence of a vector field measures the magnitude of a source or sink at a given point in the field. It is a scalar operation that describes how much the vector field is spreading out or converging at a point. The divergence of a vector field (\mathbf{F} = (F_x, F_y, F_z)) is defined as:

[ \text{div} , \mathbf{F} = \nabla \cdot \mathbf{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} ]

where (\nabla \cdot \mathbf{F}) denotes the divergence of (\mathbf{F}).

In summary:

  • Curl describes the rotation of a vector field and is itself a vector.
  • Divergence describes the net flow or "spread" of a vector field from a point and is a scalar.

These concepts are fundamental in fields like fluid dynamics, electromagnetism, and more, providing insight into the behavior of vector fields.
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