Solenoidal Field

A solenoidal Vector Field satisfies

$$\nabla\cdot{\bf B} = 0$$ (1)

for every Vector ${\bf B}$, where $\nabla\cdot{\bf B}$ is the Divergence. If this condition is satisfied, there exists a vector ${\bf A}$, known as the Vector Potential, such that

$${\bf B}\equiv\nabla\times{\bf A},$$ (2)

where $\nabla\times{\bf A}$ is the Curl. This follows from the vector identity

$$\nabla\cdot {\bf B} = \nabla\cdot(\nabla \times {\bf A}) = 0.$$ (3)

If ${\bf A}$ is an Irrotational Field, then

$${\bf A}\times {\bf r}$$ (4)

is solenoidal. If ${\bf u}$ and ${\bf v}$ are irrotational, then

$${\bf u}\times {\bf v}$$ (5)

is solenoidal. The quantity

$$(\nabla u)\times (\nabla v),$$ (6)

where $\nabla u$ is the Gradient, is always solenoidal. For a function $\phi$ satisfying Laplace's Equation

$$\nabla^2\phi = 0,$$ (7)

it follows that $\nabla\phi$ is solenoidal (and also Irrotational).

See also Beltrami Field, Curl, Divergence, Divergenceless Field, Gradient, Irrotational Field, Laplace's Equation, Vector Field

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, pp. 1084, 1980.