# Vector tube

A closed set $\Phi$ of points of a space $\Omega$ in which there has been specified a vector field $\mathbf a ( M)$ such that the normal vector $\mathbf n$ is orthogonal to $\mathbf a$ everywhere on its boundary surface $S$. The vector tube $\Phi$ consists of vector lines $\Gamma$ of the field $\mathbf a$, i.e. curves in $\Omega$ at each point of which the tangent direction coincides with the direction of $\mathbf a$. A line $\Gamma$ is completely contained in $\Phi$ if one point of $\Gamma$ is contained in $\Phi$. If $\mathbf a$ is the field of velocities of a stationary liquid flow, then $\Gamma$ is the trajectory of the liquid particles, while $\Phi$ is the part of $\Omega$" swept along" by the motion of a given amount of liquid particles.

The intensity $I$ of the tube $\Phi$ in the cross-section $S ^ \prime$ is the flux (cf. Vector analysis) of $\mathbf a$ across $S ^ \prime$:

$$I( S ^ \prime ) = \int\limits \int\limits S ^ \prime ( \mathbf a , \mathbf n ) d \sigma ,$$

where $\mathbf n$ is the unit normal vector to $S ^ \prime$. If the field $\mathbf a$ is solenoidal ( $\mathop{\rm div} \mathbf a = 0$), the law of preservation of the intensity of the vector tube holds:

$$I( S ^ \prime ) = I( S ^ {\prime\prime} ) .$$

Let $a _ {1} ( x, y, z), a _ {2} ( x, y, z), a _ {3} ( x, y, z)$ be the orthogonal Cartesian coordinates of the vector $\mathbf a = \mathbf a ( M),$ and let $x, y, z$ be the coordinates of the point $M$. Then the boundary of $\Phi$ is locally defined by an equation $F( x, y, z) = \textrm{ const }$, where $F( x, y, z)$ satisfies the partial differential equation

$$( \mathbf a , \nabla F ) =$$

$$= \ a _ {1} ( x, y, z) \frac{\partial F }{\partial x } + a _ {2} ( x, y, z) \frac{\partial F }{\partial y } + a _ {3} ( x, y, z) \frac{\partial F }{\partial z } = 0.$$