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For a vector with components $X,Y,Z$ we sometimes also write $(X,Y,Z)$.

e. If $\phi$ is a scalar magnitude, then we understand by ${\dot {\phi }}$ the derivative with respect to time t. The letter ${\dot {\mathfrak {A}}}$ denotes a vector with components: ${\dot {\mathfrak {A}}}_{x},{\dot {\mathfrak {A}}}_{y},{\dot {\mathfrak {A}}}_{z}$, or ${\tfrac {\partial {\mathfrak {A}}_{x}}{\partial t}}$ etc.

f. The expression

$\int {\mathfrak {A}}_{n}\ d\ \sigma$

we call the "integral of vector ${\mathfrak {A}}$ over the surface $\sigma$", and the magnitude

$\int {\mathfrak {A}}_{s}\ d\ s$

the "line integral of line s".

g. If a vector ${\mathfrak {A}}$ in any point of space is given, then

${\frac {\partial {\mathfrak {A}}_{x}}{\partial x}}+{\frac {\partial {\mathfrak {A}}_{y}}{\partial y}}+{\frac {\partial {\mathfrak {A}}_{z}}{\partial z}}$

has everywhere a certain value, independent of the choice of coordinate system. We call this magnitude "divergence" of vector ${\mathfrak {A}}$ and denote it by

$Div\ {\mathfrak {A}}$.

For any space limited by a surface $\sigma$, the relation is given

$\int Div\ {\mathfrak {A}}\ d\ \tau =\int {\mathfrak {A}}_{n}\ d\ \sigma {,}$

when, as already mentioned, the perpendicular n will be drawn into the outside.

h. The magnitudes

${\frac {\partial {\mathfrak {A}}_{z}}{\partial y}}-{\frac {\partial {\mathfrak {A}}_{y}}{\partial z}}{,}\ {\frac {\partial {\mathfrak {A}}_{x}}{\partial z}}-{\frac {\partial {\mathfrak {A}}_{z}}{\partial x}}{,}\ {\frac {\partial {\mathfrak {A}}_{y}}{\partial x}}-{\frac {\partial {\mathfrak {A}}_{x}}{\partial y}}$

can be interpreted as the components of vector ${\mathfrak {B}}$, which (independent from the choses coordinate system) is defined by the distribution of ${\mathfrak {A}}$. We call this vector the rotation of ${\mathfrak {A}}$ and denote it by

$Rot\ {\mathfrak {A}}{,}$