For arbitrary C , this integral implies the relation (8.53). That is, one can factor
out the constant vector C as long as this vector is different from zero. Under these
conditions the integral relation (8.53) must hold.
Region of Integration
Green’s, Gauss’ and Stokes theorems are valid only if certain conditions are
satisfied. In these theorems it has been assumed that the integrands are continuous
inside the region and on the boundary where the integrations occur. Also assumed
is that all necessary derivatives of these integrands exist and are continuous over the
regions or boundaries of the integration. In the study of the various vector and scalar
fields arising in engineering and physics, there are times when discontinuities occur
at points inside the regions or on the boundaries of the integration. Under these
circumstances the above theorems are still valid but one must modify the theorems
slightly. Modification is done by using superposition of the integrals over each side
of a discontinuity and under these circumstances there usually results some kind of
a jump condition involving the value of the field on either side of the discontinuity.
If a region of space has the property that every simple closed curve within the
region can be deformed or shrunk in a continuous manner to a single point within
the region, without intersecting a boundary of the region, then the region is said
to be simply connected. If in order to shrink or reduce a simply closed curve to a
point the curve must leave the region under consideration, then the region is said
to be a multiply connected region. An example of a multiply connected region is
the surface of a torus. Here a circle which encloses the hole of this doughnut-shaped
region cannot be shrunk to a single point without leaving the surface, and so the
region is called a multiply-connected region.
If a region is multiply connected it usually can be modified by introducing imag-
inary cuts or lines within the region and requiring that these lines cannot be crossed.
By introducing appropriate cuts, one can usually modify a multiply connected re-
gion into a simply connected region. The theorems of Gauss, Green, and Stokes are
applicable to simply connected regions or multiply connected regions which can be
reduced to simply connected regions by introducing suitable cuts.