Mathematical Physics Vol 1

2.2 Integration

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Depending on the nature of the function ϕ and the meaning of the ◦ symbol for the product the following three cases are possible - ϕ ( r ) is a scalar function, the symbol ◦ stands for the multiplication of a vector by a scalar, and the line integral is a vector, - ϕ ( r ) is a vector function, the symbol ◦ stands for the scalar product, and the line integral is a scalar function, - ϕ ( r ) is a vector function, the symbol ◦ stands for the vector product, and the line integral is a vector function.

2.2.4 Surface integral

Surface orientation

Definition Asurface S is said to be orientable (two-sided) (Fig. 2.8), if it can be partitioned into pieces which can be oriented in such a way that along each curve that is a common boundary of two different pieces, the directions of the curve relative to the two pieces are mutually opposite. If such orientation of the pieces of the surface cannot be established by any division, then the surface is said to be nonorientable (one-sided) (Fig. 2.9)).

Figure 2.8: Surface orientation.

An example of nonorientable surface is the Möbius 4 strip. It is obtained by twisting a rectangular strip ABCD and joining its end points as follows: A with C and B with D , as shown in Fig. 2.9. If a triangulation (division into triangles) of the ABCD strip is performed and the resulting triangles are oriented (Fig. 2.9c), then when the Möbius strip is formed, the boundary lines AD and BC have the same and not opposite orientations, which makes the Möbius strip a one-sided surface by definition.

4 Möbius, August Ferdinand (1790–1868), German mathematician. Known for his works in the theory of surfaces, projective geometry and mechanics, as well as number theory.