Page 163 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes The transformations (5), (7), and (12) subject to equation (14) or (15) are known as orthogonal
transformations. Three quantities u are the components of a vector if, under orthogonal
i
transformation, they transform according to equation (12). This may be taken as a definition of
a vector. According to this definition, equations (5) and (7) imply that the representation x of a
point is a bound vector since its origin coincides with the origin of the coordinate system.
If the transformation rule (7) holds for coordinate transformations from right-handed systems
to left-handed systems (or vice versa), the vector is known as a polar vector. There are scalars
and vectors known as pseudo scalars and pseudo or axial vectors; there have transformation
rules that involve a change in sign when the coordinate transformation is from a right-handed
system to a left-handed system (or vice versa), that is, when det[a ] = –1. The transformation rule
ij
for a pseudo scalar is
’ = det [a ], ...(17)
ij
and for a pseudo vector

'
u  det[a ]a u . j ...(18)
ij
i
ij
A pseudo scalar is not a true scalar if a scalar is defined as a single quantity invariant under all
coordinate transformations. An example of a pseudo vector is the vector product u × v of two
polar vectors u and v. The moment of a force about a point and the angular momentum of a
particle about a point are pseudo vectors. The scalar product of a polar vector and a pseudo
vector is a pseudo scalar; an example is the moment of a force about a line. The distinction
between pseudo vectors and scalars and polar vectors and true scalars disappears when only
right- (or left-) handed coordinate systems are considered. For the development of continuum
mechanics presented in this book, only right-handed systems are used.

Example: Show that a rotation through angle  about an axis in the direction of the unit
vector n has the transformation matrix

a = –  + 2n n, det[aij] = 1.
ij
i
ij
j
Solution. The position vector of point A has components x and point B has position vector with
i
components x .
'
i
14.4 Summary
A rectangular Cartesian coordinate system consists of an orthonormal basis of unit vectors

(e , e , e ) and a point 0 which is the origin. Right-handed Cartesian coordinate systems are
3
1
2
considered, and the axes in the (e , e , e ) directions are denoted by 0x , 0x , and 0x ,
3
2
3
1
2
1
respectively, rather than the more usual 0x, 0y, and 0z. A right-handed system is such that
a 90º right-handed screw rotation along the 0x direction rotates 0x to 0x , similarly a
2
1
3
right-handed rotation about 0x rotates 0x to 0x , and a right-handed rotation about 0x 3
1
3
2
rotates 0x to 0x .
1 2
Suffixes are used to denote components of tensors, of order greater than zero, referred to

a particular rectangular Cartesian coordinate system. Tensor equations can be expressed
in terms of these components; this is known as suffix notation. Since a tensor is independent
of any coordinate system but can be represented by its components referred to a particular
coordinate system, components of a tensor must transform in a definite manner under
transformation of coordinate systems. This is easily seen for a vector. In tensor analysis,
involving oblique Cartesian or curvilinear coordinate systems, there is a distinction
between what are called contra-variant and covariant components of tensors but this
distinction disappears when rectangular Cartesian coordinates are considered exclusively.
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