Page 164 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 14: Notation and Summation Convention

Bold lower- and uppercase letters are used for the symbolic representation of vectors and Notes
second-order tensors, respectively. Suffix notation is used to specify the components of
tensors, and the convention that a lowercase letter suffix takes the values 1, 2, and 3 for
three-dimensional and 1 and 2 for two-dimensional Euclidean spaces, unless otherwise
indicated, is adopted. The number of distinct suffixes required is equal to the order of the
tensor. An example is the suffix representation of a vector u, with components (u , u , u )
3
1
2
or ui, i  {1, 2, 3}. The vector is then given by
3
u   u e .
i i
i 1

14.5 Keywords
Rectangular Cartesian coordinate: A rectangular Cartesian coordinate system consists of an
orthonormal basis of unit vectors (e , e , e ) and a point 0 which is the origin.
2
3
1
Right-handed Cartesian coordinate systems are considered, and the axes in the (e , e , e )
3
2
1
directions are denoted by 0x , 0x , and 0x , respectively.
1
2
3
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.
14.6 Self Assessment
1. A ................. system consists of an orthonormal basis of unit vectors (e , e , e ) and a point
2
3
1
0 which is the origin.
2. ................. coordinate systems are considered, and the axes in the (e , e , e ) directions are
3
2
1
denoted by 0x , 0x , and 0x , respectively.
1
2
3
3. ................. are used to denote components of tensors, of order greater than zero, referred to
a particular rectangular Cartesian coordinate system.
4. Tensor equations can be expressed in terms of these components; this is known as ..................
5. Suffix notation is used to specify the components of tensors, and the convention that a
lowercase letter suffix takes the values 1, 2, and 3 for three-dimensional and 1 and 2 for
two-dimensional ................., unless otherwise indicated, is adopted.
14.7 Review Questions

1. Discuss the concept rectangular Cartesian coordinate systems.

2. Describe the suffix and symbolic notation.
3. Discuss the orthogonal transformation.

Answers: Self Assessment

1. rectangular Cartesian coordinate 2. Right-handed Cartesian
3. Suffixes 4. suffix notation
5. Euclidean spaces

LOVELY PROFESSIONAL UNIVERSITY 157