Tensors for Physics

4 1 Introduction Fig. 1.1 Vector addition a+b=b+a. (1.1) As a side remark, one may ask: how was the rule for the vector addition ...

1.1 Preliminary Remarks on Vectors 5 Furthermore, for any real numberkand two vectorsaandbone has: k(a+b)=ka+kb. (1.6) Mathemati ...

6 1 Introduction The distance is translationally invariant. This means: addition of the same vector xto bothaandbdoes not change ...

1.1 Preliminary Remarks on Vectors 7 1.1.4 Vectors for Special Relativity. Vectors with four components are used in special rela ...

8 1 Introduction 1.3 Remarks on History and Literature Cartesiantensors, as they are used here for the description of material p ...

1.3 Remarks on History and Literature 9 used a geometric representation of complex numbers in a way which was essentially equiva ...

Chapter 2 Basics Abstract This chapter is devoted to the basic features needed for Cartesian tensors: the components of a positi ...

12 2Basics Fig. 2.1 Position vector in a Cartesian coordinate system. Thedashed linesare guides for the eye rule for the additio ...

2.1 Coordinate System and Position Vector 13 The vectorr, divided by its lengthr, is the dimensionlessunit vector̂r: ̂r=r−^1 r, ...

14 2Basics holds true. Or in words: the scalar product of two vectors is equal to the product of their lengths times the cosine ...

2.2 Vector as Linear Combination of Basis Vectors 15 e(i)·e(j)=δij. (2.11) Hereδijis the Kronecker symbol, i.e.δij=1fori=jandδij ...

16 2Basics Fig. 2.3 Components of the position vector in a non-orthogonal coordinate system of the dashed lines perpendicular to ...

2.3 Linear Transformations of the Coordinate System 17 Fig. 2.4 Components of the position vectorrin shifted coordinate system T ...

18 2Basics r 1 ′=T 11 r 1 +T 12 r 2 +T 13 r 3. (2.20) More general, forμ=1, 2, 3, one has rμ′=Tμ 1 r 1 +Tμ 2 r 2 +Tμ 3 r 3 , (2. ...

2.3 Linear Transformations of the Coordinate System 19 Similarly, insertion of (2.24)into(2.23) leads to T·T−^1 =δ, (2.28) or in ...

20 2Basics where it is understood that the 1 on the right hand side stands for the unit matrix. Comparison of (2.32) and (2.33) ...

2.4 Rotation of the Coordinate System 21 Fig. 2.5 The components of the position vectorrin the original coordinate system and in ...

22 2Basics This is explained as follows. When first a rotationU( 3 |α)about the 3-axis is performed, the subsequent rotation ind ...

2.5 Definitions of Vectors and Tensors in Physics 23 2.5.2 What is a Tensor?. Tensors are important “tools” for the characteriza ...

24 2Basics 2.5.4 Remarks on Notation. The Cartesian components of tensors are unambiguously specified by Greek sub- scripts, e.g ...