Tensors for Physics

6 1 Introduction

The distance is translationally invariant. This means: addition of the same vector

xto bothaandbdoes not change their distance:

d(a+x,b+x)=d(a,b). (1.11)

Furthermore, the distance is homogeneous. This means: multiplication of both vec-

torsaandbby the same real numberkimplies the multiplication of the distance by

the absolute value|k|:

d(ka,kb)=|k|d(a,b). (1.12)

As stressed before, in many applications in physics, the notionvectorrefers to a more

special mathematical object. Before details are discussed in the following section,

here a short answer is given to the question: what is special about vectors in physics?

Vectors in two and three dimensions, as used in classical physics, have to be distin-

guished from the four-dimensional vectors of special relativity theory.

1.1 Exercise: Complex Numbers as 2D Vectors
Convince yourself that the complex numbersz=x+iyare elements of a vector
space, i.e. that they obey the rules (1.1)–(1.6). Make a sketch to demonstrate that
z 1 +z 2 =z 2 +z 1 , withz 1 = 3 + 4 iandz 2 = 4 + 3 i, in accord with the vector
addition in 2D.

1.1.3 Vectors for Classical Physics

Thepositionof a physical object is represented by an arrow pointing from the origin

of a coordinate system to the center of mass of this object. Thisposition vector

is specified by the coordinates of the arrowhead. This ordered set of two or three

numbers, in two-dimensional (2D) or three-dimensional (3D) spaceR^3 , is referred

to asthe componentsof the position vector. It is convenient to use aCartesian

coordinate systemwhich has rectangular axes. Then the length (norm or magnitude)

of a vector is just the square root of the sum of the components squared.

In a coordinate system rotated with respect to the original one, the same position

vector has different components. There are well defined rules to compute the com-

ponents in the rotated system from the original components. This is referred to as

transformation of the components upon rotation of the coordinate system.

Now we are in the position to state what is special about vectors in physics:

A vector is a quantity with two or three components which transform like those of

the position vector, upon a rotation of the coordinate system.

The vectors used in classical physics like the velocity or the force are elements

of a vector space, do have a norm, and they possess an additional property, viz. a

specific transformation behavior of their components.

Ascalaris a quantity which does not change upon a rotation of the coordinate

system. Examples for scalars are the mass or the length of the position vector.