Tensors for Physics

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=0fori=j.
The position vectorrcan be written as a linear combination of these unit vectors
e(i)according to
r=r 1 e(^1 )+r 2 e(^2 )+r 3 e(^3 ). (2.12)

Since thebasis vectorsare not only orthogonal, but also normalized to 1, the Cartesian
components are equal to the scalar product ofrwith the basis vectors, e.g.r 1 =e(^1 )·r.

2.2.2 Non-orthogonal Basis

Three vectorsa(i), withi=1, 2, 3, which are not within one plane, can be used as
basis vectors. Then the vectorrcan be represented by the linear combination

r=ξ^1 a(^1 )+ξ^2 a(^2 )+ξ^3 a(^3 ), (2.13)

with the coefficientsξi. Scalar multiplication of (2.13) with the basis vectorsa(i)
yields

ξi=a(i)·r=

∑^3

j= 1

gijξj. (2.14)

The coefficient matrix
gij=a(i)·a(j)=gji, (2.15)

is determined by the scalar products of the basis vectors. The coefficientsξiandξi
are referred to ascontra-andco-variantcomponents of the vector in a coordinate
system with axes specified by the basis vectorsa(i).
In this basis, the square of the length or of the magnitude of the vector is given by

r·r=

∑

i

∑

j

ξiξja(i)·a(j)=

∑

i

∑

j

ξiξjgji=

∑

i

ξiξi. (2.16)

Thecoefficientmatrixgijcharacterizestheconnectionbetweentheco-andthecontra-
variant components and it is essential for the calculation of the norm. Thus it deter-
mines themetricof the coordinate system.
The geometric meaning of the two different types of components is demonstrated
in Fig.2.3for the 2-dimensional case.
The intersection of the dashed line parallel to the 2-axis with the 1-axis marks the
componentξ^1 , similarlyξ^2 is found at the intersection of the 2-axis with the dashed
line parallel to the 1-axis. The componentξ 1 andξ 2 are found at the intersections