Notes for Wednesday August 27, 2014
September 8, 2014
1 Qualities of Coordinate Systems
We begin by listing some specic qualities of the Cartesian coordinate system.
We note that system is
• Linear
Orthogonal
• Metric Preserving
• "Length" invariant
• Coordinate invariant
• Active/Passive
2 Denition of a vector
A vector, V~ is known to be vector, if it behaves such that
0
V
µ0
0
∂xµ µ
V = Λµµ V µ
=
∂xµ
where µ and µ0 behave as indices for each component of the vector. For example
~ , V 2 the y and so on, such that a vector
V 1 could describe the x component of V
is fully described by
~ = V µ êµ
V
We have also used the indices summation notation of summing components that
have both an upper and lower index. For example, lets assume we have somthing
that describes the tangent of a curve
Vµ =
dxµ (t)
,
dt
which we will realize transforms according to
0
0
Vµ =
dxµ dxµ (t)
dxµ dt
1
Now, lets assume we have a function f (x, y). the gradiant behaves as
∂xµ ∂f
∂f
0 =
µ
∂x
∂xµ0 ∂xµ
which is, of course, dierent from our original denition of a vector. However,
we still believe it to behave as one. We, therefore, must have two classes a
vectors which we dene as
0
0
• Contravariant - V µ = Λµµ V µ
• Covariant - Vµ0 = Λµµ0 Vµ
0
If we look closer, we also notice another useful identity
0
Λµν Λγµ0 = δνγ
3 Some denitions of products
Now that we know what a vector is, lets focus our attention to two specic cases
of multiplication
~ .W
~
• Scalar: V
~ ×W
~
• Vector: V
In order to express vector multiplation in terms of indices, we choose to use the
levi-cevita symbol ijk , which is equal to 1 if ijk are cyclical, -1 if anticyclical, 0
if neither.
~ ×W
~ )i = ijk V j W k
(V
and because the product vector behaves dierently under parity than a vector,
we dene it as a psudovector.
Scalar multiplication is dened as the sum of the products of the components
with the same indices, or rather
V1 W 1 + V2 W 2 + .... = Vi W i
Since in the most general case of multiplication we have
V µW γ
we are going to need some obeject that can convert V µ → Vγ . We call this
object the metric, gµγ , so called because it allows the measurements of objects,
and our scalar product can therefore be written as
~ .W
~ = gµγ V µ W γ = Vγ W γ
V
Though the metric g has to two indices they will always be either both top
and bottom and thus should not be considered as a matrix. However it does
have a property similar to Λ.
g µγ gγσ = δσµ
2