Covariant Derivative of a Field along a Curve
Covariant Derivative of a Field along a Curve
This Demonstration gives a visual representation of the covariant derivative of a field along a curve , induced by a field , where is a Peano surface with parametric equations given by the chart
X(t)
α(t)=σ(u(t),v(t))
Y:SS
T
p
S
σ(u,v)=(u,v,(2-v)(v-))
2
u
2
u
This covariant derivative has been computed from its representation in terms of the given chart, the field and the corresponding Christoffel symbols:
X(t)
D
t
1
Γ
11
1
Γ
12
1
Γ
12
1
Γ
22
σ
u
2
Γ
11
2
Γ
12
2
Γ
12
2
Γ
22
σ
v
where are the coefficients of .
a(t),b(t)
X(t)=a(t)+b(t)
σ
u
σ
v