Covariant Derivative of a Field along a Curve

t

1

tangent vector
field X(t)
derivative X'(t)
covariant derivative of X

This Demonstration gives a visual representation of the covariant derivative of a field

X(t)

along a curve

α(t)=σ(u(t),v(t))

, induced by a field

Y:S

T

p

S

, where

S

is a Peano surface with parametric equations given by the chart

σ(u,v)=(u,v,(2

2

u

-v)(v-

2

u

))

.

This covariant derivative has been computed from its representation in terms of the given chart, the field

X(t)

and the corresponding Christoffel symbols:

D

t

X=a'+

1

Γ

11

au'+

1

Γ

12

av'+

1

Γ

12

bu'+

1

Γ

22

bv'

σ

u

+b'+

2

Γ

11

au'+

2

Γ

12

av'+

2

Γ

12

bu'+

2

Γ

22

bv'

σ

v

,

where

a(t),b(t)

are the coefficients of

X(t)=a(t)

σ

u

+b(t)

σ

v

.