Functional Differential Geometry (True PDF) by Unknown

Author:Unknown
Language: eng
Format: epub

110

Chapter 7

Directional Derivatives

(define (tilted-path tilt)

(define (coords t)

((transform tilt) (up :pi/2 t)))

(compose (point S2-spherical)

coords

(chart R1-rect)))

A southward pointing vector, with components (up 1 0), is trans-

formed to an initial vector for the tilted path by multiplying by

the derivative of the tilt transform at the initial point. We then

parallel transport this vector by numerically integrating the dif-

ferential equations. In this example we tilt by 1 radian, and we

advance for π/2 radians. In this case we know the answer: by

advancing by π/2 we walk around the circle a quarter of the way

and at that point the transported vector points south:

((state-advancer (g (tilted-path 1) sphere-Cartan))

(up 0 (* ((D (transform 1)) (up :pi/2 0)) (up 1 0)))

pi/2)

(up 1.5707963267948957

(up .9999999999997626 7.376378522558262e-13))

However, if we transport by 1 radian rather than π/2, the numbers

are not so pleasant, and the transported vector no longer points

south:

((state-advancer (g (tilted-path 1) sphere-Cartan))

(up 0 (* ((D (transform 1)) (up :pi/2 0)) (up 1 0)))

1)

(up 1. (up .7651502649360408 .9117920272006472))

But the transported vector can be obtained by tilting the orig-

inal southward-pointing vector after parallel-transporting along

the equator:16

(* ((D (transform 1)) (up :pi/2 1)) (up 1 0))

(up .7651502649370375 .9117920272004736)

16A southward-pointing vector remains southward-pointing when it is parallel-transported along the equator. To do this we do not have to integrate the differential equations, because we know the answer.

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