In this instalment on the introductory aspects of differential geometry we examine its application to a few surfaces and also the associated Jacobians in the process. As usual we assume that for a 3-d system R 3 we have a system of coordinates (x 1 , x 2 , x 3 ). Then any point in space is uniquely identified by its position vector; x = (x 1 , x 2 , x 2 ).
Now, to deal with surfaces and curves on them we start with a real, single-valued vector function and parametric angle variables (using superscripts)* u1 , u 2 :
(1)
x ( u1 , u 2 ) =
{ x 1 ( u1 , u 2 ), x 2 (
u1 ,
u 2 ) , x 3
( u1 ,
u 2 ) }
With the two angular variables u1 , u 2 defined in a
bounded domain B of the
u1 u 2 -plane Then by eqn (1) to
any point ( u1 , u 2 ) in B there is associated in R 3 a point
with position vector x (u1 , u 2). Then the point set
in R 3 obtained when u1 , u 2 vary is
called a parametric representation of the set and u1 , u 2 are the parameters of
this representation.
In these descriptions we will
also make use of the Jacobian matrix of rank 2:
J =
(¶ x 1 /¶ u1 ¶ x 1 /¶ u 2 )
(¶ x 2 /¶ u1 ¶ x 2 /¶ u 2 )
(¶ x 3 /¶ u1 ¶ x 3 /¶ u 2 )
Partial derivatives of x ( u1 , u 3 ) can be denoted:
x 1 = ¶ x/ ¶ u1 And: x 13 = ¶ 2 x/ ¶ u1 ¶ u3
Which can also be more generally denoted:
x a = ¶ x/ ¶ u1
And:
x ab = ¶ 2 x/ ¶ ua1 ¶ u b3
Or in the 2nd case:
x ab = ¶ 2 x/ ¶ u a ¶ u b
Note there are three determinants of 2nd order in the Jacobian J which bear components of the vector products , i.e. of the vectors:
x 1 = ¶ x/ ¶ u1
and x 2 = ¶ x/ ¶ u2
These are different from the null vector IFF x 1 and x 2 are linearly independent vectors . The assumption that the matrix for Jacobian J is of rank 2 is thus the necessary and sufficient condition that said vectors are linearly independent.
Note the x 1 x 2 - plane can be represented in one of two forms, Cartesian (i): x ( u1 , u 2 ) = (u1 , u 2, 0), or polar (ii) x ( u1 , u 2 ) = ( u1 cos u 2 , u1 sin u 2, 0), for which the respective matrices are:
(i) J =
(1 ....0)
(0 ......1)
(0......0)
And (ii), J =
( cos u 2 , -u1 sin u 2 )
( sin u 2 , u1 cos u 2 )
(0 ....................0 )
Recall from Part 3, the sphere can be obtained by extending the circle by one dimension ( x 2 ), i.e. to obtain:
x 3 = + Ö ( r 2 - x 1 2 - x 2 2)
And depending on the choice of sign this is a representation of one of the two hemispheres, x 3 > 0 or x 3 < o. The parametric angular representation of the preceding can then be expressed:
x ( u1 , u 2 ) = (r cos u 2 cos u1, r sin u 2 sin u1 , r sin u 2)
or writing out in terms of x 1 , x 2 , x 3 .
x 1 = r cos u 2 cos u1
x 2 = r sin u 2 sin u1
x 3 = r sin u 2
The geometrical configuration is shown below:
The similarity to the celestial sphere of positional astronomy will not be lost on regular readers, i.e. the angular parameter u1 plays an analogous role to Right Ascension while u 2 plays a role analogous to the zenith distance (z) which is related to the coordinate of declination ( d = 90 - z). Via this analogy we can also see the corresponding Jacobian matrix at the poles is:
(r cos u 2 cos u1 -r sin u 2 sin u1)
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