Wednesday, July 6, 2022

An Introduction To Differential Geometry (Part 5)

 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  3  we have a system of coordinates (x 1x 2 , x 3 ).  Then any point in space is uniquely identified by its position vector;  x =  (x 1x 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 ,  2 :

 

(1)               x ( u12=

 

{ x 1  ( u12 ),  x 2 ( u12 ) , x 3 ( u12 ) }

 

With the two angular variables  u12  defined in a bounded domain B of the

u1  2 -plane   Then by eqn (1) to any point ( u12 )   in B  there is associated in  R 3  a  point  with position vector x (u12). Then the point set in  R 3  obtained when u12  vary is called a parametric representation of the set and u12   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 / 2  ) 

( x 2 / u1     x 2 / 2

( x 3 / u1     x 3 / 2

Partial derivatives of x ( u13 )  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 ,  2 ) =   (u1 ,  2,  0),   or polar (ii) x ( u1 ,  2 ) = u1 cos 2 , u1 sin 2, 0),  for which the respective matrices are:

(i) J =  

(1   ....0)

(0 ......1)

(0......0)

And (ii),  J =

cos 2 , -u1  sin 2 )

( sin 2 ,  ucos 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 ,  2 )  =  (r cos 2  cos u1r sin 2 sin u1 , r sin  2)

or writing out in terms of  x 1 ,  x 2 , x 3 .

x 1 = r cos 2  cos u1

x 2  =  r sin 2 sin u1 

x 3 = r sin  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 u 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:

J =

(-r cos 2  sin u1        -r sin 2 cos u1)
(r cos 2  cos  u1        -r sin 2 sin u1)

(0 .........................r cos 2 )

Example 2: A cone of revolution with apex x= (0, 0 ,0) and with x 3 - axis as axis of revolution i.e.


Can be represented in the form:

x 12 +   x 2 2) -  x 32    = 0  

And:   x 3    = +Ö (x 1 2 +   x 2 2)

represents one of two graphics, x 3  >  o   and  x 3    <  0 of the cone depending on the sign of the square root.  In the case shown then we have;  x 3  = aÖ (x 1 2 +   x 2 2).  The parametric representation can be written:

x ( u1,  2 )  =  ( u1  cos u2u1 sin 2 , a2)

Note the curves u1  = const. are curves parallel to the x 1 x 2 - plane while the curves u1  = const. are the generating straight lines of the cone.
---
N.B.  These need not necessarily be angles.

Suggested Problems:

1) For the given cone of revolution (Example 2) write the corresponding Jacobian (matrix).

2) Specify a parametric representation for a cylinder generated by a straight line, L, which moves along a curve,

C: x(s)  =   ( h 1 (s)h 2 (s)h 3 (s))

And which is always parallel to the x 3 - axis.  Give the corresponding matrix (Jacobian).

3) Find a parametric representation for each of the following:

a) Ellipsoid:  x u1,  2 )  =  
(a cos ucos u1 , b cos u2 sin u1c sin u 2)

b) Hyperbolic paraboloid:  x ( u1 ,  u 2)  = 
(a u1 cosh u2,  b u1 sinh u2,  (u1 ) 2)