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  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/ ¶ u_a¹  ¶ u _b³          

Or in the 2^nd case: 

x _ab  =   ¶ ² x/ ¶ u ^a ¶ u ^b          

Note there are three determinants of 2^nd 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:

J =

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

(0 .........................r cos u 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:

a ( x ₁² +   x ₂ ²) -  x ₃²    = 0  

And:   x ₃    = + a 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,  u ² )  =  ( u¹  cos u², u¹ sin u ² , au ²)

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.

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*  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,  u ² )  =  

(a cos u² cos u¹ , b cos u² sin u¹, c sin u ²)

b) Hyperbolic paraboloid:  x ( u¹ ,  u ²)  = 

(a u¹ cosh u²,  b u¹ sinh u²,  (u¹ ) ²)

 See Also:

 An Introduction To Cylinders, Quadric Surfaces And Their Analytic Geometry