Friday, August 12, 2022

Solutions To Differential Geometry Problems - Part 6

 1)  Find the first fundamental form of the cylinder: 

x ( u1 ,  ) =  ( h 1 (u1)h 2 (u1) 2)

2) Find the first fundamental forms corresponding to Cartesian and spherical coordinates in space.  

 3) What differential equation would give rise to the family of curves shown below?

Solutions:

1)  g 11 =  h'12   + h'22    and:  g 12 = 0,   g 22 = 1,  

and a prime denotes derivative with respect to u1.  Then the fundamental form is written:

ds2   =   ( h'12  + h'22 ) (du1)  (du2)2 

2)  Let  x1 x2 , x3  be Cartesian coordinates.  Then:

ds2   =   d x12  + dx22 +  dx32

Set:  x1  =   u1  cos ucos u3
 
x =   u1  cos usin u3

x =   u1  sin u

So we obtain spherical coordinates:  u,u2 ,u3

And then from ds2  for Cartesian coordinates, we obtain:

å3 a =1   [å 3 b =1   ( x a / ub )  dub  

+( du1)2  + (u1)2  (du2)2   +(u1)2  + cos 2 u (du

3)  For the differential equation: dy/dx = -x/4y  

we can sketch the curve which passes through the point (1,1)

Re-arrange the DE  to obtain: -4y dy = xdx

Integrate to get: -2y 2 = x2/2

Or: x
2/2 + 2y2 = c

From this one can substitute in a set of different values for c to generate the family of curves appropriate to the equation.

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