Solutions To Differential Geometry Problems - Part 6

 1)  Find the first fundamental form of the cylinder: 

x ( u¹ ,  u ² ) =  ( h ₁ (u¹), h ₂ (u¹),  u 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)² +   (du²)² 

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

ds²   =   d x₁²  + dx₂² +  dx₃²

Set:  x₁  =   u¹  cos u² cos u³

 

x₂  =   u¹  cos u² sin u³

x₃  =   u¹  sin u³ 

So we obtain spherical coordinates:  u¹ ,u² ,u³

And then from ds²  for Cartesian coordinates, we obtain:

å³ _a =1   [å ³ _b =1   (¶ x _a /¶ u_b )  du_b ] ²  

+( du¹)²  + (u¹)²  (du²)²   +(u¹)²  + cos ² u²  (du³ ) ² 

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: x²/2 + 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.