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