- #1
Fellowroot
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Homework Statement
Consider the function f= sin(4pix)cos(6pix) on torus T^2=R^2/Z^2
a) prove this is a morse function and calculate min, max, saddle.
b) describe the evolution of sublevel sets f^-1(-inf, c) as c goes from min to max
Homework Equations
grad(f)= <partial x, partial y>
show hessian matrix not equal to zero
The Attempt at a Solution
From what I understand
1st need to find critical points. so take grad and set equal to zero
2nd use hessian matrix with those critical values that i found before and see if non zero
BUT, i don't know what torus T^2=R^2/Z^2 looks like. What does the T^2 mean? I believe R^2/Z^2 is just the xy graph because z has been removed. so its like 3D but if remove z then 2D
so is this a square flat torus?
once I know the shape then I can do the part b part since all you have to do is fill the shape with "water" and see how the topology changes within the critical values.
So is this correct? Since its cos and sin how do i know which critical values to pick and are within the domain.