# Maping from R2 to R3 torus, finding max min and saddle points

• Daveyboy
In summary, the conversation is about a mapping function of several variables, where the goal is to find local min, max, and saddle points. The solution involves finding four points that correspond to (s,t) values and using the second derivative test on each component of the function. However, the results are not as expected, with zeros appearing in all partial derivatives. The speaker is seeking advice on how to fix this issue.

## Homework Statement

this is from ch9 (functions of several variables)of baby rudin

a,b real b>a>0 define a mapping f=(f1,f2),f3) of R2 into R3 by
f1(s,t)=(b+acos(s))cos(t)
f2(s,t)=(b+acos(s))sin(t)
f2(s,t)=asin(s)
I showed that there are exactly 4 points p in K=image(f) such that

I am having trouble finding which of these points are local min, max, and saddle points.

## The Attempt at a Solution

The points in question correspond to the touple (s,t) when s=(pi)k and t=(pi)j k,j integers.

the points being +/-b +/-a

Okay great, I do not have any developed criteria for the second derivative test, I have no Hessin matrix to work with. I can develop that in my problem, but that seems rather complicated, I have been trying for a while now, and am having trouble.

after applying the second derivative test to each component of f I find that at each point of the second derivative of f2 and f3 that the value is zero. So I have zeros at all point of the partial derivatives which is bad because I should have 2 saddle points 1 max and one min. Even more confusing is that in f1 I get two min and two max, no saddle points...
Any ideas