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I need to show that if we put a closed curve on earth's surface, number of maxima plus number of minima inside it will be equal the number of saddle points plus 1.
A hint was to use Poincare-Hopf Theorem.
The problem with this is that Poincare-Hopf requires that on the boundaries the vector field should be pointing inwards.
For example, as in the image I attached, some vectors point inwards and some point outwards.
Also, what bothers me is that in the image that is attached, there is one maxima, one minima, but no saddle points (or maybe the drawing is decieving?), so what I was asked to show seems not hold.

Any hints or directions will be appreciated

A crater and a mountain.

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Right, to apply Poincare Hopf requires your curve to not pass through any critical points of the function. Consider the latitude function on the surface of the earth, and choose your circle to be a longitude line (passing through the north and south poles). In the open hemisphere bounded by the longitude line, there are no critical points at all of the lattitude function. In the picture you drew, there are critical points sitting along the curve (if $\nabla f$ points inward somewhere and outward somewhere else, it must be zero somewhere in between). –  Willie Wong Jul 2 '11 at 13:59
    
@Willie Got it, thank you. –  Artium Jul 2 '11 at 23:05

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up vote 3 down vote accepted

(I'm copying my comment to an answer since it looks like it addresses the OP's question, and this way we can get the question off the unanswered list.)

Right, to apply Poincare Hopf requires your curve to not pass through any critical points of the function. Consider the latitude function on the surface of the earth, and choose your circle to be a longitude line (passing through the north and south poles). In the open hemisphere bounded by the longitude line, there are no critical points at all of the latitude function. In the picture you drew, there are critical points sitting along the curve (if $\nabla f$ points inward somewhere and outward somewhere else, it must be zero somewhere in between).

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