2-manifold.Parametrization/Integration (please, help with part b)

Let be the manifold in consisting of all points such that

and

Then $\partial M$ is the circle consisting of all points such that

and

The map , where given by

is local parametrization on that covers . Orient so that belongs to the orientation, and give the induced orientation.

a)What normal vector corresponds to the orientation of ?

b)Let be the form

Evaluate

directly.

c)Evaluate directly , by expressing it as an integral over the unit disk in the plane.

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Please, help with part b)!

Re: 2-manifold.Parametrization/Integration (please, help with part b)

Quote:

Originally Posted by

**vercammen**
b)Let

be the

form

Evaluate

directly.

c)Evaluate

directly , by expressing it as an integral over the unit disk

in the

plane.

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Please, help with part b)!

We have and so

.

Now for part c), I'd say use the Gauss theorem:

.