1 Show by a direct construction that two charts suce to cover the 2-sphere S

1 Show by a direct construction that two charts suce to cover the 2-sphere S

. Hint:

Let S

be dened by x

+ y

+ z

= 1 in the euclidean space and consider stereographic projections that map a point P on S

into a point (x, y) on the z = 0 plane (by straight lines passing through P and the north and south poles, respectively).

2 Consider a curve in R

given by

x(t) = cos

t, y(t) = cos t sin t, z(t) = sin t .

(a) Find the components of the tangent vector to this curve in the cartesian basis.

(b) Express the curve in the spherical coordinates and nd the components of the tangent vector in the spherical basis.

(c) Show by a direct calculation that the results of (a) and (b) are related by the trans- formation law for the vector.

3 A vector eld u

on R

has components u

= xy , u

= x

+ y

in the cartesian basis.

(a) Find the components of u

in the polar basis {∂

, ∂

} .

(b) The vecor v

has components v

= y

, v

= x

. Find the commutator [u, v].

4 Let g

be a metric on a manifold M (a nondegenerate symmetric tensor of type (0, 2)) (a) Show that around each point p ∈ M one can nd coordinates x

such that g

(p) is

diagonal with ±1 on the diagonal.

(b) Show that among the coordinates from (a), one can nd coordinates such that

∂g

∂x

(p) = 0.

Hint: compute the metric in the new coordinate system

y

(x

) = a

+ b

x

+ c

x

x

+ O(x

).

and choose the constants a

, b

, and c

suitably.

5 Given a covector A

, show that:

(a) ∂

A

does not transform as a tensor.

(b) ∂

A

does transform as a tensor.