# Frankel 1.3 Problems

## (1)

What would be wrong in defining

$$||X||^2 = \sum_i (X_U^i)^2 \,\,\, ?$$

Defining the magnitude this way would lead to a result that depends on the choice of coordinates.

## (2)

Let the torus $$T^2$$ be embedded in $$\mathbb{R}^3$$ such that it lies flat atop the xy plane. The map $$F : T^2 \to \mathbb{R}^2$$ project points on the torus to the xy plane.  By inspection, the critical points of $$F$$ are those along the inner and outer circles in the middle of the torus. At these points the tangent space is perpendicular to the xy plane and therefore the differential $$F_*$$ is not onto.