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strictly increasing lists of indices, and if this is done, you can always have only C (p,k) =p!

k!(p−k)! terms in the sum where this denotes combinations of p things taken k at a time.It will always be assumed that aI is as smooth as desired to make everything work. As

to r ∈C1 ([a,b] ,Rp) or Lipschitz, it is not required to have DrI (u) be nonzero for any I.This means r ([a,b]) can be various sets which have points and edges.

Note that if p < k, then the functional ∑I aI (x)dxI should equal the zero functionbecause you would have xp+1 = · · · = xk = 0 and so there would be at least one row ofzeros in the above determinant. Thus, I will suppose that k ≤ p in what follows.

Example 18.0.3 Consider the ball B(0,r). Spherical coordinates are r = (x,y,z) where

x = ρ sin(φ)cos(θ) , y = ρ sin(φ)sin(θ) ,z = ρ cos(φ)

Let (ρ,φ ,θ) ∈ [0,r]× [0,π]× [0,2π] which is a box like what was just described. Thismapping is C1 and onto the ball. However, it is clearly not one to one. However, r is one toone on (0,r]× [0,π]× [0,2π) off a closed set S of m3 measure zero. Also r (S) has measurezero by Sard’s theorem. Recall from calculus that ∂ (x,y,z)

∂ (ρ,φ ,θ) = ρ2 sinφ which is positive ifρ > 0 and θ ∈ (0,π).

In a similar manner, you could obtain a solid ellipse.

x = asin(φ)cos(θ) , y = bsin(φ)sin(θ) ,z = ccos(φ)

for a,b,c > 0.

Example 18.0.4 Consider the boundary of the ball B(0,r). Spherical coordinates are

x = r sin(φ)cos(θ) , y = r sin(φ)sin(θ) ,z = r cos(φ)

Let (φ ,θ)∈ [0,π]× [0,2π] which is a box like what was just described. This mapping is C1

and onto the boundary of this ball. It is clearly not one to one. However, off a set S of m2measure zero, the mapping is indeed one to one on what remains of this box and r (S) hasH 2 measure zero.

∂ (x,y)∂ (φ ,θ)

=

∣∣∣∣ r cosφ cosθ −r sinφ sinθ

r cosφ sinθ r sinφ cosθ

∣∣∣∣= r2 cosφ sinφ

∂ (x,z)∂ (φ ,θ)

=

∣∣∣∣ r cosφ cosθ −r sinφ

r cosφ sinθ 0

∣∣∣∣= r2 cos(φ)sin(φ)sin(θ)

∂ (y,z)∂ (φ ,θ)

=

∣∣∣∣ −r sinφ sinθ −r sinφ

r sinφ cosθ 0

∣∣∣∣= r2 sin2 (φ)cos(θ)

Similarly, you could obtain the boundary of an ellipse in the same way.

Example 18.0.5 One can obtain the cylinder of radius r of height h as follows.

x = r cos(θ) ,y = r sin(θ) ,z

where (θ ,z) ∈ [0,2π]× [0,h] a box. The mapping is clearly C1 but is not one to one.

More generally,