18.7. ORIENTATION AND DEGREE 487

Proof: Let (a,b)≡∏kj=1 (a j,b j). Now from the area formula

∫r

ω ≡∫[a,b]

a(r (u))∂(xi1 , ...,xik

)∂ (u1, ...,uk)

(u)du

=∫[â,b̂]

a(r(r−1◦r̂

)(t)) ∂(xi1 , ...,xik

)∂ (u1, ...,uk)

(r−1 ◦r̂ (t)

)det(D(r−1◦r̂

)(t))

dt

=∫[â,b̂]

a(r(r−1◦r̂

)(t)) ∂(xi1 , ...,xik

)∂ (t1, ..., tk)

(t)dt ≡∫r̂

ω ■

An application of the area formula gives the following corollary. I will use r−1◦r̂ todenote a Lipschitz function which is one to one off a set S which equals the Lipschitzfunction r−1◦r̂ on SC. In particular if S has measure 0 and r−1◦r̂ is Lipschitz on SC, thenyou could extend to a Lipschitz function which would map S to a set of measure zero, thusbeing in the situation of this corollary.

Corollary 18.7.2 Suppose r−1◦r̂ is one to one off a set S⊆[â, b̂

]and that r−1 ◦r̂ (S)

has measure zero. Then the above would hold with no change.

Since this allows for Lipschitz functions, this is slightly more general than the usualsituation from Calculus even in one dimension. However, more can be said. Orientation isreally a statement about the degree of the map r−1◦r̂, a concept which makes perfect sensewithout any direct reference to differentiability.

Recall that with a smooth curve C having points p,q and a one to one map to this curve,there are two ways to move over the curve, from p to q or from q to p. One defines equiva-lence classes on the continuous mappings r which map a closed interval to C. Two of theser, r̂ are equivalent if r−1◦r̂ is increasing. It follows from the intermediate value theoremof Bolzano and a simple argument that this composition of maps is either increasing ordecreasing. Thus, from the theorem about differentiation of monotone functions,

(r−1◦r̂

)′is nonnegative a.e. exactly when the two parametrizations give the same orientation. In theabove, this is determined by det

(D(r−1◦r̂

)). In addition, this reduces to a topological no-

tion having to do with the degree. Instead of “increasing” we say that d(r−1 ◦r̂,Ω,(a,b)

)is 1. The notion of “increasng” is not available.

Recall that from Proposition 15.6.7, Corollary 15.6.6, d(r−1◦r̂,

(â, b̂

),(a,b)

)is ei-

ther 1 or −1. This is the topological degree of the mapping r−1◦r̂ which is constant onthe connected component (a,b) of Rk \r−1 ◦r̂

(∂

([â, b̂

])). Suppose then that this de-

gree d(r−1◦r̂,

(â, b̂

),y)

is 1. Then from Proposition 17.7.4 on Page 469, it follows thatwhenever f ∈Cc (a,b)∫

f (u)d(r−1◦r̂,Ω,u

)du =

∫(â,b̂)

f(r−1 ◦r̂ (t)

)det(D(r−1◦r̂

)(t))

dt

You could take an increasing sequence fn (u)→X(a,b) (u) of the above sort. From thearea formula, ∫

fn (u)du =∫(â,b̂)

f(r−1 ◦r̂ (t)

)∣∣det(D(r−1◦r̂

)(t))∣∣dt