Multivariable Advanced Calculus

362 LINE INTEGRALSLet x t ≡ tx 1 + (1 − t) x 2 and y t ≡ ty 1 + (1 − t) y 2 . Then x t < y t for all t ∈ [0, 1] becausetx 1 ≤ ty 1 and (1 − t) x 2 ≤ (1 − t) y 2with strict inequality holding for at least one of these inequalities since not both t and(1 − t) can equal zero. Now defineh (t) ≡ (ϕ (y t ) − ϕ (x t )) (y t − x t ) .Since h is continuous and h (0) < 0, while h (1) > 0, there exists t ∈ (0, 1) such thath (t) = 0. Therefore, both x t and y t are points of (a, b) and ϕ (y t ) − ϕ (x t ) = 0 contradictingthe assumption that ϕ is one to one. It follows ϕ is either strictly increasing orstrictly decreasing on (a, b) .This property of being either strictly increasing or strictly decreasing on (a, b) carriesover to [a, b] by the continuity of ϕ. Suppose ϕ is strictly increasing on (a, b) , a similarargument holding for ϕ strictly decreasing on (a, b) . If x > a, then pick y ∈ (a, x) andfrom the above, ϕ (y) < ϕ (x) . Now by continuity of ϕ at a,ϕ (a) = lim ϕ (z) ≤ ϕ (y) < ϕ (x) .x→a+Therefore, ϕ (a) < ϕ (x) whenever x ∈ (a, b) . Similarly ϕ (b) > ϕ (x) for all x ∈ (a, b).It only remains to verify ϕ −1 is continuous. Suppose then that s n → s where s nand s are points of ϕ ([a, b]) . It is desired to verify that ϕ −1 (s n ) → ϕ −1 (s) . If thisdoes not happen, there exists ε > 0 and a subsequence, still denoted by s n such that∣ ϕ −1 (s n ) − ϕ −1 (s) ∣ ≥ ε. Using the sequential compactness of [a, b] there exists a furthersubsequence, still denoted by n, such that ϕ −1 (s n ) → t 1 ∈ [a, b] , t 1 ≠ ϕ −1 (s) . Then bycontinuity of ϕ, it follows s n → ϕ (t 1 ) and so s = ϕ (t 1 ) . Therefore, t 1 = ϕ −1 (s) afterall. This proves the lemma. Now suppose γ and η are two parameterizations of the simple curve γ ∗ as describedabove. Thus γ ([a, b]) = γ ∗ = η ([c, d]) and the two continuous functions γ, η are oneto one on their respective open intervals. I need to show the two definitions of lengthyield the same thing with either parameterization. Since γ ∗ is compact, it followsfrom Theorem 5.1.3 on Page 88, both γ −1 and η −1 are continuous. Thus γ −1 ◦ η :[c, d] → [a, b] is continuous. It is also uniformly continuous because [c, d] is compact. LetP ≡ {t 0 , · · · , t n } be a partition of [a, b] , t 0 < t 1 < · · · < t n such that for L < V (γ, [a, b]) ,L