An elementary treatise on elliptic functions by Arthur Cayley

By Arthur Cayley

This quantity is made out of electronic pictures from the Cornell collage Library old arithmetic Monographs assortment.

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5. Differentiable and Lipschitz maps ✐ 17 then the divergence of φ at x is the real number div φ(x) := n � Di fi (x) i=1 where Di := ∂/∂ξi is the usual partial derivative operator . Let E ⊂ Rn be any set, and let φ : E → Rm . The Lipschitz constant of φ is the extended real number �� � � �φ(x) − φ(y)� Lip φ := sup : x, y ∈ E and x �= y . |x − y| When Lip φ < ∞, the map φ is called Lipschitz . If Ω ⊂ Rn is an open set, we call a map φ : Ω → Rm locally Lipschitz whenever the restriction φ � U is Lipschitz for each open set U � Ω.

Proof. Let K ⊂ Rn be a compact set containing all figures Ai , and let c = sup Hn−1 (∂Ai ). 32] to find a vector field w ∈ C 1 (Rn ; Rn ) such that �v − w�L∞ (K;Rn ) ≤ ε. 7, �� � � � � � n−1 � �F (Ai � Aj )� ≤ � (v − w) · ν dH Ai �Aj � � ∂(Ai �Aj ) � �� � � w · νAi �Aj dHn−1 �� + �� ∂(Ai �Aj ) �� � � � � � div w(x) dx�� ≤ εHn−1 ∂(Ai � Aj ) + �� Ai �Aj ≤ 2cε + �div w�L∞ (K) |Ai � Aj |; since ∂(Ai � Aj ) ⊂ ∂Ai ∪ ∂Aj . 1, � �� � � � �� �F (Ai ) − F (Aj )� = �� F (Ai � Aj ) + F (Ai ⊙ Aj) − F (Aj � Ai ) + F (Aj ⊙ Ai ) �� � � � � ≤ �F (Ai � Aj )� + �F (Aj � Ai )� � � ≤ �div w�L∞ (K) |Ai � Aj | + |Aj � Ai | + 4cε = �div w�L∞ (K) |Ai � Aj | + 4cε.

Cp , xp ) such that [P ] = A and � �� � � � p n−1 � � f (xi )|Ci | − v · νA dH �<ε. � i=1 Proof. 3, the flux � F : B �→ ∂A ∂B v · νB dHn−1 is defined on the family of all figures B ⊂ A. 2, there are numbers 0 ≤ sk < 1 and disjoint, possibly empty, sets Ek ⊂ A such that �∞ v is pointwise Lipschitz in A − i=1 Ek , and for k = 1, 2, . . , the following conditions hold: (i*) Hn−1+sk (Ek ) < ∞, and Hsk v(x) < ∞ for each x ∈ Ek ; (ii) Hn−1+sk (Ek ) > 0 implies Hsk v(x) = 0 for each x ∈ Ek . 3.

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