By Percival Frost Joseph Wolstenholme

The Authors of the subsequent Treatise have endeavoured to give prior to scholars as complete a view of the topic as yes boundaries have allowed them to do. the need of those boundaries has built itself during getting ready the paintings in the course of a interval of 4 years. The learn of innumerable papers, via the main celebrated mathematicians of all international locations, has confident the authors that the topic is sort of inexhaustible, and that, to have taken care of all elements of it with whatever imminent to the fulness with which they've got handled the 1st element, may have swelled their paintings in a nervous share to what it has already attained.

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**Example text**

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.