By G. H. Hardy

There could be few textbooks of arithmetic as famous as Hardy's natural arithmetic. considering that its e-book in 1908, it's been a vintage paintings to which successive generations of budding mathematicians have grew to become first and foremost in their undergraduate classes. In its pages, Hardy combines the passion of a missionary with the rigor of a purist in his exposition of the basic principles of the differential and indispensable calculus, of the houses of endless sequence and of alternative subject matters regarding the suggestion of restrict.

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To discuss smoothness properties of 1i in J let us next show that both grad J T r and Yo" depend smoothly on J E E(M, IR n ). To approach our goal, we consider a smoothly parameterized family J(t) E E(M, IRn ) with t varying in IR. We assume that J(O) coincides with a fixed I E E(M, IRn ). 19) holds for any choice of X, Y E rT M. Since V (J (t)) is torsion-free for any t E IR, the following equation is valid for all X, Y E rT M: V(I)y(A(dJ(t), dI))X = V(I)x(A(dJ(t), dI))Y. 20) With these formulas we deduce immediately gradJ(t) T = A( dJ(t), dI)-1 .

The directions are tangent vectors to E(M, lRn). [3]), a tangent vector is thus nothing else but a function in COO(M, lRn) and vice versa. In the following we take F, which is an one-form on E(M, lRn), as a constitutive law. We do not discuss the question whether F characterizes the material fully or not. Throughout these not es we assurne that Fis smooth. To allow only internal physical properties of the material to enter F, we have to specify the constitutive law somewhat more precisely. Basic to this specification is the fact that these constitutive properties should not be affected by the particular location of the body in lR n.

F O shall denote the constitutive law on E(M, IRn ) determined by 1ia. hIRn - R* ha and 'IjJ show how the material forming the boundary of the body is affected by the fact that the boundary material is implemented into the body: Without loss of generality we may think of R* h a being an additive part of hIRn. This motivates us to write only h instead of hIRn in the sequel. (JIßM)h(dJ) + 'IjJ(dJ). 19) h is unique up to IRn-valued smooth maps of E(M, IRn ), and 'IjJ is unique. 20) Deformable Media 49 Eq.

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