The kinematics of three-dimensional continuum bodies can be developed elegantly via the polar decomposition theorem and the "relative description of motion" (see ยง ยง6-9 of Noll [1], [21][22][23][24][25], and Chapter II of Truesdell [3].) In this approach coordinate systems play no essential role in the theoretical development; indeed, they need not be introduced at all. As a result, kinematical concepts are no longer shaded by swarms of indices, subscripts and superscripts, but are laid bare as notions of geometry.
The purpose of this note is to develop a parallel, coordinate-free treatment for the kinematics of surfaces. Being an extension of Noll's approach, * such a treatment will share whatever merit one would attribute to that approach after seeing it in action in the continuum mechanics of bodies. Moreover, for students who are already familiar * We called it "the Noll-Truesdell approach" in an earlier draught of this note. We are indebted to Professor C. Truesdell for pointing out our erroneous attribution and for granting us permission to quote from his letter (dated August 17, 1984) the following historical information: "References for use of a reference placement that varies arbitrarily with time are given in ยง96 of CFT [10] .... Perhaps the correct name to mention is Eckart. "Use of a placement infinitesimally different from the present one, usually interpreted as 'just before', goes back to Euler, Cauchy,. and Stokes. See CFT [10], ~82, 82A, 85, 86 .... In connection with hypo-elasticity I mentioned 'strain with respect to the immediately preceding state', cf. ยง304 of CFT [10] and the literature cited there. The topic I suggested to Noll for his thesis was to develop my work on hypo--elasticity, 1954.
"I do not recall any example of what you call 'the NolI-Truesdell approach' before the paper of Noll [1] in 1958, and I think you are safe in attributing the approach to him. I cannot recall having had any part in introducing or developing it. It was easy for me to adopt and use it because of my familiarity with Eckart's approach and with the implications of Euler's procedure, adopted by Cauchy, Stokes, Lamb, etc. Thus I cannot see that I deserve any credit.
"The co-ordinate-free approach and the use of the polar decomposition theorem comes from Noil's thesis [11] .... He was much influenced by the paper of Richter ..."
The polar decomposition theorem appears as Eq. (2.2) in Richter's paper on finite elasticity [12]. In Professor Truesdell's preface to his English translation of Richter's paper ([13], p. 16), he wrote: "This paper is the first to apply the concepts of modem linear algebra to the mechanics of continua .... Stress and deformation are envisaged directly as linear transformations of vectors, independent of co-ordinates and components.