Tensor Analysis Schaum Series Pdf 38

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For the sake of simplicity of analysis, we suppose that the curved surface that will be used in the remaining of this paper is two-dimensional manifold and locally Euclidean. Roughly stated, a curved surface is a two-dimensional manifold in the sense that, for the neighborhood of each point on the curved surface, there exists a one-to-one representation of each curved element onto a simply connected region of the Euclidean plane [46]. Moreover, the curved surface is locally Euclidean in the sense that the curved element can be represented as a small domain of the Euclidean space in a small neighborhood of any point [47]. Rigorous mathematical definitions will not be repeated here but can be found in the cited references. For example, the surface of revolution, or any surface that is isometric to the surface of revolution, is always manifold and locally-Euclidean. This supposition yields the direct consequences of the following properties of the surface that we frequently use in this paper. (i) The first is the existence of a pair of orthogonal curved axes at every point, which are referred to as a surface of an orthogonal net and (ii) the second is that scalars, vectors and tensors are locally continuous and differentiable.

where g is the determinant of the metric tensor gij. The variable u is an activator and represents the membrane potential, while the variable v is an inhibitor and represents the ion channel openness or the refractoriness. The following analysis is independent of choice of the functions F(u, v) and G(u, v) for the shape of the cardiac action potential. For example, we may use the following functions [56]:6 2b1af7f3a8