By Vladimir G Ivancevic

This graduate-level monographic textbook treats utilized differential geometry from a contemporary clinical standpoint. Co-authored through the originator of the area s best human movement simulator Human Biodynamics Engine , a posh, 264-DOF bio-mechanical method, modeled through differential-geometric instruments this is often the 1st booklet that mixes smooth differential geometry with a large spectrum of purposes, from sleek mechanics and physics, through nonlinear keep an eye on, to biology and human sciences. The e-book is designed for a two-semester direction, which supplies mathematicians numerous functions for his or her conception and physicists, in addition to different scientists and engineers, a powerful concept underlying their versions.

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**Sample text**

A Riemannian manifold is an analytic manifold in which each tangent space is equipped with an inner product g = ·, · , in a manner concepts of scalar, vector and linear operator in a way that is independent of any chosen frame of reference. While tensors can be represented by multi-dimensional arrays of components, the point of having a tensor theory is to explain the further implications of saying that a quantity is a tensor, beyond that specifying it requires a number of indexed components. In particular, tensors behave in special ways under coordinate transformations.

5 Brane Dynamics . . . . . . . . . . 7 Application: Topological String Theory . . . . . 1 Quantum Geometry Framework . . . . . 2 Green–Schwarz Bosonic Strings and Branes . . 3 Calabi–Yau Manifolds, Orbifolds and Mirror Symmetry . . . . . . . . . . . 4 More on Topological Field Theories . . . . 5 Topological Strings . . . . . . . . . 6 Geometrical Transitions . . . . . . . 7 Topological Strings and Black Hole Attractors . 8 Application: Advanced Geometry and Topology of String Theory .

Algebraic varieties and schemes: An algebraic variety is glued together from affine algebraic varieties, which are zero sets of polynomials over algebraically closed fields. Schemes are likewise glued together from affine schemes, which are a generalization of algebraic varieties. Both April 19, 2007 16:57 14 WSPC/Book Trim Size for 9in x 6in ApplDifGeom Applied Differential Geometry: A Modern Introduction are related to manifolds, but are constructed using sheaves 13 instead of atlases. Because of singular points one cannot assume a variety is a manifold.