Furthermore, the new composite path and the initial path form a triangular region, called geodesic triangle, which satisfies the local

Gauss-Bonnet Theorem. This theorem serves as guidance for selecting a better replanned path and terrain at the planning stage.

The classical

Gauss-Bonnet Theorem implies that the only compact Riemann surface with positive curvature is the Riemann sphere.

Two new sections look at a recent development concerning the

Gauss-Bonnet theorem and scalar curvature for curved noncommutative tori, and Hopf cyclic cohomology.

Chapter 5 studies the intrinsic geometry of n-dimensional submanifolds in pseudo-Euclidean m-dimensional space and Chapter 6 establishes the required machinery and gives the proof of the

Gauss-Bonnet theorem. Finally, Chapter 7 is dedicated to various aspects of extrinsic geometry, such as Frenet formulas for curves, ruled and developable surfaces, the shape operator, principal curvatures and curvature lines, etc.

Hence, in the compact case, equation (2) reduces to[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which is the well known

Gauss-Bonnet theorem for a Riemannian metric on the torus [T.sup.2].

Topics increase in mathematical complexity as the book progresses to encompass dozens of examples including a derivation of the Euler-Lagrange equation, heat flow and analytic functions, and a bicycle wheel and the

Gauss-Bonnet theorem.

Consequently, the contents are accessible to a wider audience and can be used to prepare students for the study of the Divergence Theorem, Green's Theorem, and even

Gauss-Bonnet Theorem in Differential Geometry.

(Dominic Voyer); (15) Probabilities / Probabilites (Egan Chernoff and Annie Savard); (16) First Steps toward an Archeology of Gesture in Graphing (Susan Gerofsky); (17) Function Modelling Using Secondary Data from Statistics Canada's E-STAT Database (Jennifer Hall); (18) Explanation and Proof in Mathematics and Mathematics Education (Gilla Hanna, Ella Kaye, and Riaz Saloojee); (19) Exploring Our Embodied Knowing of the

Gauss-Bonnet Theorem: Barn-Raising and Endo-Pentakis-Icosi-Dodecahedron (Eva Knoll); (20) Mathematical Biography (John Grant McLoughlin); (21) This Is Mathematics; This Isn't Mathematics; But That...

Along the way he covers the geometry of curves, surfaces, curvatures, constant mean curvature surfaces, geodesics, metrics, isometries, holonomy and the

Gauss-Bonnet theorem, the calculations of variations and geometry, and higher dimensions, just for fun.

This is the famous

Gauss-Bonnet theorem for compact surface.

The crucial fact here is the

Gauss-Bonnet theorem 1.5/1.8, which enables us to fulfill the hypothesis of the uniqueness theorem 3.2.

Among the topics are nearby cycles and periodicy in cyclic homology, the

Gauss-Bonnet theorem for the noncommutative two torus, zeta phenomenology, absolute modular forms, the transcendence of values of transcendental functions at algebraic points, and the Hopf algebraic structure of perturbative quantum gauge theories.