In the past decade there has been a significant change in the freshman/sophomore mathematics curriculum as taught at many， if not most， of our colleges. This has been brought about by the introduction of linear algebra into the curriculum at the sophomore level. The advanta8;es of using linear algebra both in the teaching of differential equations and in the teaching of multivariate calculus are by now widely recognized. Several textbooks adopting this point of view are now available and have been widely adopted. Students completing the sophomore year now have a fair preliminary understanding of spaces of many dimensions.
Chapter 1
Graphs and Level Sets
Chapter 2
Vector Fields
Chapter 3
The Tangent Space
Chapter 4
Surfaces
Chapter 5
Vector Fields on Surfaces; Orientation
Chapter 6
The Gauss Map
Chapter 7
Geodesics
Chapter 8 Chapter 1
Graphs and Level Sets
Chapter 2
Vector Fields
Chapter 3
The Tangent Space
Chapter 4
Surfaces
Chapter 5
Vector Fields on Surfaces; Orientation
Chapter 6
The Gauss Map
Chapter 7
Geodesics
Chapter 8
Parallel Transport
Chapter 9
The Weingarten Map
Chapter 10
Curvature of Plane Curves
Chapter 11
Arc Length and Line Integrals
Chapter 12
Curvature of Surfaces
Chapter 13
Convex Surfaces
Chapter 14
Parametrized Surfaces
Chapter 15
Local Equivalence of Surfaces and Parametrized Surfaces
Chapter 16
Focal Points
Chapter 17
Surface Area and Volume
Chapter 18
Minimal Surfaces
Chapter 19
The Exponential Map
Chapter 20
Surfaces with Boundary
Chapter 21
The Gauss-Bonnet Theorem
Chapter 22
Rigid Motions and Congruence
Chapter 23
Isometries
Chapter 24
Riemannian Metrics
Bibliography
Notational Index
Subject Index
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