Multivariate Calculus and Geometry

✍ Scribed by Seán Dineen (auth.)

• Publisher: Springer-Verlag London
• Year: 2014
• Tongue: English
• Leaves: 256
• Series: Springer Undergraduate Mathematics Series
• Edition: 3
• Category: Library

✦ Synopsis

Multivariate calculus can be understood best by combining geometric insight, intuitive arguments, detailed explanations and mathematical reasoning. This textbook has successfully followed this programme. It additionally provides a solid description of the basic concepts, via familiar examples, which are then tested in technically demanding situations.

In this new edition the introductory chapter and two of the chapters on the geometry of surfaces have been revised. Some exercises have been replaced and others provided with expanded solutions.

Familiarity with partial derivatives and a course in linear algebra are essential prerequisites for readers of this book. Multivariate Calculus and Geometry is aimed primarily at higher level undergraduates in the mathematical sciences. The inclusion of many practical examples involving problems of several variables will appeal to mathematics, science and engineering students.

✦ Table of Contents

Front Matter....Pages i-xiv
Introduction to Differentiable Functions....Pages 1-12
Level Sets and Tangent Spaces....Pages 13-23
Lagrange Multipliers....Pages 25-34
Maxima and Minima on Open Sets....Pages 35-45
Curves in $${\mathbb {R}}^n$$ ....Pages 47-53
Line Integrals....Pages 55-67
The Frenet–Serret Equations....Pages 69-81
Geometry of Curves in $${\mathbb R}^3$$ ....Pages 83-92
Double Integration....Pages 93-102
Parametrized Surfaces in $${\mathbb R}^3$$ ....Pages 103-120
Surface Area....Pages 121-134
Surface Integrals....Pages 135-147
Stokes’ Theorem....Pages 149-159
Triple Integrals....Pages 161-178
The Divergence Theorem....Pages 179-191
Geometry of Surfaces in $${\mathbb {R}}^3$$ ....Pages 193-205
Gaussian Curvature....Pages 207-215
Geodesic Curvature....Pages 217-227
Back Matter....Pages 229-257

✦ Subjects

Mathematics, general