Vector calculus /
Jerrold E. Marsden and Anthony J. Tromba.
- Sixth edition.
- New York : W.H. Freeman and Company, c2012
- xxv, 545 pages : illustrations ; 27 cm.
Includes bibliographical references and index.
1. The Geometry of Euclidean Space -- 1.1. Vectors In Two- and Three-Dimensional Space -- 1.2. The Inner Product, Length, and Distance -- 1.3. Matrices, Determinants, and the Cross Product -- 1.4. Cylindrical and Spherical Coordinates -- 1.5. n-Dimensional Euclidean Space -- Review Exercises for Chapter 1 -- 2. Differentiation -- 2.1. The Geometry of Real-Valued Functions -- 2.2. Limits and Continuity -- 2.3. Differentiation -- 2.4. Introduction to Paths and Curves -- 2.5. Properties of the Derivative -- 2.6. Gradients and Directional Derivatives -- Review Exercises for Chapter 2 -- 3. Higher-Order Derivatives: Maxima and Minima -- 3.1. Iterated Partial Derivatives -- 3.2. Taylor's Theorem -- 3.3. Extrema of Real-Valued Functions -- 3.4. Constrained Extrema and Lagrange Multipliers -- 3.5. The Implicit Function Theorem (Optional) -- Review Exercises for Chapter 3 -- 4. Vector-Valued Functions -- 4.1. Acceleration and Newton's Second Law -- 4.2. Arc Length -- 4.3. Vector Fields -- 4.4. Divergence and Curl -- Review Exercises for Chapter 4 -- 5. Double and Triple Integrals -- 5.1. Introduction -- 5.2. The Double Integral Over a Rectangle -- 5.3. The Double Integral Over More General Regions -- 5.4. Changing the Order of Integration -- 5.5. The Triple Integral -- Review Exercises for Chapter 5 -- 6. The Change of Variables Formula and Applications of Integration -- 6.1. The Geometry of Maps from R2 to R2 -- 6.2. The Change of Variables Theorem -- 6.3. Applications -- 6.4. Improper Integrals (Optional) -- Review Exercises for Chapter 6 -- 7. Integrals Over Paths and Surfaces -- 7.1. The Path Integral -- 7.2. Line Integrals -- 7.3. Parametrized Surfaces -- 7.4. Area of a Surface -- 7.5. Integrals of Scalar Functions Over Surfaces -- 7.6. Surface Integrals of Vector Fields -- 7.7. Applications to Differential Geometry, Physics, and Forms of Life -- Review Exercises for Chapter 7 -- 8. The Integral Theorems of Vector Analysis -- 8.1. Green's Theorem -- 8.2. Stokes' Theorem -- 8.3. Conservative Fields -- 8.4. Gauss' Theorem -- 8.5. Differential Forms -- Review Exercises for Chapter 8.
Vector Calculus helps students gain an intuitive and solid understanding of this important subject. The book's careful account is a contemporary balance between theory, application, and historical development, providing it's readers with an insight into how mathematics progresses and is in turn influenced by the natural world.