Calculus in 3D Geometry vectors and multivariate calculus 1st ediiton by Nitecki ISBN 1470443600 ‎978-1470443603

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ISBN 10: 1470443600

ISBN 13: ‎ 978-1470443603

Author: Nitecki

 Calculus in 3D is an accessible, well-written textbook for an honors course in multivariable calculus for mathematically strong first- or second-year university students. The treatment given here carefully balances theoretical rigor, the development of student facility in the procedures and algorithms, and inculcating intuition into underlying geometric principles. The focus throughout is on two or three dimensions. All of the standard multivariable material is thoroughly covered, including vector calculus treated through both vector fields and differential forms. There are rich collections of problems ranging from the routine through the theoretical to deep, challenging problems suitable for in-depth projects. Linear algebra is developed as needed. Unusual features include a rigorous formulation of cross products and determinants as oriented area, an in-depth treatment of conics harking back to the classical Greek ideas, and a more extensive than usual exploration and use of parametrized curves and surfaces. Zbigniew Nitecki is Professor of Mathematics at Tufts University and a leading authority on smooth dynamical systems. He is the author of Differentiable Dynamics, MIT Press; Differential Equations, A First Course (with M. Guterman), Saunders; Differential Equations with Linear Algebra (with M. Guterman), Saunders; and Calculus Deconstructed, AMS.

Calculus in 3D Geometry vectors and multivariate calculus 1st Table of contents:

• • Overview of Multivariable Calculus
  • Importance of Vectors in 3D Geometry
  • Basic Concepts of 3D Coordinate Systems

• Vectors and Vector Operations

  • Introduction to Vectors
  • Vector Addition and Scalar Multiplication
  • Dot Product and Cross Product
  • Applications of Vectors in 3D Geometry

• Vector Functions and Curves in 3D

  • Parametric Equations of Curves
  • Velocity and Acceleration Vectors
  • Tangent Vectors and Curvature

• Partial Derivatives

  • Functions of Several Variables
  • Concept of Partial Derivatives
  • Gradient and Directional Derivatives
  • Higher-Order Partial Derivatives

• Multiple Integrals

  • Double Integrals
  • Triple Integrals
  • Applications of Multiple Integrals in 3D Geometry
  • Change of Variables: Polar, Cylindrical, and Spherical Coordinates

• Vector Calculus: Theorems and Applications

  • The Divergence and Curl of a Vector Field
  • Green’s Theorem, Stokes’ Theorem, and Divergence Theorem
  • Applications in Fluid Mechanics and Electromagnetism

• Surface and Line Integrals

  • Parametric Equations for Surfaces
  • Surface Integrals and Flux
  • Line Integrals and Work Done by a Force
  • Applications of Surface and Line Integrals

• Gradients, Divergence, and Curl

  • The Gradient of a Scalar Field
  • The Divergence of a Vector Field
  • The Curl of a Vector Field
  • Physical Interpretations in 3D Geometry

• Applications of Multivariate Calculus in 3D Geometry

  • Optimization in 3D: Extrema and Lagrange Multipliers
  • Curve and Surface Modeling in 3D
  • Applications in Physics, Engineering, and Economics

• Coordinate Systems in Multivariable Calculus

  • Cartesian, Cylindrical, and Spherical Coordinates
  • Transformations Between Coordinate Systems
  • Applications to Surface and Volume Integrals

• Advanced Topics in Multivariable Calculus (Optional)

  • Tensor Calculus
  • Differential Forms and Manifolds
  • Advanced Vector Analysis in 3D

• Conclusion

  • Summary of Key Concepts
  • The Role of Calculus in 3D Geometry and Real-World Applications

• Appendices

  • Mathematical Tables and Formulae
  • Suggested Problems and Exercises
  • Further Reading and Resources
  • Index

 

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