The ABC’s of Calculus – Multi Variable

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The ABC’s of Calculus (Volume 2)

Angelo B. Mingarelli

About the Book

Multivariable or Volume 2

Table of Contents

1. Vectors in the Plane
   1. Vectors in the Plane
   2. Multiplication of a Vector by a Scalar
   3. Vectors and Points
   4. Sums and Differences of Vectors
   5. The Inclination of a Vector
   6. Vectors in R³ (Three Dimensions)
   7. The Dot or Scalar Product of Two Vectors
   8. Direction Cosines and Unit Vectors
   9. The Vector or Cross Product of Two Vectors
   10. The Geometry of Cross-products
   11. Applications
2. Curves and Surfaces
   1. Straight Lines
   2. Planes in Space
   3. The Geometry of Planes
   4. The Angle Between Two Planes
   5. Equations of Lines in Space
      1. The Parametric Equations of a Line Segment
   6. Rotations in the Plane
      1. Coordinate Changes for an Active Rotation
      2. Coordinate Changes for a Passive Rotation
      3. Transformations With a Translated Origin
   7. Parametric Representation of Curves in R²
      1. Parametric Forms of Curves Defined by Functions
      2. The Parametric Form of Circles
      3. The Parametric Form of an Ellipse
      4. The Parametric Form of General Conic Sections
      5. The Parametric Form of Other Curves
      6. Eliminating the Parameter
   8. Sketching Plane Parametric Curves
   9. Applications to Area Problems
   10. The Length of a Parametric Curve
   11. Polar Coordinates
       1. Polar Plots
   12. Describing Regions Defined by Polar Coordinates
   13. Area of Regions in the Polar Plane
       1. The Area Enclosed by Polar Curves
   14. Length of Curves in the Polar Plane
3. Vector Calculus
   1. Limits and Continuity
      1. Functions of Three or More Variables
      2. Discontinuity at a Point
   2. Partial Derivatives
      1. Functions of Three or More Variables
   3. Higher Order Partial Derivatives
   4. Differentiability of Functions of Two Variables (Optional)
      1. Functions of Three or More Variables
   5. Directional Derivatives and Gradients
      1. Functions of Three Variables
      2. A Mean Value Theorem for Functions of Two Variables (Optional)
   6. The Chain Rule in R² or R³
      1. The General Form of the Chain Rule (Optional)
      2. Implicit Differentiation
      3. Tangent Planes and Normal Lines
   7. Conservative Fields
      1. Finding a Scalar Field f Such That ∇f = F
4. Line Integrals
   1. Line Integrals With Respect to Arc Length
   2. Line Integrals of Vector Fields
   3. Line Integrals of Conservative Vector Fields
5. Double Integrals
   1. Describing Planar Regions
   2. The Definition of a Double Integral
      1. Iterated Integrals
      2. Finding R from the Knowledge of an Iterated Integral
   3. Volume Under a Surface
   4. Change of Variables in Double Integrals
   5. Three Dimensional Plots
      1. Contour Lines
      2. Enumeration of Quadric Surfaces
   6. Parametric Equations of Surfaces
6. The Three Big Theorems
   1. Surface Integrals
   2. Some Applications of Surface Integrals
   3. Green’s Theorem
   4. Stokes’ Theorem
   5. Triple Integrals
   6. Change of Variables in Triple Integrals
   7. Solids in Spherical and Cylindrical Coordinates
   8. The Divergence Theorem
7. Applications
   1. Taylor Polynomials
   2. Maxima and Minima
   3. Lagrange Multipliers
   4. Volumes of Solids of Revolution
   5. Centroids and Centers of Mass
   6. Area of a Surface
   7. Coordinate Free Form of the Divergence
   8. Sources and Sinks
      1. Proof of a Mean Value Theorem for Triple Integrals
   9. Coordinate Free Form of the Curl
   10. Maxwell’s Equations
   11. Continuity Equations

Teacher Resources/Student Resources

Teacher resources may include a question bank in the form of multiple choice questions covering every section of the text along with a Comprehensive Solutions Manual (also available to students).

Philosophy

The idea behind this text is to introduce students in the Sciences or Engineering to Calculus sequentially and by means of many solved examples. It has been used by tens of thousands of students over a period of 20 years and the text is reviewed and updated every year by students through in class comments and correspondence and their names are included in perpetuity in subsequent editions. The author simplifies the exposition of all the material to a point where it all appears quite normal and easy to read. Almost all the steps in reaching a solution to a given problem are outlined and this is one of the major strengths of this text.

Online Ebook

The ebook is available for lease only at https://books.mingarelli.com for a period of 120 days (the duration of a standard semester in North American colleges and universities) and all for a low price of $50 CAD. This textbook uses  hypertext for easy access and reference purposes and notes can be taken and stored online for easy retrieval later on.

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About the Author

Angelo B. Mingarelli, Ph.D. (Toronto), is Professor of Mathematics at Carleton University, Canada where he’s been an active teacher and researcher for the past 25 years. He is the author of over 100 scientific articles and 6 books two of which are educational, the rest being research oriented. He was awarded a Faculty of Science Teaching Award on two occasions as well as a   University-wide Teaching Achievement Award. Prior to joining Carleton, he was Professor of Mathematics at the University of Ottawa for 12 years. As an academic he was President of the Carleton University Academic Staff Association (a certified trade union in Ontario) for five terms and served on its Board for 10 years. As well, he sat as a faculty member on the Carleton University Board of Governors for a three year term. Among his extracurricular interests one finds his passion in creating art for sentient AI beings.