This is the pre-alpha of the multivariable calculus site. The new site is under development for use in all browsers, touchpads, and smartphones. It will be released in Beta by January 2014.

The pre-alpha of the site can be accessed at .  



Multivariable Calculus Online

(supported in part by National Science Foundation grant DUE-9950600)


1: Vector-Valued Functions

1.1 Vectors in 2 and 3 dimensions
1.2 The Dot Product
1.3 The Cross Product

1.4 Lines and Planes

1.5 Parametric Equations
1.6 Velocity and Acceleration

1.7 Speed and Arclength

1.8 Components of Acceleration

Summary and Review



2: Partial Differentiation

2.1 Functions of 2 Variables
2.2 Limits and Continuity
2.3 Partial Derivatives
2.4 Partial Differential Equations
2.5 Linearization and the Hessian 
2.6 The Chain Rule

2.7 Gradients and Level Curves

2.8 Optimization

2.9 Lagrange Multipliers

Summary and Review


3: Surfaces and Transformations

3.1 Coordinate Transformations
3.2 Polar Coordinates
3.3 The Jacobian
3.4 Surfaces
3.5 Curvilinear Coordinates

3.6 Tangent Planes

3.7 DIFF GEOM: Curves on Surfaces 

3.8 DIFF GEOM: Intrinsic Geometry 

 Summary and Review



4: Multiple Integrals

4.1 Iterated Integrals
4.2 Double Integrals
4.3 Applications of the Double Integral

4.4 Change of Variables

4.5 Double Integrals in Polar Coordinates 
4.6 Triple Integrals

4.7 Cylindrical and Spherical Coordinates

     Summary and Review



5: Fundamental Theorems

5.1 Vector Fields
5.2 Line Integrals
5.3 Potentials of Conservative Fields
5.4 Green's Theorem
5.5 Surface Integrals

5.6 The Divergence Theorem

5.7 Stoke's Theorem

5.8 Differential Forms

Summary and Review


Capstone: The Inverse Square Law

C.1. Uniform Circular Motion
C.2. Conservation Laws
C.3. Kepler's Laws
C.4. Satellites and Planets
C.5. Electric and Magnetic Fields

C.6. Maxwell's Equations

C.7. Special Relativity

C.8. General Relativity - An Overview




Individual Sections
Maple Worksheets
Online Tools

Answers to Selected Odd Exercises

The Entire Course





Instructor's Guide



Browser Considerations

Optimized for 1024 X 768 resolution 
Developed for Windows IE 6.0 or 7.0
Patch for Netscape and Mozilla
MacIntosh Users
MATHML Version -- Coming Soon   

Mathematica Notebooks -- Coming Soon   




Copyright 2001 by Jeff Knisley,  Dept. of Math., East Tennessee State University.  Free for individual use only.  No portion of this textbook may be altered or sold. 

Multivariable Calculus Online is adapted from the textbook Calculus: A Modern Approach by Kevin Shirley and Jeff Knisley.

Development of Multivariable Calculus Online was funded in part by National Science Foundation grant DUE-9950600.  


Many of the applets and images were prepared with Javaview, an excellent web-based visualization tool which is free for download and can be used with Maple 6 through 9. 

Additional applets were prepared with LiveGraphics3d, an excellent complement to Mathematica featuring parameterized graphics.

Sections written in LaTeX2e and converted to html with TTHGold.  Final editing was performed with Microsoft Frontpage and Eversoft 1stPage2000.  

A special thanks to my wife, Debra, for all her support, suggestions, guidance, and encouragement. 

Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the
views of the National Science Foundation.

Javaview Eversoft 1stPage2000.