Advanced calculus demystified

Preface

Acknowledgments

1: Functions Of Multiple Variables

Functions

Three dimensions

Introduction to graphing

Graphing level curves

Putting it all together

Functions of three variables

Parameterized curves

Quiz

2: Fundamentals Of Advanced Calculus

Limits of functions of multiple variables

Continuity

Quiz

3: Derivatives

Partial derivatives

Composition and the chain rule

Second partials

Quiz

4: Integration

Integrals over rectangular domains

Integrals over nonrectangular domains

Computing volume with triple integrals

Quiz

5: Cylindrical And Spherical Coordinates

Cylindrical coordinates

Graphing cylindrical equations

Spherical coordinates

Graphing spherical equations

Quiz

6: Parameterizations

Parameterized surfaces

Importance of the domain

This stuff can be hard!

Parameterized areas and volumes

Quiz

7: Vectors And Gradients

Introduction to vectors

Dot products

Gradient vectors and directional derivatives

Maxima, minima, and saddles

Application: optimization problems

LaGrange multipliers

Determinants

Cross product

Quiz

8: Calculus With Parameterizations

Differentiating parameterizations

Arc length

Line integrals

Surface area

Surface integrals

Volume

Change of variables

Quiz

9: Vector Fields And Derivatives

Definition

Gradients, revisited

Divergence

Curl

Quiz

10: Integrating Vector Fields

Line integrals

Surface integrals

Quiz

11: Integration Theorems

Path independence

Green's theorem on rectangular domains

Green's theorem over more general domains

Stokes' theorem

Geometric interpretation of curl

Gauss' theorem

Geometric interpretation of divergence

Quiz

Final Exam

Answers To Problems

Index.