Readings for Math 2374

IT Multivariable Calculus and Vector Analysis

These are web readings designed to introduce some topics in Multivariable Calculus and Vector analysis. Many readings were designed to be read before students attended lecture, so they are intended to be somewhat readable introductions to the basic ideas. The readings err on the side of qualitative description rather than getting all technical details precise.

You can also view the readings organized by lecture as we present them in Math 2374.

1. Introduction to the online readings

An introduction
Interactive concept-visualization tools

2. Geometry in higher dimensions

3. Differential Calculus

3a. The partial derivative

Introduction to partial derivatives
Partial derivatives by limit definition
Partial derivative examples

3b. The derivative and differentiability

Differentiability in higher dimensions
More on differentiability
Differentiability examples

3c. The chain rule, directional derivative and gradient

The chain rule
Chain rule examples
The directional derivative and the gradient
Directional derivative and gradient examples

3d. Applications of differential calculus

Taylor's theorem
Taylor polynomial example
Local extrema
Extrema examples

4. Integral calculus

4a. Double integrals

Double integrals
Double integrals as iterated integrals
Double integrals examples
Double integrals as volume or area
Change order of integration examples
Area of region example
Changing order of integration is not always feasible

4b. Triple Integrals

Triple integrals
Triple integral examples

4c. Changing variables

Change of variables in double integrals
Change of variables examples
Change of variables in triple integrals
Change of variables examples

5. Vector calculus

5a. Vector field basics

Vector fields
The idea of divergence and curl
The components of the curl
More details on the components of the curl
Divergence and curl notation
Divergence and curl example
Subtleties about divergence
Subtleties about curl

5b. Parametrized curves

Parametrized curves and their derivatives
Tangent lines to parametrized curves
Tangent line to parametrized curve examples
Physical interpretation of a parametrization and its derivative
The length of a path
Length of a path examples

5c. Parametrized surfaces

Parametrized surfaces
Parametrized surface examples
Surface area of parametrized surfaces
Surface area calculation
Normal vector of parametrized surfaces
Orienting surfaces
A surface that is not orientable
Surface area example

5d. Path integrals and line integrals

Path integral of a scalar-valued function
Path integral examples
Line integral of a vector field
Alternate notation for line integrals
Line integrals as circulation
Line integral examples

5e. Surface integrals

Surface integrals
Surface integral examples

6. The fundamental theorems of vector calculus

6a. Green's theorem

The idea behind Green's theorem
Other ways of writing Green's theorem
Green's theorem with multiple boundary components
Using Green's theorem to find area
Sketch of proof for circulation per unit area
Green's theorem examples

6b. The theorem for conservative vector fields (a.k.a. the fundamental theorem of calculus for line integrals)

Path-independent or conservative vector fields
An example of a conservative vector field
Path-independence implies no circulation
Understanding the conditions for path-independence
Finding the potential function for path-independent vector fields
Path-independence example in three dimensions

6c. Stokes' theorem

The idea of Stokes' theorem
Stokes theorem examples

6d. The divergence theorem (a.k.a Gauss's theorem)

The idea of the divergence theorem
Divergence theorem examples

7. Review material

The integrals
The fundamental theorems
Length, area, and volume factors

© Copyright 2004-2007 Duane Nykamp. This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 2.5 License. Though not required, I'm always interested in hearing from people who want to use this work. I ask that you obtain permission before making a derivative work as this material is typically undergoing revision.

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