Basic Insights in Vector Calculus with a Supplement on Mathematical Understanding

Title Page
Copyright Page
Preface
	P.1 General context
	P.2 Beginning with the coordinate definition of the 3-d vector curl
	P.3 Beginning with descriptions of imagined paddle wheels and integral definitions of circulation
	P.4 Do vector fields represent force or velocity? Or other?
	P.5 Historical context and pedagogy
	P.6 Prerequisites, the lessons, and follow up
	P.7 Our experience
	P.8 Acknowledgments
Contents
1. Reviewing Some Calculus Essentials
	1.1 Rates of change in one variable
	1.2 The fundamental theorem of calculus in one variable
	1.3 Applications by which the theorem earns the name “fundamental”
	1.4 Linear approximation and area
	1.5 Intimations of Green’s theorem in one dimension
	1.6 Dot-product or scalar product: a solution to a problem about projection length, in coordinates
	1.7 Areas and volumes, in coordinates
	1.8 Vector cross-product: a mathematical solution to a problem about rotation, from classical physics
	1.9 The chain rule in one variable
	1.10 An elementary example of ratios in integration in one variable
	1.11 The general case: applications of the chain rule in integration in one variable
	1.12 The chain rule in two dimensions and one independent variable
	1.13 The chain rule in two dimensions and two independent variables
	1.14 Implications for integration over a region in two variables
Fluid Motion in 2 Dimensions
	2. 2-D Mass-Flow Velocity Fields
		2.1 Surface vector fields for fluid flow
		2.2 Water flows, streamlines and integral curves
		2.3 Mass flow rates
		2.4 Units for mass flow rates
	3. Circulation and Green’s Theorem
		3.1 Circulation (mass flow rate) along a parallel line segment
		3.2 Circulation (mass flow rate) of a constant velocity field along an arbitrary line segment in 2-d
		3.3 Circulation along a curve in 2-d
		3.4 Invariance of the circulation integral
		3.5 Abbreviated notation for the circulation integral
		3.6 Adding circulations from different curves
		3.7 Preparing for Green’s Theorem
		3.8 Green’s Theorem
		3.9 Rotation and circulation density
		3.10 A general case
	4. 2-D Flux and Divergence
		4.1 2-dimensional flux
		4.2 Complementarity of 2-d flux and 2-d circulation
		4.3 Divergence in 2-d and rate of relative change in areas
	5. Unifying 2-d Results
		5.1 Unifying results
		5.2 The divergence and the Jacobian
		5.3 The curl and the Jacobian
Fluid Motion in 3 Dimensions
	6. Flux and Divergence
		6.1 Beginnings
		6.2 Flux across a surface S
		6.3 Flux across a surface that bounds a volume
	7. Stokes’ Theorem
		7.1 Stokes’ Theorem: The Question
		7.2 Stokes’ Theorem: First approach: elementary derivation of “Green’s theorem” for a two-parameter surface
	8. Relative Change in Volumes and in Increments
		8.1 Rate of change of relative change in volume
		8.2 Rate of change of relative change of increments
		8.3 Pure rotation (Jv)A represents an isometry
		8.4 Flux density and relative change in volume
Supplement: Mathematical Understanding
	Introduction to the Supplement
	Part A Mathematical Understanding
		A.1 A diagram
		A.2 Some puzzles
		A.3 Descriptive and explanatory understanding, and judgment in mathematics
		A.4 Descriptive definition and explanatory definition
		A.5 Proofs
		A.6 Algebra from arithmetic, “and so on”: that is, sequences of “higher viewpoints”
		A.7 Correlations, concepts and other fruit of understanding
		A.8 The historical context for teachers (and scholars)
	Part B A Few Implications for Teaching
	Part C Observations Regarding Modern Mathematics Education
Index