Multivariable calculus and vector analysis

Readings organized by lecture

These are web readings designed to introduce some topics in multivariable calculus and vector analysis. Those readings marked with an asterisk (*) 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. The quizzes are available (on WebVista) only to students enrolled in Math 2374 as an incentive to read the initial readings before lecture.

You can also view the readings organized by topic.

Part 1

• Introduction
• Quiz 1: Practice Quiz (not graded)
• Interactive concept-visualization tools
• Parametrization of a line
• Matrices and determinants
• The cross product
• The component formula for the cross product
• The scalar triple product
• The relationship between determinants and area or volume
• Parametrization of a line examples
• Cross product examples
• Scalar triple product example

Part 2

• Forming planes*
• Quiz 2: Forming Planes quiz*
• A line or a plane or a point?
• Distance from point to plane
• Forming plane examples
• Distance from point to plane example
• Intersecting planes example

Part 3

• Multiplying matrices and vectors*
• Quiz 3: Matrix-vector multiplication quiz*
• Dot product in matrix notation
• Matrices and linear functions
• Examples of n-dimensional vectors
• Matrix and vector multiplication examples

Part 4

• Level sets*
• Quiz 4: Level sets quiz*
• Function notation
• Translation, rescaling, and reflection
• Quadric surfaces
• Level set examples

Part 5

• Introduction to partial derivatives*
• Quiz 5: Partial derivatives intro quiz*
• Partial derivatives by limit definition
• Partial derivative examples

Part 6

• Differentiability in higher dimensions*
• Quiz 6: Differentiability quiz*
• More on differentiability
• The linear approximation
• Differentiability examples

Part 7

• The chain rule*
• Quiz 7: Chain rule quiz*
• Chain rule examples
• Parametrized curves and their derivatives
• Tangent lines to parametrized curves
• Tangent line to parametrized curve examples

Part 8

• The directional derivative and the gradient*
• Quiz 8: Gradient and directional derivative quiz*
• Directional derivative and gradient examples

Part 9

• Double integrals*
• Quiz 9: Double integral quiz*
• Double integrals as iterated integrals
• Double integral examples

Part 10

• Quiz 10: Gradient and directional derivative review quiz*
• Double integrals as volume or area
• Change order of integration examples
• Area of region example
• Changing order of integration is not always

Part 11

• Triple integrals*
• Quiz 11: Triple integral quiz*
• Triple integral examples

Part 12

• The length of a path*
• Quiz 12: Length of path quiz*
• Physical interpretation of a parametrization and its derivative
• Length of a path examples

Part 13

• The idea of divergence and curl*
• Quiz 13: Divergence and curl quiz*
• Vector fields
• 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

Part 14

• Path integral of a scalar-valued function*
• Quiz 14: Path integral quiz*
• Path integral examples
• Line integral of a vector field
• Alternate notation for line integrals
• Line integrals as circulation
• Line integral examples

Part 15

• The idea behind Green's theorem*
• Quiz 15: Green's theorem quiz*
• 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

Part 16

• Path-independent or conservative vector fields*
• Quiz 16: Path-independence quiz*
• An example of a conservative vector field
• Path-independence implies no circulation
• Understanding the conditions for path-independence
• Finding the potential function
• Path-independence example in three dimensions

Part 17

• Change of variables in double integrals*
• Quiz 17: Change of variables 2D quiz*
• Double integral change of variables examples

Part 18

• Change of variables in triple integrals*
• Quiz 18: Change of variables 3D quiz*
• Spherical Coordinates
• Triple integral change of variables examples

Part 19

• Parametrized surfaces*
• Quiz 19: Parametrized surface quiz*
• Parametrized surface examples
• Parametrization of a plane
• Plane parametrization example

Part 20

• Surface area of parametrized surfaces*
• Quiz 20: Surface area quiz*
• Surface area calculation
• Surface area example
• Normal vector of parametrized surfaces
• Orienting surfaces
• A surface that is not orientable

Part 21

• Surface integrals*
• Quiz 21: Surface integral quiz*
• Surface integral examples

Part 22

• The idea of Stokes' theorem*
• Quiz 22: Stokes' theorem quiz*
• Stokes' theorem examples

Part 23

• Quiz 23: Bonus quiz*

Part 24

• The idea of the divergence theorem*
• Quiz 24: Divergence theorem quiz*
• Divergence theorem examples

Part 25

• Taylor's theorem*
• Quiz 25: Taylor theorem quiz*
• Taylor polynomial example

Part 26

• Local extrema*
• Quiz 26: Extrema quiz*
• Extrema examples

Part 27

• Quiz 27: Review quiz*
• The integrals
• The fundamental theorems
• Length, area, and volume factors