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source: James Cook 2017年1月16日

These are the Lectures from my section of Math 231 from the Spring 2017 Semester. While my notes are based on several calculus texts (Stewart, Thomas, Salas and Hille and Etgen etc.) these Lectures follow my notes primarily. I am doing my best to make the notes self-contained so there is no need to purchase additional materials. That said, it's probably wise to get an old edition of Salas and Hille or something for the sake of having many additional homework examples to ponder.

My understanding of multivariate calculus stems in part from my background in physics. I have much higher expectations for what multivariate calculus should encompass. On the other hand, I may spend a bit less time on certain issues of analysis than other math professors. I take up those issues in greater generality in the advanced calculus class (audience permitting).

Topics: vectors, calculus and geometry of paths, Frenet frames, limits of functions of several variables, partial derivatives, chain rules, integration over areas or volumes, optimization, Lagrange multipliers, line integrals, surface integrals, theorems of vector calculus including Green, Stokes and Divergence. Time permitting, the theory of scalar and vector potentials. We also hope to see how to present various constructions in noncartesian coordinate frames.

Multivariate Calculus: vectors, components, distance: 1-16-17 51:43

dot-product, 1-17-17, part 1 59:51

dot-product, 1-17-17, part 2 20:05

end of dot-product, start of cross-product, 1-18-17 47:50

cross product identities, planes, 1-19-17, part 1 59:51

cross product identities, planes, 1-19-17, part 2 8:51

curves and surfaces big picture, 1-20-17 51:37

graphing surfaces and parametrizations, 1-23-17 50:53

graphing surfaces and curvelinear coordinates, 1-24-17, part 1 59:51

graphing surfaces and curvelinear coordinates, 1-24-17, part 2 16:36

distance to pts, lines and planes, calculus of paths intro, 1-25-17 51:57

the way of calculus, or calculus of paths, 1-26-17,part 1 59:51

the way of calculus, or calculus of paths, 1-26-17,part 2 16:06

Isaac's Dizzy Dance 0:08

tangent lines, arclength, TNB intro, 1-27-17 51:17

Frenet Serret Equations, curvature, torsion, 1-30-17 50:49

motion, Kepler's Laws in brief, int. w.r.t. arclength, 1-31-17, part 1 59:51

motion, Kepler's Laws in brief, int. w.r.t. arclength, 1-31-17, part 2 12:57

find center of mass of variable mass helix, 2-1-17 35:17

review for Test 1, 2-6-17 45:03

directional derivative via the gradient, 2-13-17 46:52

gradient and contour plots, 2-14-17, part 1 59:51

gradient and contour plots, 2-14-17, part 2 13:50

discussion of Test 1, general concept of differentiation, 2-15-17 48:43

continuous diff dangers, chain rule, 2-16-17, part 1 59:51

more chain rules and discussion of tangent plane, 2-17-17 50:36

tangent planes from many viewpoints, 2-20-17 45:27

constrained partial differentiation, 2-21-17, part 1 59:51

constrained partial differentiation, 2-21-17, part 2 9:55

gradient in polar or spherical coordinates, 2-22-17 50:32

optimization, Lagrange Multipliers, 2-23-17, part 1 59:51

optimization, Lagrange Multipliers, 2-23-17, part 2 14:45

the quadratic form uber-example, power series, 2-24-17 49:45

second derivative test for several variables, 2-27-17 44:47

absolute extrema, 2-28-17, part 1 59:51

absolute extrema, 2-28-17, part 2 14:09

review for Test 2 on differentiation, 3-6-17 43:45

double integrals, 3-7-17, part 1 59:51

double integrals, 3-7-17, part 2 14:28

triple integrals in xyz, 3-8-17 50:42

integration in polars, 3-20-17 38:44

change of variables theorem, 3-21-17, part 1 59:51

change of variables theorem, 3-21-17, part 2 20:32

spherical volume element via wedges, integration, 3-22-17 48:15

integration, moment of intertia for solid sphere, 3-23-17, part 1 59:51

integration, moment of intertia for solid sphere, 3-23-17, part 2 12:25

integration examples, 3-24-17 49:10

integration advice, 3-27-17 45:37

the three derivatives of vector calculus, 3-28-17, part 1 59:51

the three derivatives of vector calculus, 3-28-17, part 2 12:26

review for Test 3, 4-3-17 30:42

definition of line integral, 4-5-17 47:05

line integral calculation and notation, 4-6-17, part 1 59:51

line integral calculation and notation, 4-6-17, part 2 13:52

conservative vector field theorems, 4-7-17 46:10

Green's Theorem introduction, 4-10-17 46:56

more on Green's Theorem and locally conservative, 4-11-17, part 1 59:51

more on Green's Theorem and locally conservative, 4-11-17, part 2 14:11

surface integrals over sphere, 4-12-17 43:21

surface integrals on cylinders, cones, planes, graphs, 4-13-17, part 1 59:51

surface integrals on cylinders, cones, planes, graphs, 4-13-17, part 2 13:50

Stokes Theorem polyhedral argument, two examples, 4-14-17 49:13

proof of Stokes' for graph, intuitive div. thm. , 4-18-17 42:30

Stokes and Divergence with holes, deformation thms, 4-19-17 49:25

examples of vector calculus, 4-20-17, part 1 59:51

examples of vector calculus, 4-20-17, part 2 11:24

Theory of Harmonic Functions part I, 4-21-17 46:54

Green's Third Identity and Physics, 4-24-17 48:45

differential forms and generalized Stokes' Thm, 4-25-17, part 1 59:51

differential forms and generalized Stokes' Thm, 4-25-17, part 2 7:11

comments for Test 4, 5-1-17 39:21

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