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source: James Cook 2015年8月24日

Multivariate Calculus

These are lectures captured from Math 231 at Liberty University in Lynchburg VA. Much more for this course can be found at the course website: http://www.supermath.info/MultivariateCalculus.html

This is the third semester in the usual calculus sequence. In a nutshell, we study of vector algebra, curves in 3D including Frenet-Serret equations, limits of several variables, partial differentiation, local extrema, Lagrange multipliers, closed set test, double and triple integrals, Jacobians, curvelinear coordinates, gradients, level curves, line and surface integrals, Green's Theorem, Stokes' Theorem and Gauss' Theorem. Some attention is paid to a few nontraditional topics: we do discuss the general derivative, cylindrical and spherical coordinate frame fields and the formulae for calculation of grad, div, curl and Laplacian in noncartesian setting. Finally, I'll probably include some discussion of differential forms if time permits. Some proofs are given in this course, but, deeper theorems are just sketched. Past this course, for the student who also has some experience with linear algebra, I recommend study of my advanced calculus course if you like my take on calculus III.

Multivariate Calculus: Lecture 1: points and vectors 26:40 here we set the terms R2 and R3 and explain the geometric meaning of xhat, yhat and zhat (other texts call these i, j and k respective due to their historic correspondence with the versors in Hamilton's quaternions). We derive the 3D-distance formula from the 2D-formula and a little picture. Next time, the dot-product ! See http://www.supermath.info/Multivariat... for more resources on multivariate calculus including my lecture notes and oodles of solutions from previous years.

Lecture 2 part 1: the dot product and vector length 59:51

Lecture 2 part 2: projections and dot products 11:03

Lecture 3: the cross product 42:12

Lecture 4 part 1: lines and planes 59:51

Lecture 4 part 2: lines and planes 14:39

Lecture 5: problem solving session 47:57

Lecture 6: curves and surfaces 50:34

Lecture 7 part 1: surfaces and coordinates 59:51

Lecture 7 part 2: surfaces and coordinates 10:35

Lecture 8: frames and calculus of curves 43:37

Lecture 9 part 1: product rules and arclength 59:51

Lecture 9 part 2: product rules and arclength 12:40

Lecture 10: Frenet Serret Frame and Equations 51:47

Lecture 11: Frenet Serret non-unit speed 50:12

Lecture 12 part 2: kinematics in 3D 9:20

Lecture 13: help with Mission 2 46:44

Lecture 14: problem solving session 56:31

Lecture 15: mission 2 solution 30:45

Lecture 16: questions for Test 1 47:39

Lecture 17: limits and concepts of topology 49:15

Lecture 18 part 1: directional derivatives 59:51

Lecture 18 part 2: directional derivatives 14:24

Lecture 19: partial derivatives 48:49

Lecture 20: sick examples and gradients and contour plots 2D 45:58

Lecture 21 part 1: gradients in 3D, deriving unit-vector fields 59:51

Lecture 21 part 2: gradients in 3D, general derivative 12:26

Lecture 22: linearization and tangent planes 47:14

Lecture 23 part 1: chain rules 59:51

Lecture 23 part 2: chain rules 17:45

Lecture 24: Mission 3 help session 48:23

Lecture 25 Part 1: Normal Vector Fields to Surfaces in R3 14:34

Lecture 25 Part 2: Normal Vector Fields to Surfaces in R3 55:27

Lecture 26: curvelinear calculus, gradient in polar coordinates 49:28

Lecture 27 Part 1: exact differential equations 59:51

Lecture 27 Part 2: exact differential equations 10:06

Lecture 28 Part 1: help with Mission 4 59:51

Lecture 28 Part 2: help with Mission 4 6:53

Lecture 29: questions before Test 2 and Mission 4 solution 44:16

Lecture 30: Lagrange Multiplier Technique 47:17

Lecture 31 Part 1: Lagrange Multiplier Examples, Multivariate Taylor 59:51

Lecture 31 Part 2: Lagrange Multiplier Examples, Multivariate Taylor 15:45

Lecture 32: Inuition and examples of the Second Derivative Test 48:32

Lecture 32: closed set test for two variables 45:11

Lecture 33 Part 1: more extreme examples, help on Mission 5 59:51

Lecture 33 Part 2: more extreme examples, help on Mission 5 8:46

Lecture 34: definition and concept of multivariate integrals 45:26

Lecture 35 Part 1: integrals over nonrectangular regions 59:51

Lecture 35 Part 2: integrals over nonrectangular regions 12:39

Lecture 36: nontrivial triple integrals 51:12

Lecture 37 Part 1 integration variable change 59:51

Lecture 37 Part 2 integration variable change 13:04

Lecture 38: volume of hypersphere, more integration examples 45:06

Lecture 39 part 1: help with mission 6 59:51

Lecture 39 part 2: help with mission 6 16:57

Lecture 40: differential forms and Jacobians, centroids 52:47

Lecture 41: Review for Test 3 43:03

Lecture 42 part 1: vector fields, curl and divergence 59:51

Lecture 42 part 2: vector fields, curl and divergence 10:11

Lecture 43: paths, curves, reparametrization, arclength 50:57

Lecture 44 part 1: line integrals 59:51

Lecture 44 part 2: line integrals 14:50

Lecture 45: conservative vector fields, path independence 51:45

Lecture 46: review for Test 3 49:03

Lecture 47 part 1: circulation, flux and Green's Theorem 59:51

Lecture 47 part 2: circulation, flux and Green's Theorem 14:20

Lecture 48: electrostatics in 2D 45:41

Lecture 49: help with homeworks 47:06

Lecture 50 part 1: surface integration 59:51

Lecture 50 part 2: surface integration 12:41

Lecture 51: surface integration examples 50:26

Lecture 52 part 1: Stokes and Gauss Theorems 59:51

Lecture 52 part 2: Stokes and Gauss Theorems 3:54

Lecture 53: deformation theorem, examples using Div. and Stokes' 52:19

Lecture 54 part 1: harmonic functions, a bit of potential theory 59:51

Lecture 54 part 2: harmonic functions, a bit of potential theory 14:40

Lecture 55: centroid of cone, homework help 48:44

Lecture 56: examples of integral vector calculus 46:21

Lecture 57: introduction to differential forms 49:45

Lecture 58: review for Test 4 42:46

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