Multivariable Calculus

Home Page for Multivariable Calculus

Please note that this is not the CTY course of the same name, though I teach that one too. This is a selection of resources I have collected over a long period of time for my past courses. I have taught MVC for more than 20 years in a real class room format at various places (Pakistan, Mexico, USA) and various langauges (Urdu, Spanish, English). The list includes Tec de Monterrey, Santa Fe Campus, Mexico, and Hamdard University, Karachi, Pakistan.

You may also go back to my main page, http://pages.jh.edu/~maliyou1/

Syllabus

Topics covered varied from course to course but generally included:

• Vectors and the Geometry of Space
• Vector Functions
• Partial Derivatives
• Multiple Integrals
• Vector Calculus

The course used various texts, including, most recently:

Multivariable Calculus (7th or 8th edition) by James Stewart.
ISBN-13 for 7th edition: 978-0538497879
ISBN-13 for 8th edition: 978-1285741550

Lecture Set 1

Currently there are two sets of lecture slides avaibalble. First are from my MVC course offered in Mexico (download as single zip file) in 2006. It used a different textbook and you will find Spanish here and there but otherwise the notes are in English.

Lecture Set 2

The second set is from the textbook iteslf. I hvae not used them in a real classroom setting and online students have access to another set of video and written lectures. (All presentations copyright by the publisher).

Chapter 12: Vectors and the Geometry of Space

• 12.1 - Three-Dimensional Coordinate Systems
• 12.2 - Vectors
• 12.3 - The Dot Product
• 12.4 - The Cross Product
• 12.5 - Equations of Lines and Planes
• 12.6 - Cylinders and Quadric Surfaces

Chapter 13: Vector Functions

• 13.1 - Vector Functions and Space Curves
• 13.2 - Derivatives and Integrals of Vector Functions
• 13.3 - Arc Length and Curvature
• 13.4 - Motion in Space: Velocity and Acceleration

Chapter 14: Partial Derivatives

• 14.1 - Functions of Several Variables
• 14.2 - Limits and Continuity
• 14.3 - Partial Derivatives
• 14.4 - Tangent Planes and Linear Approximations
• 14.5 - The Chain Rule
• 14.6 - Directional Derivatives and the Gradient Vector
• 14.7 - Maximum and Minimum Values
• 14.8 - Lagrange Multipliers

Chapter 15: Multiple Integrals

• 15.1 - Double Integrals over Rectangles
• 15.2 - Iterated Integrals
• 15.3 - Double Integrals over General Regions
• 15.4 - Double Integrals in Polar Coordinates
• 15.5 - Applications of Double Integrals
• 15.6 - Surface Area
• 15.7 - Triple Integrals
• 15.8 - Triple Integrals in Cylindrical Coordinates
• 15.9 - Triple Integrals in Spherical Coordinates
• 15.10 - Change in Variables in Multiple Integrals

Chapter 16: Vector Calculus

• 16.1 - Vector Fields
• 16.2 - Line Integrals
• 16.3 - The Fundamental Theorem for Line Integrals
• 16.4 - Green's Theorem
• 16.5 - Curl and Divergence
• 16.6 - Parametric Surfaces and Their Areas
• 16.7 - Surface Integrals
• 16.8 - Stoke's Theorem
• 16.9 - The Divergence Theorem

Khan Academy has a set of 175+ lectures on Multivariable Calculus. The set covers the following topics but you need to go to their site to find direct links:

1. Multivariable functions
2. Representing points in 3d
3. Introduction to 3d graphs
4. Interpreting graphs with slices
5. Contour plots
6. Parametric curves
7. Parametric surfaces
8. Vector fields, introduction
9. Fluid flow and vector fields
10. 3d vector fields, introduction
11. 3d vector field example
12. Transformations, part 1
13. Transformations, part 2
14. Transformations, part 3
15. Partial derivatives, introduction
16. Partial derivatives and graphs
17. Formal definition of partial derivatives
18. Symmetry of second partial derivatives
19. Gradient
20. Gradient and graphs
21. Directional derivative
22. Directional derivative, formal definition
23. Directional derivatives and slope
24. Why the gradient is the direction of steepest ascent
25. Gradient and contour maps
26. Position vector valued functions
27. Derivative of a position vector valued function
28. Differential of a vector valued function
29. Vector valued function derivative example
30. Multivariable chain rule
31. Multivariable chain rule intuition
32. Vector form of the multivariable chain rule
33. Multivariable chain rule and directional derivatives
34. More formal treatment of multivariable chain rule
35. Curvature intuition
36. Curvature formula, part 1
37. Curvature formula, part 2
38. Curvature formula, part 3
39. Curvature formula, part 4
40. Curvature formula, part 5
41. Curvature of a helix, part 1
42. Curvature of a helix, part 2
43. Curvature of a cycloid
44. Computing the partial derivative of a vector-valued function
45. Partial derivative of a parametric surface, part 1
46. Partial derivative of a parametric surface, part 2
47. Partial derivatives of vector fields
48. Partial derivatives of vector fields, component by component
49. Divergence intuition, part 1
50. Divergence intuition, part 2
51. Divergence formula, part 1
52. Divergence formula, part 2
53. Divergence example
54. Divergence notation
55. 2d curl intuition
56. 2d curl formula
57. 2d curl example
58. 2d curl nuance
59. Describing rotation in 3d with a vector
60. 3d curl intuition, part 1
61. 3d curl intuition, part 2
62. 3d curl formula, part 1
63. 3d curl formula, part 2
64. 3d curl computation example
65. Laplacian intuition
66. Laplacian computation example
67. Explicit Laplacian formula
68. Harmonic Functions
69. Jacobian prerequisite knowledge
70. Local linearity for a multivariable function
71. The Jacobian matrix
72. Computing a Jacobian matrix
73. The Jacobian Determinant
74. What is a tangent plane
75. Controlling a plane in space
76. Computing a tangent plane
77. Local linearization
78. What do quadratic approximations look like
79. Quadratic approximation formula, part 1
80. Quadratic approximation formula, part 2
81. Quadratic approximation example
82. The Hessian matrix
83. Expressing a quadratic form with a matrix
84. Vector form of multivariable quadratic approximation
85. Multivariable maxima and minima
86. Saddle points
87. Warm up to the second partial derivative test
88. Second partial derivative test
89. Second partial derivative test intuition
90. Second partial derivative test example, part 1
91. Second partial derivative test example, part 2
92. Constrained optimization introduction
93. Lagrange multipliers, using tangency to solve constrained optimization
94. Finishing the intro lagrange multiplier example
95. Lagrange multiplier example, part 1
96. Lagrange multiplier example, part 2
97. The Lagrangian
98. Meaning of Lagrange multiplier
99. Proof for the meaning of Lagrange multipliers
100. Introduction to the line integral
101. Line integral example 1
102. Line integral example 2 (part 1)
103. Line integral example 2 (part 2)
104. Line integrals and vector fields
105. Using a line integral to find the work done by a vector field example
106. Parametrization of a reverse path
107. Scalar field line integral independent of path direction
108. Vector field line integrals dependent on path direction
109. Path independence for line integrals
110. Closed curve line integrals of conservative vector fields
111. Example of closed line integral of conservative field
112. Second example of line integral of conservative vector field
113. Double integral 1
114. Double integrals 2
115. Double integrals 3
116. Double integrals 4
117. Double integrals 5
118. Double integrals 6
119. Triple integrals 1
120. Triple integrals 2
121. Triple integrals 3
122. Introduction to parametrizing a surface with two parameters
123. Determining a position vector-valued function for a parametrization of two parameters
124. Partial derivatives of vector-valued functions
125. Introduction to the surface integral
126. Example of calculating a surface integral part 1
127. Example of calculating a surface integral part 2
128. Example of calculating a surface integral part 3
129. Surface integral example part 1: Parameterizing the unit sphere
130. Surface integral example part 2: Calculating the surface differential
131. Surface integral example part 3: The home stretch
132. Surface integral ex2 part 1: Parameterizing the surface
133. Surface integral ex2 part 2: Evaluating integral
134. Surface integral ex3 part 1: Parameterizing the outside surface
135. Surface integral ex3 part 2: Evaluating the outside surface
136. Surface integral ex3 part 3: Top surface
137. Surface integral ex3 part 4: Home stretch
138. Conceptual understanding of flux in three dimensions
139. Constructing a unit normal vector to a surface
140. Vector representation of a surface integral
141. Green's theorem proof part 1
142. Green's theorem proof (part 2)
143. Green's theorem example 1
144. Green's theorem example 2
145. Constructing a unit normal vector to a curve
146. 2D divergence theorem
147. Conceptual clarification for 2D divergence theorem
148. Stokes' theorem intuition
149. Green's and Stokes' theorem relationship
150. Orienting boundary with surface
151. Orientation and stokes
152. Conditions for stokes theorem
153. Stokes example part 1
154. Stokes example part 2: Parameterizing the surface
155. Stokes example part 3: Surface to double integral
156. Stokes example part 4: Curl and final answer
157. Evaluating line integral directly - part 1
158. Evaluating line integral directly - part 2
159. 3D divergence theorem intuition
160. Divergence theorem example 1
161. Stokes' theorem proof part 1
162. Stokes' theorem proof part 2
163. Stokes' theorem proof part 3
164. Stokes' theorem proof part 4
165. Stokes' theorem proof part 5
166. Stokes' theorem proof part 6
167. Stokes' theorem proof part 7
168. Type I regions in three dimensions
169. Type II regions in three dimensions
170. Type III regions in three dimensions
171. Divergence theorem proof (part 1)
172. Divergence theorem proof (part 2)
173. Divergence theorem proof (part 3)
174. Divergence theorem proof (part 4)
175. Divergence theorem proof (part 5)

Other Good MVC Courses and their Websites

• Multivariable Calculus at MIT
• Multivariable Calculus at Harvard University
• Solutions Manual for the textbook, from Amazon
• Online Answers to ALL problems
• Krista King's Math Channel

Free online books and notes

• Free books at Community Calculus.
• Another textbook for a course in Multivariable Calculus.
• Animated Demonstrations for Multivariable Calculus.

Technical Help

The objective of this section is to provide resources on online graphing and computational tools:

• Wolfram Alpha
• Wolfram Alpha has a complete collection of apps for Multivariable Calculus
• Desmos Graphing Calculator
• Desmos based slope field generator
• Symbolab (Symbolic Maths with steps)
• CalcPlot3D - New (Works in Firefox)
• 3D Graphing Calculator of GeoGebra
• CPM 3D Plotter to plot points and plains

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