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Table of Contents
I. Vectors
Points & Vectors
28m 23s
 Intro0:00
 Points and Vectors1:02
 A Point in a Plane1:03
 A Point in Space3:14
 Notation for a Space of a Given Space6:34
 Introduction to Vectors9:51
 Adding Vectors14:51
 Example 116:52
 Properties of Vector Addition18:24
 Example 221:01
 Two More Properties of Vector Addition24:16
 Multiplication of a Vector by a Constant25:27
Scalar Product & Norm
30m 25s
 Intro0:00
 Scalar Product and Norm1:05
 Introduction to Scalar Product1:06
 Example 13:21
 Properties of Scalar Product6:14
 Definition: Orthogonal11:41
 Example 2: Orthogonal14:19
 Definition: Norm of a Vector15:30
 Example 319:37
 Distance Between Two Vectors22:05
 Example 427:19
More on Vectors & Norms
38m 18s
 Intro0:00
 More on Vectors and Norms0:38
 Open Disc0:39
 Close Disc3:14
 Open Ball, Closed Ball, and the Sphere5:22
 Property and Definition of Unit Vector7:16
 Example 114:04
 Three Special Unit Vectors17:24
 General Pythagorean Theorem19:44
 Projection23:00
 Example 228:35
 Example 335:54
Inequalities & Parametric Lines
33m 19s
 Intro0:00
 Inequalities and Parametric Lines0:30
 Starting Example0:31
 Theorem 15:10
 Theorem 27:22
 Definition 1: Parametric Equation of a Straight Line10:16
 Definition 217:38
 Example 121:19
 Example 225:20
Planes
29m 59s
 Intro0:00
 Planes0:18
 Definition 10:19
 Example 17:04
 Example 212:45
 General Definitions and Properties: 2 Vectors are Said to Be Paralleled If14:50
 Example 316:44
 Example 420:17
More on Planes
34m 18s
 Intro0:00
 More on Planes0:25
 Example 10:26
 Distance From Some Point in Space to a Given Plane: Derivation10:12
 Final Formula for Distance21:20
 Example 223:09
 Example 3: Part 126:56
 Example 3: Part 231:46
II. Differentiation of Vectors
Maps, Curves & Parameterizations
29m 48s
 Intro0:00
 Maps, Curves and Parameterizations1:10
 Recall1:11
 Looking at y = x2 or f(x) = x22:23
 Departure Space & Arrival Space7:01
 Looking at a 'Function' from ℝ to ℝ210:36
 Example 114:50
 Definition 1: Parameterized Curve17:33
 Example 221:56
 Example 325:16
Differentiation of Vectors
39m 40s
 Intro0:00
 Differentiation of Vectors0:18
 Example 10:19
 Definition 1: Velocity of a Curve1:45
 Line Tangent to a Curve6:10
 Example 27:40
 Definition 2: Speed of a Curve12:18
 Example 313:53
 Definition 3: Acceleration Vector16:37
 Two Definitions for the Scalar Part of Acceleration17:22
 Rules for Differentiating Vectors: 119:52
 Rules for Differentiating Vectors: 221:28
 Rules for Differentiating Vectors: 322:03
 Rules for Differentiating Vectors: 424:14
 Example 426:57
III. Functions of Several Variables
Functions of Several Variable
29m 31s
 Intro0:00
 Length of a Curve in Space0:25
 Definition 1: Length of a Curve in Space0:26
 Extended Form2:06
 Example 13:40
 Example 26:28
 Functions of Several Variable8:55
 Functions of Several Variable8:56
 General Examples11:11
 Graph by Plotting13:00
 Example 116:31
 Definition 118:33
 Example 222:15
 Equipotential Surfaces25:27
 Isothermal Surfaces27:30
Partial Derivatives
23m 31s
 Intro0:00
 Partial Derivatives0:19
 Example 10:20
 Example 25:30
 Example 37:48
 Example 49:19
 Definition 112:19
 Example 514:24
 Example 616:14
 Notation and Properties for Gradient20:26
Higher and Mixed Partial Derivatives
30m 48s
 Intro0:00
 Higher and Mixed Partial Derivatives0:45
 Definition 1: Open Set0:46
 Notation: Partial Derivatives5:39
 Example 112:00
 Theorem 114:25
 Now Consider a Function of Three Variables16:50
 Example 220:09
 Caution23:16
 Example 325:42
IV. Chain Rule and The Gradient
The Chain Rule
28m 3s
 Intro0:00
 The Chain Rule0:45
 Conceptual Example0:46
 Example 15:10
 The Chain Rule10:11
 Example 2: Part 119:06
 Example 2: Part 2  Solving Directly25:26
Tangent Plane
42m 25s
 Intro0:00
 Tangent Plane1:02
 Tangent Plane Part 11:03
 Tangent Plane Part 210:00
 Tangent Plane Part 318:18
 Tangent Plane Part 421:18
 Definition 1: Tangent Plane to a Surface27:46
 Example 1: Find the Equation of the Plane Tangent to the Surface31:18
 Example 2: Find the Tangent Line to the Curve36:54
Further Examples with Gradients & Tangents
47m 11s
 Intro0:00
 Example 1: Parametric Equation for the Line Tangent to the Curve of Two Intersecting Surfaces0:41
 Part 1: Question0:42
 Part 2: When Two Surfaces in ℝ3 Intersect4:31
 Part 3: Diagrams7:36
 Part 4: Solution12:10
 Part 5: Diagram of Final Answer23:52
 Example 2: Gradients & Composite Functions26:42
 Part 1: Question26:43
 Part 2: Solution29:21
 Example 3: Cos of the Angle Between the Surfaces39:20
 Part 1: Question39:21
 Part 2: Definition of Angle Between Two Surfaces41:04
 Part 3: Solution42:39
Directional Derivative
41m 22s
 Intro0:00
 Directional Derivative0:10
 Rate of Change & Direction Overview0:11
 Rate of Change : Function of Two Variables4:32
 Directional Derivative10:13
 Example 118:26
 Examining Gradient of f(p) ∙ A When A is a Unit Vector25:30
 Directional Derivative of f(p)31:03
 Norm of the Gradient f(p)33:23
 Example 234:53
A Unified View of Derivatives for Mappings
39m 41s
 Intro0:00
 A Unified View of Derivatives for Mappings1:29
 Derivatives for Mappings1:30
 Example 15:46
 Example 28:25
 Example 312:08
 Example 414:35
 Derivative for Mappings of Composite Function17:47
 Example 522:15
 Example 628:42
V. Maxima and Minima
Maxima & Minima
36m 41s
 Intro0:00
 Maxima and Minima0:35
 Definition 1: Critical Point0:36
 Example 1: Find the Critical Values2:48
 Definition 2: Local Max & Local Min10:03
 Theorem 114:10
 Example 2: Local Max, Min, and Extreme18:28
 Definition 3: Boundary Point27:00
 Definition 4: Closed Set29:50
 Definition 5: Bounded Set31:32
 Theorem 233:34
Further Examples with Extrema
32m 48s
 Intro0:00
 Further Example with Extrema1:02
 Example 1: Max and Min Values of f on the Square1:03
 Example 2: Find the Extreme for f(x,y) = x² + 2y²  x10:44
 Example 3: Max and Min Value of f(x,y) = (x²+ y²)⁻¹ in the Region (x 2)²+ y² ≤ 117:20
Lagrange Multipliers
32m 32s
 Intro0:00
 Lagrange Multipliers1:13
 Theorem 11:14
 Method6:35
 Example 1: Find the Largest and Smallest Values that f Achieves Subject to g9:14
 Example 2: Find the Max & Min Values of f(x,y)= 3x + 4y on the Circle x² + y² = 122:18
More Lagrange Multiplier Examples
27m 42s
 Intro0:00
 Example 1: Find the Point on the Surface z² xy = 1 Closet to the Origin0:54
 Part 10:55
 Part 27:37
 Part 310:44
 Example 2: Find the Max & Min of f(x,y) = x² + 2y  x on the Closed Disc of Radius 1 Centered at the Origin16:05
 Part 116:06
 Part 219:33
 Part 323:17
Lagrange Multipliers, Continued
31m 47s
 Intro0:00
 Lagrange Multipliers0:42
 First Example of Lesson 200:44
 Let's Look at This Geometrically3:12
 Example 1: Lagrange Multiplier Problem with 2 Constraints8:42
 Part 1: Question8:43
 Part 2: What We Have to Solve15:13
 Part 3: Case 120:49
 Part 4: Case 222:59
 Part 5: Final Solution25:45
VI. Line Integrals and Potential Functions
Line Integrals
36m 8s
 Intro0:00
 Line Integrals0:18
 Introduction to Line Integrals0:19
 Definition 1: Vector Field3:57
 Example 15:46
 Example 2: Gradient Operator & Vector Field8:06
 Example 312:19
 Vector Field, Curve in Space & Line Integrals14:07
 Definition 2: F(C(t)) ∙ C'(t) is a Function of t17:45
 Example 418:10
 Definition 3: Line Integrals20:21
 Example 525:00
 Example 630:33
More on Line Integrals
28m 4s
 Intro0:00
 More on Line Integrals0:10
 Line Integrals Notation0:11
 Curve Given in Nonparameterized Way: In General4:34
 Curve Given in Nonparameterized Way: For the Circle of Radius r6:07
 Curve Given in Nonparameterized Way: For a Straight Line Segment Between P & Q6:32
 The Integral is Independent of the Parameterization Chosen7:17
 Example 1: Find the Integral on the Ellipse Centered at the Origin9:18
 Example 2: Find the Integral of the Vector Field16:26
 Discussion of Result and Vector Field for Example 223:52
 Graphical Example26:03
Line Integrals, Part 3
29m 30s
 Intro0:00
 Line Integrals0:12
 Piecewise Continuous Path0:13
 Closed Path1:47
 Example 1: Find the Integral3:50
 The Reverse Path14:14
 Theorem 116:18
 Parameterization for the Reverse Path17:24
 Example 218:50
 Line Integrals of Functions on ℝn21:36
 Example 324:20
Potential Functions
40m 19s
 Intro0:00
 Potential Functions0:08
 Definition 1: Potential Functions0:09
 Definition 2: An Open Set S is Called Connected if…5:52
 Theorem 18:19
 Existence of a Potential Function11:04
 Theorem 218:06
 Example 122:18
 Contrapositive and Positive Form of the Theorem28:02
 The Converse is Not Generally True30:59
 Our Theorem32:55
 Compare the nth Term Test for Divergence of an Infinite Series36:00
 So for Our Theorem38:16
Potential Functions, Continued
31m 45s
 Intro0:00
 Potential Functions0:52
 Theorem 10:53
 Example 14:00
 Theorem in 3Space14:07
 Example 217:53
 Example 324:07
Potential Functions, Conclusion & Summary
28m 22s
 Intro0:00
 Potential Functions0:16
 Theorem 10:17
 In Other Words3:25
 Corollary5:22
 Example 17:45
 Theorem 211:34
 Summary on Potential Functions 115:32
 Summary on Potential Functions 217:26
 Summary on Potential Functions 318:43
 Case 119:24
 Case 220:48
 Case 321:35
 Example 223:59
VII. Double Integrals
Double Integrals
29m 46s
 Intro0:00
 Double Integrals0:52
 Introduction to Double Integrals0:53
 Function with Two Variables3:39
 Example 1: Find the Integral of xy³ over the Region x ϵ[1,2] & y ϵ[4,6]9:42
 Example 2: f(x,y) = x²y & R be the Region Such That x ϵ[2,3] & x² ≤ y ≤ x³15:07
 Example 3: f(x,y) = 4xy over the Region Bounded by y= 0, y= x, and y= x+319:20
Polar Coordinates
36m 17s
 Intro0:00
 Polar Coordinates0:50
 Polar Coordinates0:51
 Example 1: Let (x,y) = (6,√6), Convert to Polar Coordinates3:24
 Example 2: Express the Circle (x2)² + y² = 4 in Polar Form.5:46
 Graphing Function in Polar Form.10:02
 Converting a Region in the xyplane to Polar Coordinates14:14
 Example 3: Find the Integral over the Region Bounded by the Semicircle20:06
 Example 4: Find the Integral over the Region27:57
 Example 5: Find the Integral of f(x,y) = x² over the Region Contained by r= 1  cosθ32:55
Green's Theorem
38m 1s
 Intro0:00
 Green's Theorem0:38
 Introduction to Green's Theorem and Notations0:39
 Green's Theorem3:17
 Example 1: Find the Integral of the Vector Field around the Ellipse8:30
 Verifying Green's Theorem with Example 115:35
 A More General Version of Green's Theorem20:03
 Example 222:59
 Example 326:30
 Example 432:05
Divergence & Curl of a Vector Field
37m 16s
 Intro0:00
 Divergence & Curl of a Vector Field0:18
 Definitions: Divergence(F) & Curl(F)0:19
 Example 1: Evaluate Divergence(F) and Curl(F)3:43
 Properties of Divergence9:24
 Properties of Curl12:24
 Two Versions of Green's Theorem: Circulation  Curl17:46
 Two Versions of Green's Theorem: Flux Divergence19:09
 CirculationCurl Part 120:08
 CirculationCurl Part 228:29
 Example 232:06
Divergence & Curl, Continued
33m 7s
 Intro0:00
 Divergence & Curl, Continued0:24
 Divergence Part 10:25
 Divergence Part 2: Right Normal Vector and Left Normal Vector5:28
 Divergence Part 39:09
 Divergence Part 413:51
 Divergence Part 519:19
 Example 123:40
Final Comments on Divergence & Curl
16m 49s
 Intro0:00
 Final Comments on Divergence and Curl0:37
 Several Symbolic Representations for Green's Theorem0:38
 CirculationCurl9:44
 Flux Divergence11:02
 Closing Comments on Divergence and Curl15:04
VIII. Triple Integrals
Triple Integrals
27m 24s
 Intro0:00
 Triple Integrals0:21
 Example 12:01
 Example 29:42
 Example 315:25
 Example 420:54
Cylindrical & Spherical Coordinates
35m 33s
 Intro0:00
 Cylindrical and Spherical Coordinates0:42
 Cylindrical Coordinates0:43
 When Integrating Over a Region in 3space, Upon Transformation the Triple Integral Becomes..4:29
 Example 16:27
 The Cartesian Integral15:00
 Introduction to Spherical Coordinates19:44
 Reason It's Called Spherical Coordinates22:49
 Spherical Transformation26:12
 Example 229:23
IX. Surface Integrals and Stokes' Theorem
Parameterizing Surfaces & Cross Product
41m 29s
 Intro0:00
 Parameterizing Surfaces0:40
 Describing a Line or a Curve Parametrically0:41
 Describing a Line or a Curve Parametrically: Example1:52
 Describing a Surface Parametrically2:58
 Describing a Surface Parametrically: Example5:30
 Recall: Parameterizations are not Unique7:18
 Example 1: Sphere of Radius R8:22
 Example 2: Another P for the Sphere of Radius R10:52
 This is True in General13:35
 Example 3: Paraboloid15:05
 Example 4: A Surface of Revolution around zaxis18:10
 Cross Product23:15
 Defining Cross Product23:16
 Example 5: Part 128:04
 Example 5: Part 2  Right Hand Rule32:31
 Example 637:20
Tangent Plane & Normal Vector to a Surface
37m 6s
 Intro0:00
 Tangent Plane and Normal Vector to a Surface0:35
 Tangent Plane and Normal Vector to a Surface Part 10:36
 Tangent Plane and Normal Vector to a Surface Part 25:22
 Tangent Plane and Normal Vector to a Surface Part 313:42
 Example 1: Question & Solution17:59
 Example 1: Illustrative Explanation of the Solution28:37
 Example 2: Question & Solution30:55
 Example 2: Illustrative Explanation of the Solution35:10
Surface Area
32m 48s
 Intro0:00
 Surface Area0:27
 Introduction to Surface Area0:28
 Given a Surface in 3space and a Parameterization P3:31
 Defining Surface Area7:46
 Curve Length10:52
 Example 1: Find the Are of a Sphere of Radius R15:03
 Example 2: Find the Area of the Paraboloid z= x² + y² for 0 ≤ z ≤ 519:10
 Example 2: Writing the Answer in Polar Coordinates28:07
Surface Integrals
46m 52s
 Intro0:00
 Surface Integrals0:25
 Introduction to Surface Integrals0:26
 General Integral for Surface Are of Any Parameterization3:03
 Integral of a Function Over a Surface4:47
 Example 19:53
 Integral of a Vector Field Over a Surface17:20
 Example 222:15
 Side Note: Be Very Careful28:58
 Example 330:42
 Summary43:57
Divergence & Curl in 3Space
23m 40s
 Intro0:00
 Divergence and Curl in 3Space0:26
 Introduction to Divergence and Curl in 3Space0:27
 Define: Divergence of F2:50
 Define: Curl of F4:12
 The Del Operator6:25
 Symbolically: Div(F)9:03
 Symbolically: Curl(F)10:50
 Example 114:07
 Example 218:01
Divergence Theorem in 3Space
34m 12s
 Intro0:00
 Divergence Theorem in 3Space0:36
 Green's FluxDivergence0:37
 Divergence Theorem in 3Space3:34
 Note: Closed Surface6:43
 Figure: Paraboloid8:44
 Example 112:13
 Example 218:50
 Recap for Surfaces: Introduction27:50
 Recap for Surfaces: Surface Area29:16
 Recap for Surfaces: Surface Integral of a Function29:50
 Recap for Surfaces: Surface Integral of a Vector Field30:39
 Recap for Surfaces: Divergence Theorem32:32
Stokes' Theorem, Part 1
22m 1s
 Intro0:00
 Stokes' Theorem0:25
 Recall CirculationCurl Version of Green's Theorem0:26
 Constructing a Surface in 3Space2:26
 Stokes' Theorem5:34
 Note on Curve and Vector Field in 3Space9:50
 Example 1: Find the Circulation of F around the Curve12:40
 Part 1: Question12:48
 Part 2: Drawing the Figure13:56
 Part 3: Solution16:08
Stokes' Theorem, Part 2
20m 32s
 Intro0:00
 Example 1: Calculate the Boundary of the Surface and the Circulation of F around this Boundary0:30
 Part 1: Question0:31
 Part 2: Drawing the Figure2:02
 Part 3: Solution5:24
 Example 2: Calculate the Boundary of the Surface and the Circulation of F around this Boundary13:11
 Part 1: Question13:12
 Part 2: Solution13:56
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For more information, please see full course syllabus of Multivariable Calculus
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1 answer
Last reply by: Professor Hovasapian
Mon Dec 21, 2015 7:42 PM
Post by Jamal Tischler on December 21, 2015
We could say at the projection of a vector just, a cos(tetha)* b/b. Where a cos (tetha) is the legth of the projection( in the right triangle, I applied cos) and b/b is the init vector along b.
1 answer
Last reply by: Professor Hovasapian
Wed Jan 28, 2015 12:38 PM
Post by Jacob Kalbfleisch on January 25, 2015
Prof. Hovasapian is hands down the best lecturer on Educator. So clear and succinct, and offers fantastic advice very bluntly. I've so much respect for this man!
4 answers
Last reply by: Professor Hovasapian
Fri Jun 20, 2014 4:50 PM
Post by abc def on June 11, 2014
Good morning Professor Hovasapian,
I was playing around with some AngleDifference Identities to wrap my mind around the dot product in 23D. I observed that A*B = ABcos(@) can be proven by using the AngleDifference Identity in respect of cos(@),
A*B=ABcos(@a  @b) = AB[cos(@a)cos(@b)+sin(@a)sin(@b)] = [Acos(@a)Bcos(@b)+Asin(@a)Bsin(@b)] = (Ax)(Bx)+(Ay)(By) QED.
Is there a operation that _____=ABsin(@) would work?
1 answer
Last reply by: Professor Hovasapian
Wed Oct 2, 2013 2:50 PM
Post by Said Sabir on October 2, 2013
how to calculate the angel if I know already the Cos of that angel ?