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Raffi Hovasapian
Differentiation of Vectors
Slide Duration: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
Wed Dec 30, 2015 12:24 AM
Post by ShihKuan Chen on December 24, 2015
In example 2, don't you have to plug pi/4 as the coefficient for (xprime of t) in and get pi/4*(1/root2, 1/root2, 1)?
2 answers
Last reply by: Jamal Tischler
Mon Dec 21, 2015 7:05 AM
Post by Hen McGibbons on August 28, 2015
@11:10, why did you write t (followed by that vector)? Don't we have to use a different variable, such as c, since t was used in the vectors x(t) and x'(t)? or are these the exact same t?
2 answers
Last reply by: Denny Yang â™• [Moderator]
Tue Aug 19, 2014 4:45 PM
Post by Denny Yang on August 12, 2014
When you did example 4, how do you know e^t and e^t gives you e^t? I understand adding the two vectors but not so much about the two functions. I know they are the same. Can you please clarify?
1 answer
Last reply by: Professor Hovasapian
Tue Feb 4, 2014 1:18 AM
Post by Alex Emil on February 3, 2014
Thank you, great lecture!!!
2 answers
Last reply by: Professor Hovasapian
Tue Sep 24, 2013 1:05 AM
Post by yaqub ali on September 22, 2013
correct me if i'm wrong but is e^2t suppose to be on the top at all??
1 answer
Last reply by: Professor Hovasapian
Wed Mar 27, 2013 7:48 PM
Post by Jawad Hassan on March 27, 2013
thank you for this lessons, i think i will pass my class thanks to this, by far the best teacher online on tis topic!
1 answer
Last reply by: Professor Hovasapian
Fri Dec 21, 2012 5:48 PM
Post by SOUFIANE LAMOUNI on December 21, 2012
Great lecture .. straight to the point! thank you
1 answer
Last reply by: Professor Hovasapian
Sun Aug 5, 2012 4:13 AM
Post by Mohammed Alhumaidi on August 4, 2012
Isn't 1/sqrt(2) = Pi/6 = 30 degrees? ( Pi/4 = 45 degrees )
you almost kept writing that mistake in the whole lecture !!