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Raffi Hovasapian

Further Examples with Gradients & Tangents

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 Non-parameterized Way: In General4:34
- Curve Given in Non-parameterized Way: For the Circle of Radius r6:07
- Curve Given in Non-parameterized 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 n-th 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 3-Space14:07
- 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 (x-2)² + y² = 4 in Polar Form.5:46
- Graphing Function in Polar Form.10:02
- Converting a Region in the xy-plane 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
- Circulation-Curl Part 120:08
- Circulation-Curl 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
- Circulation-Curl9:44
- 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 3-space, 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 z-axis18:10
- 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 3-space 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 3-Space

23m 40s

- Intro0:00
- Divergence and Curl in 3-Space0:26
- Introduction to Divergence and Curl in 3-Space0:27
- 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 3-Space

34m 12s

- Intro0:00
- Divergence Theorem in 3-Space0:36
- Green's Flux-Divergence0:37
- Divergence Theorem in 3-Space3:34
- 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 Circulation-Curl Version of Green's Theorem0:26
- Constructing a Surface in 3-Space2:26
- Stokes' Theorem5:34
- Note on Curve and Vector Field in 3-Space9:50
- 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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1 answer

Last reply by: Professor Hovasapian

Tue Oct 16, 2018 10:53 PM

Post by Mohamed Talni on October 15, 2018

Dear Professor Hovasapian,

I really enjoy your explanation and insight.

I have a question that I do not understand:

Consider the first quadrant in the OXY plane in R2. Point O is the origin and the points P and Q are chosen on the y-axis and the x-axis, respectively as it is showed in the figure below. We create a family of line segments like PQ in a way that OP+OQ=10

1.

Determine the equation of the curve which appears by drawing more and more such line segments.

2.

Suppose we want to investigate a similar problem in the three dimensional case. Consider the first octant of the OXYZ in R3. Then we create a family of plane segments such that OP+OQ+OR=10. Then what would be the equation of the surface which will arise by drawing infinitely many such planes.

1 answer

Last reply by: Professor Hovasapian

Mon Oct 13, 2014 6:32 PM

Post by sam sohirad on October 11, 2014

hi raffi for example 1, can we use the cross product to find a vector in the direction of the space curve (intersection curve of the planes)?

I dont know if my thought process is correct but as i understood the two planes intersect and then we have a curve of intersection. we are looking for the tangent line at a point on this curve. in order to find the equation of a line we need 2 things; 1 a point on the line (3,2,-6) and a vector pointing in the direction of this line. since the 2 surfaces meet and create a curve, we can find the gradient vector at a certain point on the line. this will give us 2 normal vectors at a certain point of intersection. we can then go on to find the cross product of these 2 normal vectors at a mutual point to find a perpendicular vector at this point, thereby giving us the direction of the line. we then plug into our equation X=tv+P and have the vector equation of a line at the point.

if i use this method will i get the equation of a line that touches the point (3,2,-6)making it an equation of a tangent line?

i also understand what you said about the planes and when they intersect we get a line and that is the equation of our tangent line at that point, but i dont see how we used the planes in example 1? to me it seems we found the gradients of the surfaces.

5 answers

Last reply by: Josh Winfield

Sat May 11, 2013 8:07 AM

Post by Josh Winfield on February 12, 2013

Example 2:You ask to find c'(t) but found c'(0) instead. Which one where we supposed to find?

Example 3: âˆš(152) x âˆš(149) = 150.5 not -150.5

So the angle you get arccos(-150/150.5) = 175.33 (degrees)

1 answer

Last reply by: Professor Hovasapian

Thu Sep 6, 2012 2:43 PM

Post by Ian Vaagenes on September 6, 2012

Great course Raffi, you have a real gift for making things understandable. Any chance you'll do a few lectures on measure theory/probability theory?