Section 1: Vectors 

Points & Vectors 
28:23 
 
Intro 
0:00  
 
Points and Vectors 
1:02  
 
 A Point in a Plane 
1:03  
 
 A Point in Space 
3:14  
 
 Notation for a Space of a Given Space 
6:34  
 
 Introduction to Vectors 
9:51  
 
 Adding Vectors 
14:51  
 
 Example 1 
16:52  
 
 Properties of Vector Addition 
18:24  
 
 Example 2 
21:01  
 
 Two More Properties of Vector Addition 
24:16  
 
 Multiplication of a Vector by a Constant 
25:27  

Scalar Product & Norm 
30:25 
 
Intro 
0:00  
 
Scalar Product and Norm 
1:05  
 
 Introduction to Scalar Product 
1:06  
 
 Example 1 
3:21  
 
 Properties of Scalar Product 
6:14  
 
 Definition: Orthogonal 
11:41  
 
 Example 2: Orthogonal 
14:19  
 
 Definition: Norm of a Vector 
15:30  
 
 Example 3 
19:37  
 
 Distance Between Two Vectors 
22:05  
 
 Example 4 
27:19  

More on Vectors & Norms 
38:18 
 
Intro 
0:00  
 
More on Vectors and Norms 
0:38  
 
 Open Disc 
0:39  
 
 Close Disc 
3:14  
 
 Open Ball, Closed Ball, and the Sphere 
5:22  
 
 Property and Definition of Unit Vector 
7:16  
 
 Example 1 
14:04  
 
 Three Special Unit Vectors 
17:24  
 
 General Pythagorean Theorem 
19:44  
 
 Projection 
23:00  
 
 Example 2 
28:35  
 
 Example 3 
35:54  

Inequalities & Parametric Lines 
33:19 
 
Intro 
0:00  
 
Inequalities and Parametric Lines 
0:30  
 
 Starting Example 
0:31  
 
 Theorem 1 
5:10  
 
 Theorem 2 
7:22  
 
 Definition 1: Parametric Equation of a Straight Line 
10:16  
 
 Definition 2 
17:38  
 
 Example 1 
21:19  
 
 Example 2 
25:20  

Planes 
29:59 
 
Intro 
0:00  
 
Planes 
0:18  
 
 Definition 1 
0:19  
 
 Example 1 
7:04  
 
 Example 2 
12:45  
 
 General Definitions and Properties: 2 Vectors are Said to Be Paralleled If 
14:50  
 
 Example 3 
16:44  
 
 Example 4 
20:17  

More on Planes 
34:18 
 
Intro 
0:00  
 
More on Planes 
0:25  
 
 Example 1 
0:26  
 
 Distance From Some Point in Space to a Given Plane: Derivation 
10:12  
 
 Final Formula for Distance 
21:20  
 
 Example 2 
23:09  
 
 Example 3: Part 1 
26:56  
 
 Example 3: Part 2 
31:46  
Section 2: Differentiation of Vectors 

Maps, Curves & Parameterizations 
29:48 
 
Intro 
0:00  
 
Maps, Curves and Parameterizations 
1:10  
 
 Recall 
1:11  
 
 Looking at y = x2 or f(x) = x2 
2:23  
 
 Departure Space & Arrival Space 
7:01  
 
 Looking at a 'Function' from ℝ to ℝ2 
10:36  
 
 Example 1 
14:50  
 
 Definition 1: Parameterized Curve 
17:33  
 
 Example 2 
21:56  
 
 Example 3 
25:16  

Differentiation of Vectors 
39:40 
 
Intro 
0:00  
 
Differentiation of Vectors 
0:18  
 
 Example 1 
0:19  
 
 Definition 1: Velocity of a Curve 
1:45  
 
 Line Tangent to a Curve 
6:10  
 
 Example 2 
7:40  
 
 Definition 2: Speed of a Curve 
12:18  
 
 Example 3 
13:53  
 
 Definition 3: Acceleration Vector 
16:37  
 
 Two Definitions for the Scalar Part of Acceleration 
17:22  
 
 Rules for Differentiating Vectors: 1 
19:52  
 
 Rules for Differentiating Vectors: 2 
21:28  
 
 Rules for Differentiating Vectors: 3 
22:03  
 
 Rules for Differentiating Vectors: 4 
24:14  
 
 Example 4 
26:57  
Section 3: Functions of Several Variables 

Functions of Several Variable 
29:31 
 
Intro 
0:00  
 
Length of a Curve in Space 
0:25  
 
 Definition 1: Length of a Curve in Space 
0:26  
 
 Extended Form 
2:06  
 
 Example 1 
3:40  
 
 Example 2 
6:28  
 
Functions of Several Variable 
8:55  
 
 Functions of Several Variable 
8:56  
 
 General Examples 
11:11  
 
 Graph by Plotting 
13:00  
 
 Example 1 
16:31  
 
 Definition 1 
18:33  
 
 Example 2 
22:15  
 
 Equipotential Surfaces 
25:27  
 
 Isothermal Surfaces 
27:30  

Partial Derivatives 
23:31 
 
Intro 
0:00  
 
Partial Derivatives 
0:19  
 
 Example 1 
0:20  
 
 Example 2 
5:30  
 
 Example 3 
7:48  
 
 Example 4 
9:19  
 
 Definition 1 
12:19  
 
 Example 5 
14:24  
 
 Example 6 
16:14  
 
 Notation and Properties for Gradient 
20:26  

Higher and Mixed Partial Derivatives 
30:48 
 
Intro 
0:00  
 
Higher and Mixed Partial Derivatives 
0:45  
 
 Definition 1: Open Set 
0:46  
 
 Notation: Partial Derivatives 
5:39  
 
 Example 1 
12:00  
 
 Theorem 1 
14:25  
 
 Now Consider a Function of Three Variables 
16:50  
 
 Example 2 
20:09  
 
 Caution 
23:16  
 
 Example 3 
25:42  
Section 4: Chain Rule and The Gradient 

The Chain Rule 
28:03 
 
Intro 
0:00  
 
The Chain Rule 
0:45  
 
 Conceptual Example 
0:46  
 
 Example 1 
5:10  
 
 The Chain Rule 
10:11  
 
 Example 2: Part 1 
19:06  
 
 Example 2: Part 2  Solving Directly 
25:26  

Tangent Plane 
42:25 
 
Intro 
0:00  
 
Tangent Plane 
1:02  
 
 Tangent Plane Part 1 
1:03  
 
 Tangent Plane Part 2 
10:00  
 
 Tangent Plane Part 3 
18:18  
 
 Tangent Plane Part 4 
21:18  
 
 Definition 1: Tangent Plane to a Surface 
27:46  
 
 Example 1: Find the Equation of the Plane Tangent to the Surface 
31:18  
 
 Example 2: Find the Tangent Line to the Curve 
36:54  

Further Examples with Gradients & Tangents 
47:11 
 
Intro 
0:00  
 
Example 1: Parametric Equation for the Line Tangent to the Curve of Two Intersecting Surfaces 
0:41  
 
 Part 1: Question 
0:42  
 
 Part 2: When Two Surfaces in ℝ3 Intersect 
4:31  
 
 Part 3: Diagrams 
7:36  
 
 Part 4: Solution 
12:10  
 
 Part 5: Diagram of Final Answer 
23:52  
 
Example 2: Gradients & Composite Functions 
26:42  
 
 Part 1: Question 
26:43  
 
 Part 2: Solution 
29:21  
 
Example 3: Cos of the Angle Between the Surfaces 
39:20  
 
 Part 1: Question 
39:21  
 
 Part 2: Definition of Angle Between Two Surfaces 
41:04  
 
 Part 3: Solution 
42:39  

Directional Derivative 
41:22 
 
Intro 
0:00  
 
Directional Derivative 
0:10  
 
 Rate of Change & Direction Overview 
0:11  
 
 Rate of Change : Function of Two Variables 
4:32  
 
 Directional Derivative 
10:13  
 
 Example 1 
18:26  
 
 Examining Gradient of f(p) ∙ A When A is a Unit Vector 
25:30  
 
 Directional Derivative of f(p) 
31:03  
 
 Norm of the Gradient f(p) 
33:23  
 
 Example 2 
34:53  

A Unified View of Derivatives for Mappings 
39:41 
 
Intro 
0:00  
 
A Unified View of Derivatives for Mappings 
1:29  
 
 Derivatives for Mappings 
1:30  
 
 Example 1 
5:46  
 
 Example 2 
8:25  
 
 Example 3 
12:08  
 
 Example 4 
14:35  
 
 Derivative for Mappings of Composite Function 
17:47  
 
 Example 5 
22:15  
 
 Example 6 
28:42  
Section 5: Maxima and Minima 

Maxima & Minima 
36:41 
 
Intro 
0:00  
 
Maxima and Minima 
0:35  
 
 Definition 1: Critical Point 
0:36  
 
 Example 1: Find the Critical Values 
2:48  
 
 Definition 2: Local Max & Local Min 
10:03  
 
 Theorem 1 
14:10  
 
 Example 2: Local Max, Min, and Extreme 
18:28  
 
 Definition 3: Boundary Point 
27:00  
 
 Definition 4: Closed Set 
29:50  
 
 Definition 5: Bounded Set 
31:32  
 
 Theorem 2 
33:34  

Further Examples with Extrema 
32:48 
 
Intro 
0:00  
 
Further Example with Extrema 
1:02  
 
 Example 1: Max and Min Values of f on the Square 
1:03  
 
 Example 2: Find the Extreme for f(x,y) = x² + 2y²  x 
10:44  
 
 Example 3: Max and Min Value of f(x,y) = (x²+ y²)⁻¹ in the Region (x 2)²+ y² ≤ 1 
17:20  

Lagrange Multipliers 
32:32 
 
Intro 
0:00  
 
Lagrange Multipliers 
1:13  
 
 Theorem 1 
1:14  
 
 Method 
6:35  
 
 Example 1: Find the Largest and Smallest Values that f Achieves Subject to g 
9:14  
 
 Example 2: Find the Max & Min Values of f(x,y)= 3x + 4y on the Circle x² + y² = 1 
22:18  

More Lagrange Multiplier Examples 
27:42 
 
Intro 
0:00  
 
Example 1: Find the Point on the Surface z² xy = 1 Closet to the Origin 
0:54  
 
 Part 1 
0:55  
 
 Part 2 
7:37  
 
 Part 3 
10:44  
 
Example 2: Find the Max & Min of f(x,y) = x² + 2y  x on the Closed Disc of Radius 1 Centered at the Origin 
16:05  
 
 Part 1 
16:06  
 
 Part 2 
19:33  
 
 Part 3 
23:17  

Lagrange Multipliers, Continued 
31:47 
 
Intro 
0:00  
 
Lagrange Multipliers 
0:42  
 
 First Example of Lesson 20 
0:44  
 
 Let's Look at This Geometrically 
3:12  
 
Example 1: Lagrange Multiplier Problem with 2 Constraints 
8:42  
 
 Part 1: Question 
8:43  
 
 Part 2: What We Have to Solve 
15:13  
 
 Part 3: Case 1 
20:49  
 
 Part 4: Case 2 
22:59  
 
 Part 5: Final Solution 
25:45  
Section 6: Line Integrals and Potential Functions 

Line Integrals 
36:08 
 
Intro 
0:00  
 
Line Integrals 
0:18  
 
 Introduction to Line Integrals 
0:19  
 
 Definition 1: Vector Field 
3:57  
 
 Example 1 
5:46  
 
 Example 2: Gradient Operator & Vector Field 
8:06  
 
 Example 3 
12:19  
 
 Vector Field, Curve in Space & Line Integrals 
14:07  
 
 Definition 2: F(C(t)) ∙ C'(t) is a Function of t 
17:45  
 
 Example 4 
18:10  
 
 Definition 3: Line Integrals 
20:21  
 
 Example 5 
25:00  
 
 Example 6 
30:33  

More on Line Integrals 
28:04 
 
Intro 
0:00  
 
More on Line Integrals 
0:10  
 
 Line Integrals Notation 
0:11  
 
 Curve Given in Nonparameterized Way: In General 
4:34  
 
 Curve Given in Nonparameterized Way: For the Circle of Radius r 
6:07  
 
 Curve Given in Nonparameterized Way: For a Straight Line Segment Between P & Q 
6:32  
 
 The Integral is Independent of the Parameterization Chosen 
7:17  
 
 Example 1: Find the Integral on the Ellipse Centered at the Origin 
9:18  
 
 Example 2: Find the Integral of the Vector Field 
16:26  
 
 Discussion of Result and Vector Field for Example 2 
23:52  
 
 Graphical Example 
26:03  

Line Integrals, Part 3 
29:30 
 
Intro 
0:00  
 
Line Integrals 
0:12  
 
 Piecewise Continuous Path 
0:13  
 
 Closed Path 
1:47  
 
 Example 1: Find the Integral 
3:50  
 
 The Reverse Path 
14:14  
 
 Theorem 1 
16:18  
 
 Parameterization for the Reverse Path 
17:24  
 
 Example 2 
18:50  
 
 Line Integrals of Functions on ℝn 
21:36  
 
 Example 3 
24:20  

Potential Functions 
40:19 
 
Intro 
0:00  
 
Potential Functions 
0:08  
 
 Definition 1: Potential Functions 
0:09  
 
 Definition 2: An Open Set S is Called Connected if… 
5:52  
 
 Theorem 1 
8:19  
 
 Existence of a Potential Function 
11:04  
 
 Theorem 2 
18:06  
 
 Example 1 
22:18  
 
 Contrapositive and Positive Form of the Theorem 
28:02  
 
 The Converse is Not Generally True 
30:59  
 
 Our Theorem 
32:55  
 
 Compare the nth Term Test for Divergence of an Infinite Series 
36:00  
 
 So for Our Theorem 
38:16  

Potential Functions, Continued 
31:45 
 
Intro 
0:00  
 
Potential Functions 
0:52  
 
 Theorem 1 
0:53  
 
 Example 1 
4:00  
 
 Theorem in 3Space 
14:07  
 
 Example 2 
17:53  
 
 Example 3 
24:07  

Potential Functions, Conclusion & Summary 
28:22 
 
Intro 
0:00  
 
Potential Functions 
0:16  
 
 Theorem 1 
0:17  
 
 In Other Words 
3:25  
 
 Corollary 
5:22  
 
 Example 1 
7:45  
 
 Theorem 2 
11:34  
 
 Summary on Potential Functions 1 
15:32  
 
 Summary on Potential Functions 2 
17:26  
 
 Summary on Potential Functions 3 
18:43  
 
 Case 1 
19:24  
 
 Case 2 
20:48  
 
 Case 3 
21:35  
 
 Example 2 
23:59  
Section 7: Double Integrals 

Double Integrals 
29:46 
 
Intro 
0:00  
 
Double Integrals 
0:52  
 
 Introduction to Double Integrals 
0:53  
 
 Function with Two Variables 
3:39  
 
 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 ≤ x³ 
15:07  
 
 Example 3: f(x,y) = 4xy over the Region Bounded by y= 0, y= x, and y= x+3 
19:20  

Polar Coordinates 
36:17 
 
Intro 
0:00  
 
Polar Coordinates 
0:50  
 
 Polar Coordinates 
0:51  
 
 Example 1: Let (x,y) = (6,√6), Convert to Polar Coordinates 
3:24  
 
 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 Coordinates 
14:14  
 
 Example 3: Find the Integral over the Region Bounded by the Semicircle 
20:06  
 
 Example 4: Find the Integral over the Region 
27:57  
 
 Example 5: Find the Integral of f(x,y) = x² over the Region Contained by r= 1  cosθ 
32:55  

Green's Theorem 
38:01 
 
Intro 
0:00  
 
Green's Theorem 
0:38  
 
 Introduction to Green's Theorem and Notations 
0:39  
 
 Green's Theorem 
3:17  
 
 Example 1: Find the Integral of the Vector Field around the Ellipse 
8:30  
 
 Verifying Green's Theorem with Example 1 
15:35  
 
 A More General Version of Green's Theorem 
20:03  
 
 Example 2 
22:59  
 
 Example 3 
26:30  
 
 Example 4 
32:05  

Divergence & Curl of a Vector Field 
37:16 
 
Intro 
0:00  
 
Divergence & Curl of a Vector Field 
0:18  
 
 Definitions: Divergence(F) & Curl(F) 
0:19  
 
 Example 1: Evaluate Divergence(F) and Curl(F) 
3:43  
 
 Properties of Divergence 
9:24  
 
 Properties of Curl 
12:24  
 
 Two Versions of Green's Theorem: Circulation  Curl 
17:46  
 
 Two Versions of Green's Theorem: Flux Divergence 
19:09  
 
 CirculationCurl Part 1 
20:08  
 
 CirculationCurl Part 2 
28:29  
 
 Example 2 
32:06  

Divergence & Curl, Continued 
33:07 
 
Intro 
0:00  
 
Divergence & Curl, Continued 
0:24  
 
 Divergence Part 1 
0:25  
 
 Divergence Part 2: Right Normal Vector and Left Normal Vector 
5:28  
 
 Divergence Part 3 
9:09  
 
 Divergence Part 4 
13:51  
 
 Divergence Part 5 
19:19  
 
 Example 1 
23:40  

Final Comments on Divergence & Curl 
16:49 
 
Intro 
0:00  
 
Final Comments on Divergence and Curl 
0:37  
 
 Several Symbolic Representations for Green's Theorem 
0:38  
 
 CirculationCurl 
9:44  
 
 Flux Divergence 
11:02  
 
 Closing Comments on Divergence and Curl 
15:04  
Section 8: Triple Integrals 

Triple Integrals 
27:24 
 
Intro 
0:00  
 
Triple Integrals 
0:21  
 
 Example 1 
2:01  
 
 Example 2 
9:42  
 
 Example 3 
15:25  
 
 Example 4 
20:54  

Cylindrical & Spherical Coordinates 
35:33 
 
Intro 
0:00  
 
Cylindrical and Spherical Coordinates 
0:42  
 
 Cylindrical Coordinates 
0:43  
 
 When Integrating Over a Region in 3space, Upon Transformation the Triple Integral Becomes.. 
4:29  
 
 Example 1 
6:27  
 
 The Cartesian Integral 
15:00  
 
 Introduction to Spherical Coordinates 
19:44  
 
 Reason It's Called Spherical Coordinates 
22:49  
 
 Spherical Transformation 
26:12  
 
 Example 2 
29:23  
Section 9: Surface Integrals and Stokes' Theorem 

Parameterizing Surfaces & Cross Product 
41:29 
 
Intro 
0:00  
 
Parameterizing Surfaces 
0:40  
 
 Describing a Line or a Curve Parametrically 
0:41  
 
 Describing a Line or a Curve Parametrically: Example 
1:52  
 
 Describing a Surface Parametrically 
2:58  
 
 Describing a Surface Parametrically: Example 
5:30  
 
 Recall: Parameterizations are not Unique 
7:18  
 
 Example 1: Sphere of Radius R 
8:22  
 
 Example 2: Another P for the Sphere of Radius R 
10:52  
 
 This is True in General 
13:35  
 
 Example 3: Paraboloid 
15:05  
 
 Example 4: A Surface of Revolution around zaxis 
18:10  
 
Cross Product 
23:15  
 
 Defining Cross Product 
23:16  
 
 Example 5: Part 1 
28:04  
 
 Example 5: Part 2  Right Hand Rule 
32:31  
 
 Example 6 
37:20  

Tangent Plane & Normal Vector to a Surface 
37:06 
 
Intro 
0:00  
 
Tangent Plane and Normal Vector to a Surface 
0:35  
 
 Tangent Plane and Normal Vector to a Surface Part 1 
0:36  
 
 Tangent Plane and Normal Vector to a Surface Part 2 
5:22  
 
 Tangent Plane and Normal Vector to a Surface Part 3 
13:42  
 
 Example 1: Question & Solution 
17:59  
 
 Example 1: Illustrative Explanation of the Solution 
28:37  
 
 Example 2: Question & Solution 
30:55  
 
 Example 2: Illustrative Explanation of the Solution 
35:10  

Surface Area 
32:48 
 
Intro 
0:00  
 
Surface Area 
0:27  
 
 Introduction to Surface Area 
0:28  
 
 Given a Surface in 3space and a Parameterization P 
3:31  
 
 Defining Surface Area 
7:46  
 
 Curve Length 
10:52  
 
 Example 1: Find the Are of a Sphere of Radius R 
15:03  
 
 Example 2: Find the Area of the Paraboloid z= x² + y² for 0 ≤ z ≤ 5 
19:10  
 
 Example 2: Writing the Answer in Polar Coordinates 
28:07  

Surface Integrals 
46:52 
 
Intro 
0:00  
 
Surface Integrals 
0:25  
 
 Introduction to Surface Integrals 
0:26  
 
 General Integral for Surface Are of Any Parameterization 
3:03  
 
 Integral of a Function Over a Surface 
4:47  
 
 Example 1 
9:53  
 
 Integral of a Vector Field Over a Surface 
17:20  
 
 Example 2 
22:15  
 
 Side Note: Be Very Careful 
28:58  
 
 Example 3 
30:42  
 
 Summary 
43:57  

Divergence & Curl in 3Space 
23:40 
 
Intro 
0:00  
 
Divergence and Curl in 3Space 
0:26  
 
 Introduction to Divergence and Curl in 3Space 
0:27  
 
 Define: Divergence of F 
2:50  
 
 Define: Curl of F 
4:12  
 
 The Del Operator 
6:25  
 
 Symbolically: Div(F) 
9:03  
 
 Symbolically: Curl(F) 
10:50  
 
 Example 1 
14:07  
 
 Example 2 
18:01  

Divergence Theorem in 3Space 
34:12 
 
Intro 
0:00  
 
Divergence Theorem in 3Space 
0:36  
 
 Green's FluxDivergence 
0:37  
 
 Divergence Theorem in 3Space 
3:34  
 
 Note: Closed Surface 
6:43  
 
 Figure: Paraboloid 
8:44  
 
 Example 1 
12:13  
 
 Example 2 
18:50  
 
 Recap for Surfaces: Introduction 
27:50  
 
 Recap for Surfaces: Surface Area 
29:16  
 
 Recap for Surfaces: Surface Integral of a Function 
29:50  
 
 Recap for Surfaces: Surface Integral of a Vector Field 
30:39  
 
 Recap for Surfaces: Divergence Theorem 
32:32  

Stokes' Theorem, Part 1 
22:01 
 
Intro 
0:00  
 
Stokes' Theorem 
0:25  
 
 Recall CirculationCurl Version of Green's Theorem 
0:26  
 
 Constructing a Surface in 3Space 
2:26  
 
 Stokes' Theorem 
5:34  
 
 Note on Curve and Vector Field in 3Space 
9:50  
 
Example 1: Find the Circulation of F around the Curve 
12:40  
 
 Part 1: Question 
12:48  
 
 Part 2: Drawing the Figure 
13:56  
 
 Part 3: Solution 
16:08  

Stokes' Theorem, Part 2 
20:32 
 
Intro 
0:00  
 
Example 1: Calculate the Boundary of the Surface and the Circulation of F around this Boundary 
0:30  
 
 Part 1: Question 
0:31  
 
 Part 2: Drawing the Figure 
2:02  
 
 Part 3: Solution 
5:24  
 
Example 2: Calculate the Boundary of the Surface and the Circulation of F around this Boundary 
13:11  
 
 Part 1: Question 
13:12  
 
 Part 2: Solution 
13:56  