Professor Raffi Hovasapian helps students develop their Multivariable Calculus intuition with in-depth explanations of concepts before reinforcing an understanding of the material through varied examples. This course is appropriate for those who have completed single-variable calculus. Topics covered include everything from Vectors to Partial Derivatives, Lagrange Multipliers, Line Integrals, Triple Integrals, and Stokes' Theorem. Professor Hovasapian has degrees in Mathematics, Chemistry, and Classics and over 10 years of teaching experience.

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I. 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 
II. 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 
III. 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 
IV. 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 
V. 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 
VI. 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 Non-parameterized Way: In General 4:34 
    Curve Given in Non-parameterized Way: For the Circle of Radius r 6:07 
    Curve Given in Non-parameterized 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 n-th 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 3-Space 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 
VII. 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 (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 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 
    Circulation-Curl Part 1 20:08 
    Circulation-Curl 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 
    Circulation-Curl 9:44 
    Flux Divergence 11:02 
    Closing Comments on Divergence and Curl 15:04 
VIII. 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 3-space, 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 
IX. 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 z-axis 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 3-space 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 3-Space 23:40
   Intro 0:00 
   Divergence and Curl in 3-Space 0:26 
    Introduction to Divergence and Curl in 3-Space 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 3-Space 34:12
   Intro 0:00 
   Divergence Theorem in 3-Space 0:36 
    Green's Flux-Divergence 0:37 
    Divergence Theorem in 3-Space 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 Circulation-Curl Version of Green's Theorem 0:26 
    Constructing a Surface in 3-Space 2:26 
    Stokes' Theorem 5:34 
    Note on Curve and Vector Field in 3-Space 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