Professor Raffi Hovasapian helps you develop your Multivariable Calculus intuition with clear explanations of concepts before reinforcing an understanding of the material through step-by-step examples. See why Dr. Hovasapian is one of the most loved instructors on Educator.

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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 

Duration: 23 hours, 37 minutes

Number of Lessons: 43

This online course is perfect for those who have completed single-variable calculus (Calculus 1 & 2). All topics are covered in-depth and will prepare you for even higher level college math courses.

Additional Features:

  • Free Sample Lessons
  • Closed Captioning (CC)
  • Practice Questions
  • Downloadable Lecture Slides
  • Instructor Comments

Topics Include:

  • Vectors & Planes
  • Differentiation of Vectors
  • Partial Derivatives
  • Chain Rule
  • Minima & Maxima
  • Lagrange Multipliers
  • Line Integrals
  • Double Integrals
  • Divergence & Curl
  • Triple Integrals
  • Stokes’ Theorem

Professor Hovasapian brings with him triple degrees in Mathematics, Chemistry, and Classics and over 15 years of teaching experience. Check out more of his other math & science courses at Educator.

Student Testimonials:

"Thanks so much Raffi, not only for helping me to get a good grade but for helping me to fully understand math, its applications, and its power!" — Josh W.

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“Great course Raffi, you have a real gift for making things understandable.” — Ian V.