Home » Mathematics » Multivariable Calculus
No. of
Lectures
Duration
(hrs:min)
43
23:37

• Level Intermediate
• 43 Lessons (23hr : 37min)
• Audio: English
• English

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.

## 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
Example 1 16:52
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 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

## 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 (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

## 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 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

## 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 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.

• Free Sample Lessons
• Closed Captioning (CC)
• Practice Questions

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.

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