Multi-Variable Calculus
with: Professor Jocelyn

Jocelyn C. teaches Educator's Multivariable Calculus course covering topics such as double integrals, partial derivatives, Lagrange Multipliers, and vector functions. She teaches the course from an engineering and physics standpoint using her background from MIT (B.S.) and Stanford (M.S.), as well as her 10+ years tutoring both in the classroom and online. She utilizes Educator's innovative interface to explain topics using diagrams and then spending the majority of time going over many sample questions which will likely be seen in assignments and tests. Other topics covered include continuity, triple integrals, line integrals, and Green's Theorem.

Table of Contents

I. Introduction to Vector Geometric Spaces
	Three Dimensional Coordinate System			31:42
		Intro		0:00
		Three Dimensional Coordinate Systems		0:12
			Topics Overview	0:13
		Two Dimensional to Three Dimensional Coordinate Systems		0:30
			'Addresses' of Points in Two Space	0:31
			Squares to Cubes	1:40
		Three Dimensional Coordinate Systems		2:48
			Orientation	2:49
			Octants	5:11
		Plotting Points in Three Dimensions		7:49
			Plotting Projection	7:50
		Three Dimensional Distance Formula		12:32
			Distance Formula in Two Space	12:34
			Distance Formula in Three Space	13:35
		Curves in Space		14:38
			Curves in Space	14:39
			Equations of Spheres	17:40
		Higher Order Dimensions		20:35
			Higher Order Dimensions	20:36
		Example 1: Graphing x=2 in One, Two, and Three Dimensional Space		24:07
		Example 2: Graphing y=x in Two & Three Dimensional Space		26:10
		Example 3: Finding the Distance Between P(1,1,1) & Q(2,3,4)		27:33
		Example 4: Sphere, Center and Radius		28:40
	Vectors			42:47
		Intro		0:00
		Vectors		0:25
			Topics Overview	0:26
		Vectors & Components		0:43
			Vectors	0:44
			Components	2:50
		Representations, Notation, and Position Vectors		3:30
			Representations	3:31
			Notations	6:17
			Position Vectors	9:30
		Vector Length and Addition		10:32
			Vector Length	10:33
			Vector Addition	12:25
		Triangle Law		19:54
			The Triangle Law	19:56
			Parallelogram Picture	21:11
		Scalar Multiplication		23:19
			Scalar Vector Multiplication	23:20
		Basis & Unit Vectors		26:16
			Definition of Basis Vectors	26:18
			Graphical Representation of Basis Vectors	26:56
			Unit Vectors	29:08
		Properties of Vectors		30:09
			Properties of Vectors	30:10
		Example 1: Finding Vector		34:00
		Example 2: Finding Vector		35:02
		Example 3: Expressing Vectors		38:00
		Example 4: Finding Unit Vector		40:32
	Dot Products			25:26
		Intro		0:00
		Dot Products		0:13
			Topics Overview	0:14
		Definition of Dot Product		0:29
			Definition of Dot Product	0:30
		Properties of Dot Products		3:20
			Properties of Dot Products	3:21
		Angles Between Vectors		7:00
			Angles Between Vectors, Graphically	7:01
			Definition of Angles Between Vectors	8:33
			Cosine of Angles Between Vectors	9:57
		Direction Angles		10:40
			Definition of Direction Angles	10:41
			Formulas of Direction Angles	14:16
		Projections		15:17
			Projections	15:18
			Vector Projections	17:34
			Scalar Projections	18:58
		Example 1: Dot Product		19:39
		Example 2: Vectors and Dot Product		20:30
		Example 3: Are a=2i+2j-2k & b=3i+j+4k Perpendicular?		21:16
		Example 4: Find the Direction Angles of the Vector		22:44
	The Cross Products			47:21
		Intro		0:00
		The Cross Product		0:13
			Topics Overview	0:15
		Definition of The Cross Product		0:32
			Example: Cross Product	0:51
		Determinants		2:51
			What is a Determinant?	3:26
		Determinants		4:10
			Geometrical Definition	4:11
			Simple Rectangle	4:13
			Sheared Rectangular	4:39
		Determinants		6:14
			Mathematical Definition	6:15
		Determinants		10:22
			3 x 3 Matrix	10:28
		Right Hand Rule		16:01
			Cross Product	16:03
			Non-Commutable	17:48
		Lengths of Cross Products		18:04
			Parallel Vectors	23:15
		Propertied of Cross Products		23:36
			Vectors a, b, c and Scalar C	23:40
		Parallelepipeds		26:31
			Geometry of Parallelepipeds	26:48
			Volume of Parallelepipeds	27:24
			Scalar Triple Product	27:43
			Coplanar Vectors	28:45
		Torque		29:26
			Torque Equations	30:52
		Example 1: Finding 'a x b' and 'b x a'		32:23
		Example 2: Area of Triangle		37:08
		Example 3: Coplanar		41:00
		Example 4: Magnitude of Torque		44:47
	Equations of Lines and Planes			40:18
		Intro		0:00
		Equations of Lines and Planes		0:12
			Topics	0:14
		Introduction to Vector Functions		0:39
			Properties of Vector Functions	0:59
		Introduction to Vector Functions		2:04
			Illustrating Vector Functions	2:10
		Vector Equations of Lines		5:07
			Directional Numbers	7:20
		Vector Equations of Lines		8:02
			Graph: Vector Equation of Lines	8:12
		Parametric and Symmetric Equations of Lines		11:30
			Parametric Equations	11:47
			Symmetric Equations	14:00
			Skew Lines	14:38
		Vector Equations of Planes		15:23
			Normal Vectors	16:36
		Scalar and Linear Equations of Planes		17:37
			Scalar Equations of Planes	17:47
			Linear Equations of Planes	19:50
		Example 1: Vector and Parametric Forms		20:56
		Example 2: Symmetric and Parametric Forms		26:07
		Example 3: Equation of Plane		31:50
		Example 4: Equation of Plane		33:57
	Quadratic Surfaces and Cylinders			48:47
		Intro		0:00
		Quadratic Surfaces		0:28
			Topics Overview	0:29
		Quadric Surfaces		0:50
			General Equation of Quadric Surfaces	0:51
			Equations of Common Quadratic Surface	1:35
		Cylinders		2:10
			Definition of Cylinders	2:13
			Special Cases	3:34
		Cylinders		3:53
			Circular Cylinder	3:54
			Equation of Circular Cylinder	4:50
			Example: Circular Cylinder	5:03
		Cylinders		7:20
			Elliptical Cylinder	7:26
			Equation of Elliptical Cylinder	7:54
			Example: Elliptical Cylinder	8:10
		Cylinders		8:26
			Parabolic Cylinder	8:42
			Equation of Parabolic Cylinder	8:57
			Example: Parabolic Cylinder	9:06
		Cylinders		9:24
			Hyperbolic Cylinder	9:27
			Equation of Hyperbolic Cylinder	10:06
			Example: Hyperbolic Cylinder	10:18
		Ellipsoids		10:34
			Definition of Ellipsoid	10:37
			Equation of Ellipsoid	10:43
			General Example	10:58
			Spheroid Example	11:37
			Spherical Example	12:21
		Paraboloids		13:27
			Definition of Paraboloid	13:31
			Elliptical Paraboloid	13:38
			Circular Paraboloid	14:02
			Hyperbolic Paraboloid	14:41
		Cones		15:14
			Conical Surface	15:32
			Equation of Cone	15:40
			Example: Cone and Circular Cone	15:55
		Hyperboloids		16:19
			Definition of Hyperboloids	16:22
			Hyperboloids of One Sheet	16:32
			Hyperboloids of Two Sheets	16:39
		Hyperboloids of One Sheet		17:19
			Equation of Hyperboloids of One Sheet	17:20
			Example: Hyperboloids of One Sheet	17:58
		Hyperboloids of Two Sheets		18:05
			Equation of Hyperboloids of Two Sheets	18:06
			Definition of Hyperboloids of Two Sheets	18:39
			Example: Hyperboloids of One Sheet	19:06
			Example: Hyperboloids of Two Sheets	19:56
		Example 1: Identify and Sketch the Graph		21:00
		Example 2: Identify and Sketch the Graph		25:41
		Example 3: Graph of Quadratic		32:34
		Example 4: Sketch the Graph		35:40
	Cylindrical, Spherical, and Polar Coordinates			46:35
		Intro		0:00
		Cylindrical and Spherical Coordinates		0:20
			Topics Overview	0:21
		Changing Coordinate Systems		0:32
			Different Coordinate Systems	1:18
		Cylindrical Coordinates		1:44
			Transformation Equation: Cylindrical to Rectangular	3:35
			Transformation Equation: Rectangular to Cylindrical	4:12
		Spherical Coordinates		4:46
			Terminology of Spherical Coordinates	5:00
		Spherical Coordinates		11:42
			Transformation Equation: Spherical to Rectangular	12:22
			Transformation Equation: Rectangular to Spherical	13:03
		Example 1: Cylindrical and Rectangular Coordinates		13:26
		Example 2: Identify Surfaces		21:54
		Example 3: Spherical and Rectangular Coordinates		27:45
		Example 4: Identify Surfaces		37:20
II. Vector Valued Functions
	Vector Functions			28:02
		Intro		0:00
		Vector Function		0:13
			Topics Overview	0:14
		Introduction to Vector Functions		0:27
			Definition of Vector Functions	0:28
		Introduction to Vector Functions		1:48
			Illustrating Vector Functions	1:49
		Limits and Continuity of Vector Functions		4:42
			Limit of Vector Functions	4:49
			Continuity of Vector Functions	5:58
		Space Curves		7:46
			Definition of Spaces Curves	7:47
			Equations of Space Curves	8:42
		Types of Space Curves		10:38
			Helices	10:39
			Tori	11:56
		Types of Space Curves		13:01
			Trefoil Knots	13:05
			Twisted Cubics	13:50
		Example 1: Finding Domain		14:52
		Example 2: Curve of Vector Function		20:00
		Example 3: Vector Function		22:28
		Example 4: Vector Function		25:04
	Calculus of Vector Functions			20:55
		Intro		0:00
		Vector Functions		0:15
			Topics	0:16
		Definition of the Derivative of Vector Functions		0:35
			Definition of the Derivative of Vector Functions	0:40
			Convenient Form of Equations	1:32
		Second Derivatives and Smoothness		2:33
			Smoothness	2:35
			Second Derivatives of Vector Functions	3:09
			Piecewise Smooth	3:49
		Differentiation Rules for Vector Functions		4:45
			Differentiation Rules	4:46
		Integration and Vector Functions		8:46
			Indefinite Integration of Vector Functions	8:58
			Definite Integration of Vector Functions	9:55
			Fundamental Theorem of Calculus	10:31
		Example 1: Calculate Limit		11:12
		Example 2: Compute		13:32
		Example 3: Indefinite Integral		14:42
		Example 4: Definite Integral		17:03
	Tangent, Normal, and Binormal Vectors			31:41
		Intro		0:00
		Tangent, Normal, and Binormal Vectors		0:14
			Topics Overview	0:15
		Tangent Vectors		0:22
			Definition of Unit Tangent Vector	0:24
		Normal Vector		2:56
			Definition of Normal Vector	3:00
		Binormal Vectors		7:43
			Definition of the Binormal Vector	8:06
			Normal Plane	9:18
			Osculating Plane	10:41
		Example 1: Tangent Vector		13:01
		Example 2: Vector Equation of Tangent Line		16:50
		Example 3: Normal Vector		21:09
		Example 4: Binormal Vector		27:50
	Arc Length and Curvature			34:52
		Intro		0:00
		Arc Length and Curvature		0:11
			Topics Overview	0:12
		Arc Length		0:21
			Arc Length for Parametric Equations Two Dimensions	0:35
			Arc Length for Parametric Equations Three Dimensions	1:44
		Parameterizations		4:13
			Parameterization of the Arc Length Function	4:29
			Convenient Form	7:23
		Curvature		8:16
			Definition and Example of Curvature	8:17
			Convenient Form	9:13
		Curvature: Special Cases and Equations		10:28
			Curvature in Terms of Cross Products	10:35
			Curvature of a Plane Curve	13:40
		Example 1: Curve Length		14:51
		Example 2: Particle and Arc Length		17:23
		Example 3: Curvature		22:34
		Example 4: Curvature		28:29
	Velocity and Acceleration			48:49
		Intro		0:00
		Velocity and Acceleration		0:30
			Topics	0:31
		Velocity Vectors		1:19
			Speed	1:24
			Velocity	2:41
			Velocity Vector	4:28
		Acceleration Vectors		6:26
			Acceleration Vectors	6:30
			Tangent and Normal Components of Acceleration	7:17
		Acceleration Components and Newton's 2nd Law		12:08
			Acceleration Vectors	12:09
			Other Formulas For the Tangent and Normal Components of Acceleration	13:53
			Newton's Second Law of Motion	17:02
		Example 1: Velocity, Speed, and Acceleration		19:55
		Example 2: Particle's Velocity and Position		23:44
		Example 3: Tangential and Normal Components		30:11
		Example 4: Velocity		37:56
III. Partial Derivatives
	Functions Involving More than One Variable			46:18
		Intro		0:00
		Functions Involving More than One Variable		0:30
			Topics Overview	0:31
		Introduction to Functions Involving Two Variables		0:54
			Definition of Functions of Two Variables	0:59
			Domain and Range for Functions of Two Variables	4:02
			Independent and Dependent Variables	5:19
		Tabular Representations		6:42
			Examples of Tables Involving Functions of Two Variables	7:02
		Graphical Representations		10:24
			Example Graphs	10:33
		Level Curves		12:47
		Level Curves		14:59
		Level Curves: Topological Maps		17:10
		Functions of Three Variables		18:46
			Definition of Functions of Three Variables	18:51
			Level Surfaces	20:42
		Functions of More Than Three Variables		23:16
			Definition of Functions of More Than Three Variables	23:23
			Different Perspectives for Functions of 'n' Variables	28:10
		Example 1: Domain		30:14
		Example 2: Determine Revenue		32:20
		Example 3: Heat Source		39:22
		Example 4: Domain		44:06
	Limits and Continuity			39:55
		Intro		0:00
		Limits and Continuity		0:42
			Topics Overview	0:43
		Limits of Functions of Two Variables		0:58
			Recalling Limits of Functions of One Variables	1:03
			Limits of Functions of Two Variables	3:26
		Continuity		9:02
			Example: Continuity	9:04
		Sandwich Theorem		11:36
			Recalling Sandwich Theorem of Functions of One Variable	11:47
			Squeeze Theorem of Functions of Two Variables	14:04
		When Limits Don't Exist		15:36
			Non-Existent Limits	16:21
			Polynomial and Rational Functions of More than One Variable	18:26
		Functions That Don't Have Limits at (0,0)		21:46
		Methods For Taking Limits		22:20
			Approaches to Taking Limits	22:31
		Limits of Functions of More Than One Variable		25:21
			Limits of Functions of Three Variables	25:33
			Limits of Functions of 'n' Variables	26:03
		Example 1: Limit		28:30
		Example 2: Limit		31:55
		Example 3: Limit		36:01
		Example 4: Function		38:24
	Partial Derivatives			47:31
		Intro		0:00
		Partial Derivatives		0:26
			Topics Overview	0:27
		Introduction to Partial Derivatives		0:52
			Graphical Representation of Partial Derivatives	1:11
			Comparing Partial and Total Derivatives	4:36
		Definitions of Partial Derivatives		9:59
			Partial Derivative of 'f' with respect to x at (a,b)	10:08
			Partial Derivative of 'f' with respect to 'y' at (a,b)	11:50
		Partial Derivatives of Functions of More Than Two Variables		14:35
			Partial Derivatives of Functions of Three Variables	14:43
			Partial Derivatives of Functions of 'n' Variables	17:08
		Higher Order Partial Derivatives		19:52
			Second Derivatives	20:56
			Clairaut's Theorem	24:54
		Partial Differential Equations		26:54
			General Form	27:37
			Practical Equations in Physics and Engineering: Wave Equation	30:26
			Practical Equations in Physics and Engineering: Laplace's Equation	32:37
		Example 1: Function		34:12
		Example 2: First Order of Partial Derivatives		38:30
		Example 3: First Order of Partial Derivatives		41:04
		Example 4: Laplace's Equation		44:11
	Tangent Planes and Linear Approximations			38:33
		Intro		0:00
		Tangent Planes and Linear Approximations		0:27
			Topics Overview	0:28
		Tangent Planes		0:43
			Tangent Lines of Functions of One Variable	0:50
			Tangents Planes of Multivariables Functions	1:16
			Definition of the Tangent Plane of a Function	1:50
		Linear Approximations		4:09
			Tangent Plane of a Function at (a,b)	4:35
			Linearization	5:00
			Linear Approximation	6:00
			Increments	7:43
		Differentiability and Differentials		10:16
			Conditions for Differentiability	10:24
			Differentials for Functions of One Variable	11:57
			Differentials for Functions of Two Variables	15:37
		Functions of More Than Two Variables		17:07
			Linear Approximations of Functions of Three Variables	17:31
			Increment	19:05
			Differentials	19:46
		Example 1: Tangent Plane		20:38
		Example 2: Linear Approximation		25:17
		Example 3: Find dz		30:16
		Example 4: Find dw		35:22
	Chain Rule			54:57
		Intro		0:00
		Chain Rule		0:32
			Topics Overview	0:33
		Introduction to the Chain Rule		0:48
			Chain Rule For Functions of One Variable	0:55
			Independent Variable	2:45
			Dependent Variable	3:30
			Intermediate Variable	4:09
			Tree Diagrams	6:20
		Chain Rule: Special Cases		11:31
			Case 1	11:33
			Case 2	15:12
		Chain Rule: Special Cases		18:58
			Case 3	18:59
		Chain Rule: General Case		23:23
			General Chain Rule Equation	23:39
		Implicit Differentiation: F(x,y)=F(xg(x))=0		26:27
			Example: Implicit Differentiation	27:05
		Implicit Differentiation: F(x,y)=F(xg(x))=0 Cont.		32:43
			Example: Implicit Differentiation	33:14
		Example 1: Tree Diagram		35:37
		Example 2: Chain Rule		39:42
		Example 3: Chain Rule		42:36
		Example 4: Chain Rule		49:10
	Directional Derivatives and Gradient Vectors			53:42
		Intro		0:00
		Directional Derivatives and Gradient Vectors		0:23
			Topics Overview	0:24
		Definition of Directional Derivatives		0:43
			Graphical Representation of Directional Derivatives	1:28
			Definition of Directional Derivatives	4:49
		Computing Directional Derivatives		6:40
			Unit Vectors and Directional Derivatives	7:07
			Computing Directional Derivatives	12:45
		Gradient Vectors		14:51
			Graphically Defining Gradient Vectors	15:02
			Mathematically Defining Gradient Vectors	15:52
			Directional Derivatives of Gradient Vectors	17:00
		Properties of Gradient Vectors		18:43
			Properties of Gradient Vectors	18:52
			Uses of the Gradient Vectors	22:51
		Functions of Three Variables		27:34
			Definition of the Directional Derivative	27:40
			Definition of the Gradient Vector	28:32
			Directional Derivative in Terms of Gradient Vector	29:04
		Gradient Vectors and Tangent Planes		29:35
		Example 1: Directional Derivative		37:58
		Example 2: Rate of Change		42:47
		Example 3: Rate of Change		48:08
		Example 4: Symmetric Equations		50:45
	Maxima and Minima			68:06
		Intro		0:00
		Maxima and Minima		0:17
			Topics Overview	0:18
		Minima and Maxima		0:28
			Local Maxima	2:20
			Local Minima	3:04
			Absolute Maxima and Minima	4:47
			Partial Derivatives, Minima and Maxima	7:23
		Critical Points & The Second Derivative Test		9:08
			Critical Points	9:34
			Second Derivative Test	15:53
		Extreme Value Theorem		21:56
			Closed Region Definitions	22:24
			Open Region Definitions	22:44
			Bounded Region Definitions	23:11
			Extreme Value Theorem	24:16
			Method for the Extreme Value Theorem	27:45
		Example 1: Local Maxima, Local Minima, and Saddle Points		30:41
		Example 2: Critical Points		42:37
		Example 3: Absolute Minimum and Absolute Maximum		48:54
		Example 4: Absolute Minimum and Absolute Maximum		57:14
	Lagrange Multipliers			36:00
		Intro		0:00
		Lagrange Multipliers		0:20
			Topics Overview	0:21
		Introduction to Lagrange Multipliers		0:32
			What Are Lagrange Multipliers?	0:34
			Lagrange Multiplier Relational Equation	2:01
			Applications of Lagrange Multipliers	2:36
		Method of Lagrange Multipliers		3:42
			How to Use Lagrange Multipliers	3:49
		Functions With Two Constraints		8:48
			Using Lagrange Multipliers with Functions With Two Constraints	9:01
		Example 1: Products Production		11:54
		Example 2: Dimensions of Rectangular Box		18:45
		Example 3: Extreme Values		26:37
		Example 4: Extreme Values		31:43
IV. Multiple Integrals
	Introduction to Double Integrals			38:39
		Intro		0:00
		Introduction to Double Integrals		0:20
			Topics Overview	0:21
		Riemann Sums and Double Integrals		0:32
			Reviewing Definite Integrals	0:47
			Volumes and Definite Integrals	7:18
		The Midpoint Rule and Average Values		9:34
			The Midpoint Rule	9:53
			Average Values	12:06
		Properties of Double Integrals		14:00
			Properties of Double Integrals 1	14:13
			Properties of Double Integrals 2	15:00
			Properties of Double Integrals 3	15:26
		Example 1: Volume of Solid		16:52
		Example 2: Volume of Solid		22:18
		Example 3: Midpoint Rule		28:50
		Example 4: Volume & Average Value		34:47
	Integrating Multivariable Functions			29:13
		Intro		0:00
		Integrating Multivariable Functions		0:20
			Topics Overview	0:22
		Single Integrals & Multivariable Functions		0:34
			General Forms of Single Integrals of Multivariable Functions	0:41
			Treating Non-Integrated Variable as a Constant	1:01
			Example	1:55
		Iterated Integrals		3:04
			What Are Iterated Integrals?	3:09
			Method of Iterated Integrals	4:48
		Fubini's Theorem		9:46
			Definition of Fubini's Theorem	9:54
			Reasoning Behind Fubini's Theorem	11:20
		Example 1: Integration		13:49
		Example 2: Integration		15:09
		Example 3: Integration		21:08
		Example 4: Volume of Solid		24:08
	Double Integrals Over Regions			47:58
		Intro		0:00
		Double Integrals Over Regions		0:24
			Topics Overview	0:25
		Double Integrals Type 1		0:45
			Graphical Representation	0:51
			Definition of Region	3:05
			Double Integral	4:29
		Double Integrals Type 2		6:19
			Graphical Representation	6:21
			Definition of Region	8:44
			Double Integral	9:27
		Checklist For Integration		10:57
			Rule Number 1	11:16
			Rule Number 2	12:30
			Rule Number 3	13:25
		Properties of Double Integrals Over General Regions		15:22
			Property 1	17:10
			Property 2	18:41
			Property 3	19:18
		Bound Volumes		20:24
			Method of Finding Bound Volumes	21:12
		Example 1: Evaluate		24:43
		Example 2: Evaluate		28:56
		Example 3: Volume		34:24
		Example 4: Volume		41:19
	Double Integrals in Polar Coordinates			38:46
		Intro		0:00
		Double Integrals In Polar Coordinates		0:36
			Topics Overview	0:38
		Comparing Rectangular & Polar Coordinates		0:52
			Example Problem in Rectangular Coordinates	1:07
			Same Example Problem in Polar Coordinates	4:18
		Polar Rectangles & Area Translation		6:55
			Polar Rectangles	7:05
			Defining dA	10:43
		Translation Equations		15:16
			Polar Rectangular Regions	16:11
			General Regions	18:46
		Example 1:Are Enclosed by Two Loops		20:51
		Example 2: Find the Area		24:47
		Example 3: Find the Volume		29:43
		Example 4: Integrate		34:08
	Triple Integrals			47:58
		Intro		0:00
		Triple Integrals		0:18
			Topics Overview	0:19
		Definition of Triple Integrals		0:34
			Triple Riemann Sums	0:39
			Definition	4:20
			Fubini's Theorem	5:41
		Triple Integrals (Type I)		8:26
			Graphic	8:54
			Formula	11:06
		Triple Integrals (Type II)		12:27
			Graphic	12:30
			Formula	14:15
		Triple Integrals (Type III)		15:14
			Graphic	15:18
			Formula	16:26
		Applications of Triple Integrals		17:19
			Mass	17:44
			Center of Mass	18:08
			Moments of Inertia	21:04
		Applications of Triple Integrals		22:07
			Electric Charge	22:12
			Probability Density	22:55
		Example 1: Integrate		25:37
		Example 2: Integrate		31:37
		Example 3: Integrate		34:39
		Example 4: Mass and Constant Density		40:48
	Triple Integrals in Cylindrical and Spherical Coordinates			42:05
		Intro		0:00
		Triple Integrals in Cylindrical and Spherical Coordinates		0:19
			Topics Overview	0:20
		Cylindrical Coordinates		0:34
			Graphical Presentation of Cylindrical Coordinates	0:35
			Rectangular to Polar Transformation Equations	6:53
			Rectangular to Cylindrical Coordinate Equations	7:22
			Rectangular to Cylindrical Transformation Equations	7:56
		Triple Integrals and Cylindrical Coordinates		8:26
			General Triple Integral Equation in Cylindrical Coordinates	8:28
			Triple Integral Equation in Cylindrical Coordinates for Polar 'D' Regions	10:46
			Conversion Procedures	13:44
		Spherical Coordinates		17:14
			Graphical Presentation of Spherical Coordinates	17:15
			Rectangular to Spherical Coordinate Transformation Equations	22:40
			Rectangular to Spherical Transformation Equations	23:39
		Triple Integrals and Spherical Coordinates		25:41
			Triple Integral Equation in Spherical Coordinates	25:57
			Conversion Procedure	27:03
		Example 1: Convert Cylindrical Coordinates into Rectangular Coordinates		28:51
		Example 2: Integrate		30:21
		Example 3: Write in Spherical Coordinates		36:14
		Example 4: Integrate		98:04
	Changing Variables in Multiple Integrals			37:11
		Intro		0:00
		Changing Variables in Multiple Integrals		0:21
			Topics Overview	0:22
		Introduction to Changing Variables		0:32
			Changing Variables for Single Integrals	0:33
			Changing Variables for Polar Integrals	1:59
			General Changing of Variables	3:08
		Changing Variables for Double Integrals		5:06
			Approximation of Area R	5:07
			Jacobian of Transformation	7:53
			Change of Variables for Double Integrals	13:19
		Changing Variables for Triple Integrals		14:24
			Transformation Equations	14:26
			Jacobian of Transformation	15:09
			Change of Variables for Triple Integrals	16:21
		Example 1: Transform the Equation		17:44
		Example 2: Evaluate		18:34
		Example 3: Polar Coordinate Change		25:31
		Example 4: Spherical Coordinate Change		29:11
VI. Vector Calculus
	Applications of Double Integrals			48:33
		Intro		0:00
		Applications of Double Integrals		0:12
			Topics Overview	0:13
		Density & Mass		0:29
			Density of a Small Rectangle on 'D'	0:47
			Total Mass	3:10
			Mass as a Function of Density	5:04
		Review: Moments & Centers of Mass		6:10
			Balancing Equation	6:14
			Center of Mass, Simple System	7:26
			Center of Mass, n particles	8:29
			Moments About x & y	9:57
			Centers of Mass	11:34
		Moments & Centers of Mass		13:44
			Moment Approximation	13:45
			Moment About x	15:15
			Moment About y	16:17
			Centers of mass	17:07
		Moment of Inertia		18:50
			Definition	18:51
			Procedure	19:28
			Moment of Inertia About x	20:58
			Moment of Inertia About y	21:58
			Moment of Inertia About Origin (Polar Moment of Inertia)	22:50
		Probability		24:35
			Probability Density as a Function of One Variable	24:36
			Vocabulary: Independent Random Variables	25:33
			Probability Density as a Function of Two Variables (Joint Density Function)	26:09
			Normalized Probability Density Function	27:31
		Expected Values		29:25
			Mean of Single Variable Probability Density Function	29:26
			Expected Values of x & y	29:54
			Two Variable Joint Density Function for Normal Distributions	30:27
		Example 1: Charge Density		32:04
		Example 2: Moments of Inertia		35:39
		Example 3: Joint Density		40:17
		Example 4: Joint Density		42:17
	Area, Surface Area, and Volume			31:07
		Intro		0:00
		Area, Surface Area, and Volume		0:17
			Topics Overview	0:18
		Area and Surface Area		0:26
			Area Formula	0:27
			Approximate Surface Area Formula	6:02
			Surface Area Integral Expressions	6:48
		Volume		8:33
			Triple Integral Volume Expression	8:47
			Volume Functions for z=f(x,y)	9:40
			Volume Functions for y=f(x,z)	10:34
			Volume Functions for x=f(y,z)	10:47
		Example 1: Find the Surface Area		11:11
		Example 2: Find the Surface Area of the Paraboloid		15:31
		Example 3: Find the Surface Area of a Sphere		19:58
		Example 4: Find the Volume of a Sphere		26:04
V. Vector Calculus
	Vector Fields			35:54
		Intro		0:00
		Vector Fields		0:16
			Topics Overview	0:17
		Vector Fields In Two Space		0:28
			Definition	0:31
			Examples	3:50
		Vector Fields In Three Space		7:11
			Definition	7:12
			Examples	9:08
		Gradients Fields		15:25
			Definition	15:56
			Conservative Vector Fields	18:00
		Example 1: Sketch The Vector Field		19:30
		Example 2: Sketch The Vector Field		25:40
		Example 3: Sketch The Vector Field		29:36
		Example 4: Sketch The Gradient Vector		34:41
	Line Integrals			39:42
		Intro		0:00
		Line Integrals		0:25
			Topics Overview	0:26
		Review of Parametric Equations		0:43
			Review of Parametric Formulas	0:45
			Example Curve: Smooth	7:27
			Example Curve: Parametric Equation	7:50
			Example Curve: Parameterization as a Vector Function	8:09
		Introduction to Line Integrals		8:24
			Line Integrals as a Sum	8:31
			Reviewing Arc Length	10:58
			Definition: General Line Integral	11:29
			Definition: Line Integral from a to b	11:48
			Definition: Line Integrals in Terms of Vectors	12:53
		Piecewise Smooth and Path Direction		14:48
			Piecewise Smooth	14:51
			Path Direction	17:23
		Line Integrals with Respect to Specific Variables		21:25
			Line Integral with Respect to x	21:34
			Line Integral with Respect to y	21:58
			Combined Equation	22:28
		Line Integrals in Three Space		23:31
			Line Integrals In Space	23:39
			Line Integrals in Space with Respect to Specific Variables	25:51
		Example 1: Evaluate		28:16
		Example 2: Evaluate		30:57
		Example 3: Evaluate		33:05
		Example 4: Evaluate		35:49
	Line Integrals of Vector Fields			27:41
		Intro		0:00
		Line Integrals of Vector Fields		0:28
			Topics Overview	0:29
		Introduction to Line Integrals of Vector Fields		0:40
			Reviewing Vector Fields	0:47
			Work Done by a Continuous Force	1:45
		Defining Line Integrals of Vector Fields		6:31
			General Definition	6:34
			Deriving the General Definition	8:09
		Procedure for Finding Line Integrals of Vector Functions		10:09
			Steps in Fiding Line Integrals of Vector Functions	10:14
		Example 1: Force Field		12:25
		Example 2: Evaluate		15:49
		Example 3: Evaluate		19:06
		Example 4: Evaluate		24:15
	Fundamental Theorem for Line Integrals			51:04
		Intro		0:00
		The Fundamental Theorem for Line Integrals		0:39
			Topics Overview	0:41
		Defining & Deriving the Fundamental Theorem		0:55
			Fundamental Theorem of Calculus for Definite Integrals	0:57
			Fundamental Theorem of Calculus for Line Integrals	1:23
		Reviewing & Defining Terminology		9:13
			Definitions and Review	9:19
		Path Independence		18:57
			Example: Closed Path	19:04
		Conservative Vector Fields		26:11
			Example: Conservative Vector Field	26:13
		Conservation of Energy		31:50
			Examples	32:04
		Example 1: Evaluate		37:45
		Example 2:Conservative		41:01
		Example 3: Evaluate		42:42
		Example 4: Integrate and Differentiate		46:31
	Green's Theorem			33:21
		Intro		0:00
		Green's Theorem		0:09
			Overview	0:10
		Defining Green's Theorem		0:22
			Orientation of Green's Theorem	0:23
			Definition of Green's Theorem	2:00
		Green's Theorem for Regions with Holes		5:01
			Green's Theorem for Regions with Holes	5:02
		Finding Area with Line Integrals		13:19
			Finding Area with Line Integrals	13:20
		Example 1: Evaluate		17:10
		Example 2: Evaluate		21:13
		Example 3: Find the Area		24:32
		Example 4: Evaluate		28:26
	Parametric Surfaces			44:15
		Intro		0:00
		Parametric Surfaces		0:19
			Topics Overview	0:20
		Introduction to Parametric Surfaces		0:32
			Review of Parametric Equations	0:33
			Parametric Surfaces	2:22
			Equations of Parametric Surfaces	4:18
		Surfaces of Revolution		5:59
			Surfaces of Revolution	6:01
			Revolving Surfaces about the x-axis	8:05
		Tangent Planes		8:35
			Tangent Planes & Parametric Surfaces	8:37
		Surface Areas of Parametric Surfaces		15:25
			Defining the Surface Area of Parametric Surface	15:27
		Example 1: Find the Surface Given by the Parametric Representation		22:08
		Example 2: Give the Parametric Representation of the Sphere		24:45
		Example 3: Find the Tangent Plane to the Surface with the Parametric Equations		28:17
		Example 4: Find the Surface Area of a Sphere of Radius 2		36:24
	Surface Integrals			51:43
		Intro		0:00
		Surface Integrals		0:11
			Topics Overview	0:12
		Introduction to Surface Integrals		0:27
			Defining Surface Integrals	0:28
			Surface Integral Equations	3:32
		Surface Integrals of Parametric Surfaces		8:13
			Graphically	8:14
			Parameterization	12:23
		Oriented Surfaces		13:45
			Non-oriented Surfaces	13:47
			Oriented Surfaces	15:36
		Surface Integrals of Vector Fields		21:52
			Graphically	21:53
			Equations	23:01
		Surface Integrals, Parametric Surfaces, & Vector Fields		30:10
			Equations of Parametric Surfaces	30:11
		Example 1: Evaluate		32:52
		Example 2: Evaluate		37:10
		Example 3: Use Surface Integrals to Find the Area of a Disk		42:16
		Example 4: Evaluate		46:52