The Chain Rule, Multivariable Calculus

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Post by Caleb Lear on October 11, 2012

I like your method for using the chain rule, I wonder if there's a way to adapt it for implicit differentiation? Right now I'm just deriving as before and multiplying by a partial where I need to.

3 answers

Last reply by: Professor Hovasapian
Wed Mar 13, 2013 5:34 AM

Post by Yujin Jung on March 11 at 02:17:42 AM

Hello!

Do you by any chance teach the 'Jacobian matrix' of a function?
Our lecturer referred to it as he was teaching the chain rule and was wondering how they relate to each other..

Thank you :)

The Chain Rule

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

Intro

The Chain Rule

I. Vectors
	Points & Vectors	28:23
	Scalar Product & Norm	30:25
	More on Vectors & Norms	38:18
	Inequalities & Parametric Lines	33:19
	Planes	29:59
	More on Planes	34:18
II. Differentiation of Vectors
	Maps, Curves & Parameterizations	29:48
	Differentiation of Vectors	39:40
III. Functions of Several Variables
	Functions of Several Variable	29:31
	Partial Derivatives	23:31
	Higher and Mixed Partial Derivatives	30:48
IV. Chain Rule and The Gradient
	The Chain Rule	28:03
	Tangent Plane	42:25
	Further Examples with Gradients & Tangents	47:11
	Directional Derivative	41:22
	A Unified View of Derivatives for Mappings	39:41
V. Maxima and Minima
	Maxima & Minima	36:41
	Further Examples with Extrema	32:48
	Lagrange Multipliers	32:32
	More Lagrange Multiplier Examples	27:42
	Lagrange Multipliers, Continued	31:47
VI. Line Integrals and Potential Functions
	Line Integrals	36:08
	More on Line Integrals	28:04
	Line Integrals, Part 3	29:30
	Potential Functions	40:19
	Potential Functions, Continued	31:45
	Potential Functions, Conclusion & Summary	28:22
VII. Double Integrals
	Double Integrals	29:46
	Polar Coordinates	36:17
	Green's Theorem	38:01
	Divergence & Curl of a Vector Field	37:16
	Divergence & Curl, Continued	33:07
	Final Comments on Divergence & Curl	16:49
VIII. Triple Integrals
	Triple Integrals	27:24
	Cylindrical & Spherical Coordinates	35:33
IX. Surface Integrals and Stokes' Theorem
	Parameterizing Surfaces & Cross Product	41:29
	Tangent Plane & Normal Vector to a Surface	37:06
	Surface Area	32:48
	Surface Integrals	46:52
	Divergence & Curl in 3-Space	23:40
	Divergence Theorem in 3-Space	34:12
	Stokes' Theorem, Part 1	22:01
	Stokes' Theorem, Part 2	20:32

Multi-Variable Calculus

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

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