  Raffi Hovasapian

Higher and Mixed Partial Derivatives

Slide Duration:

Section 1: Vectors
Points & Vectors

28m 23s

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
14:51
Example 1
16:52
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

30m 25s

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

38m 18s

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

33m 19s

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

29m 59s

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

34m 18s

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

29m 48s

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

39m 40s

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

29m 31s

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

23m 31s

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
20:26
Higher and Mixed Partial Derivatives

30m 48s

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

28m 3s

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

42m 25s

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

47m 11s

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

41m 22s

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
33:23
Example 2
34:53
A Unified View of Derivatives for Mappings

39m 41s

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

36m 41s

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

32m 48s

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

32m 32s

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

27m 42s

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

31m 47s

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

36m 8s

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

28m 4s

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

29m 30s

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

40m 19s

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

31m 45s

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

28m 22s

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

29m 46s

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

36m 17s

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

38m 1s

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

37m 16s

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

33m 7s

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

16m 49s

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

27m 24s

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

35m 33s

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

41m 29s

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

37m 6s

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

32m 48s

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

46m 52s

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

23m 40s

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

34m 12s

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

22m 1s

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

20m 32s

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
Bookmark & Share Embed

## Copy & Paste this embed code into your website’s HTML

Please ensure that your website editor is in text mode when you paste the code.
(In Wordpress, the mode button is on the top right corner.)
×
• - Allow users to view the embedded video in full-size.
Since this lesson is not free, only the preview will appear on your website.

• ## Related Books 1 answer Last reply by: Professor HovasapianWed Dec 30, 2015 12:27 AMPost by Micheal Bingham on December 23, 2015Hello Professor, Does the mixed partial derivative theorem hold for f: R^n --> R  f(x_1, x_2, x_3, ..., x_n) or does it only hold for n = 2 and 3  ? 1 answer Last reply by: Professor HovasapianTue Sep 8, 2015 8:58 PMPost by Jim Tang on September 7, 2015Wait I'm confused. The partial derivative of x^2+y with respect to x is 2x and not 2x+y? I think in some other lecture you wrote 2x+y but I might be mistaken. 1 answer Last reply by: Professor HovasapianTue Jul 28, 2015 11:27 PMPost by Jonathan Snow on July 25, 2015Hey, Thanks for the great lecture, I was wondering why you said "power rule" at around 27:30, did you mean product rule? 1 answer Last reply by: Professor HovasapianTue Nov 25, 2014 2:25 AMPost by Utomo Pratama on November 24, 2014Dear Prof. Hovasapian,Thank you for the great delivery. Now Math is becoming interesting to me.Just some quick clarifications, if there are 3 variables, then there would be six combinations of equal mixed partial derivatives? (3!)CMIIWRegards,Utomo 1 answer Last reply by: Professor HovasapianThu Feb 27, 2014 7:33 PMPost by Chase Lottinger on February 26, 2014Hello, I was just wondering why don't you cover multivariable limits? Or is it labeled as something else in the table of contents? 1 answer Last reply by: Professor HovasapianWed Apr 3, 2013 2:30 AMPost by Jawad Hassan on April 2, 2013Hi raffi!I was wondering if you know any good place to practise exam problems for multivariable calculus? once i am done with all the videos.Or any good books you recomend that have exam problems and good solution manual.Btw excellent video! I am realy enjoying math atm.regards-Jawad 2 answersLast reply by: Matt CWed Mar 27, 2013 9:17 PMPost by Matt C on March 25, 2013In your notes you state Assume D1, D2, D1D2, D2D1, exist and are continuous then D1D2f = D2D1f. Is it ever possible that they can be not equal? Im guessing they would be discontinuous then, but that is just a guess. I have done many problems in my book and have yet to find a case where they don't equal each other, they all seem to work out. 1 answer Last reply by: Professor HovasapianTue Aug 7, 2012 4:45 PMPost by Shahaz Shajahan on August 7, 2012Hi, I know this is very basic but anything to take part,right?when you was expanding D1D1 you forgot to expand out the 2 onto the second term so it should have read out 2cos(x^2+y) instead of just cos(x^2+y)Btw you have been a real help with my exams, as i have a month left and was really worried i'd fail but you have given me renewed hope! :D

### Higher and Mixed Partial Derivatives

Let f(x,y) = − 3x2 + 5xy − y2 Find [df/dx] and [df/dy]
• [df/dx] is the partial derivative in respect to x, similarly [df/dy] is the partial derivative in respect to y.
So [df/dx] = − 6x + 5y and [df/dy] = 5x − 2y.
Let f(x,y) = − 3x2 + 5xy − y2 Find ( [d/dx] )2f and ( [d/dy] )2f
• Note that ( [d/dx] )2f = [d/dx]( [df/dx] )f, so ( [d/dx] )2f = [d/dx]( − 6x + 5y ) = − 6.
Similarly ( [d/dy] )2f = [d/dy]( 5x − 2y ) = − 2.
Let f(x,y) = − 3x2 + 5xy − y2 Verify [d/dx][d/dy]f = [d/dy][d/dx]f
• To compute [d/dx][d/dy]f, we find the partial derivative of f in respect to y and then take the partial derivative of f in respect to x.
• So [d/dx][d/dy]f = [d/dx]( 5x − 2y ) = 5.
Similarly [d/dy][d/dx]f = [d/dy]( − 6x + 5y ) = 5. We note that both [d/dx][d/dy]f and [d/dy][d/dx]f yield the same result.
Let g(x,y) = cos(x2 + 1) − exsin(y) Find ( [d/dx] )2[d/dy]g
• To compute ( [d/dx] )2[d/dy]g, we take each partial derivative to the left of g. Note that ( [d/dx] )2 = [d/dx][d/dx].
So ( [d/dx] )2[d/dy]g = [d/dx][d/dx][d/dy]g = [d/dx][d/dx]( − excos(y) ) = [d/dx]( − excos(y) ) = − excos(y).
Let g(x,y) = cos(x2 + 1) − exsin(y) Find ( [d/dy] )2[d/dx]g
• To compute ( [d/dy] )2[d/dx]g, we take each partial derivative to the left of g. Note that ( [d/dy] )2 = [d/dy][d/dy].
So ( [d/dy] )2[d/dx]g = [d/dy][d/dy][d/dx]g = [d/dy][d/dy]( − 2xsin(x2 + 1) − exsin(y) ) = [d/dy]( − excos(y) ) = exsin(y).
Let g(x,y) = cos(x2 + 1) − exsin(y) Verify [d/dy][d/dx]g = [d/dx][d/dy]g
• Note that [d/dy][d/dx]g = − excos(y) and [d/dx][d/dy]g = − excos(y) from our previous problems.
[d/dy][d/dx]g = [d/dx][d/dy]g since [d/dy][d/dx]g and [d/dx][d/dy]g yield the same result.
Let h(x,y,z) = [1/xyz] Evaluate [d/dz][d/dx]h(1,1, − 2)
• To compute [d/dz][d/dx]h(1,1, − 2) we first find [d/dz][d/dx]h. Note that [1/xyz] = (xyz) − 1 = x − 1y − 1z − 1.
• Now [d/dz][d/dx]h = [d/dz]( − [1/(x2yz)] ) = [1/(x2yz2)].
Hence [d/dz][d/dx]h(1,1, − 2) = [1/((1)2(1)( − 2)2)] = [1/4].
Let h(x,y,z) = [1/xyz] Evaluate [d/dz][d/dy]h(1,1, − 2)
• To compute [d/dz][d/dy]h(1,1, − 2) we first find [d/dz][d/dy]h. Note that [1/xyz] = (xyz) − 1 = x − 1y − 1z − 1.
• Now [d/dz][d/dy]h = [d/dz]( − [1/(xy2z)] ) = [1/(xy2z2)].
Hence [d/dz][d/dy]h(1,1, − 2) = [1/((1)(1)2( − 2)2)] = [1/4].
Let h(x,y,z) = [1/xyz] Evaluate [d/dz][d/dy][d/dx]h(1,1, − 2)
• To compute [d/dz][d/dy][d/dx]h(1,1, − 2) we first find [d/dz][d/dy][d/dx]h. Note that [1/xyz] = (xyz) − 1 = x − 1y − 1z − 1.
• Now [d/dz][d/dy][d/dx]h = [d/dz][d/dy]( − [1/(x2yz)] ) = [d/dz]( [1/(x2y2z)] ) = − [1/(x2y2z2)].
Hence [d/dz][d/dy][d/dx]h(1,1, − 2) = − [1/((1)2(1)2( − 2)2)] = − [1/4].
Let h(x,y,z) = [1/xyz] Evaluate [d/dx][d/dy][d/dz]h(1,1, − 2)
• To compute [d/dx][d/dy][d/dz]h(1,1, − 2) we first find [d/dx][d/dy][d/dz]h. Note that [1/xyz] = (xyz) − 1 = x − 1y − 1z − 1.
• Now [d/dx][d/dy][d/dz]h = [d/dx][d/dy]( − [1/(xyz2)] ) = [d/dx]( [1/(xy2z2)] ) = − [1/(x2y2z2)].
Hence [d/dx][d/dy][d/dz]h(1,1, − 2) = − [1/((1)2(1)2( − 2)2)] = − [1/4].

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

### Higher and Mixed Partial Derivatives

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 0:00
• Higher and Mixed Partial Derivatives 0:45
• Definition 1: Open Set
• Notation: Partial Derivatives
• Example 1
• Theorem 1
• Now Consider a Function of Three Variables
• Example 2
• Caution
• Example 3

### Transcription: Higher and Mixed Partial Derivatives

Hello and welcome back to educator.com and welcome back to Multivariable Calculus.0000

Today, we are going to be talking about higher derivatives and mixed partial derivatives.0005

So, we introduced the notion of the partial derivative of a function of several variables.0010

You basically take the single variable derivative by holding the other variables constant, and you work through a series of partial derivatives, first derivatives.0015

Just like for single variable, we can go ahead and take higher derivatives, second, third, fourth, and fifth, but now we can mix and match.0024

For example, we can take the first partial derivative and then we can take the derivative with respect to the other variable of that derivative, and back and forth.0030

Let us sort of just jump in with an example and see what we can do with this.0040

Before I actually discuss that, however, I want to introduce a notion called an open set.0046

I am not going to spend a lot of time on it, I just want you to know that this idea of an open set is going to be the domain over which we are going to be defining our functions of several variables.0052

I want you to be aware of it, it is going to come up over, and over, and over again in the theorems, and the definitions -- very, very simple -- it is exactly the same as it is for single-variable calculus.0063

It is basically just a region, all the points in that region, but not including the endpoint, not including the boundaries.0072

Let us go ahead and just write the definition down and then we will move on to mixed partial derivatives.0081

So, definition. An open set in n-space -- excuse me -- is the analog, this is not really a formal definition, it is an informal definition.0086

If you want the formal definition, you know, you can go into your books... is the analog to an open set in 1-space, which is the real number line.0108

Which is what you have been dealing with all of these years. You know that the real number line... something like this... let us say this is 0, let us say we have -1, let us say we have +1.0124

Well the open set, you remember, is the notation that has the parentheses like this instead of the closed brackets, which means that we include all the numbers between -1 and 1, but we do not include the endpoints, that is it.0134

So, basically, an open set is every point within a specified boundary, but not including the boundary points.0150

That is really all we are talking about. If you have something like... let us do a 2-space example... so this is the x,y plane, and you have some region like this.0191

Actually, you know what, let us go ahead and make it so it looks... so let us say we have some region like this, I will put dotted, something like that.0203

So this region in here, that is an open set. It does not include the boundary. That is all we are talking about.0215

Just a quick version in R3, so this is the x, the y, and the z.0221

So we have some... let us just do a sphere. We call it a ball. Here, if you have a circle in the plane, we call it an open disc.0230

Again, the specific things that we call them are not important. The idea is the notion of an open set, or an open region.0240

Because you are not always going to get a perfect domain, you know a nice symmetrical domain like a ball or a disc.0247

In this case, let us just do that, and then maybe like that, so think of a sphere in n-space anywhere -- centered at the origin or centered at some place else -- but, not including the actual boundaries themselves, everything inside.0254

So again, these open sets... let me write this down.0268

So, these open sets form the domains over which we define our functions of several variables.0273

So, if we had some function f that is a mapping from R2 to R, let us go ahead and make sure this is a little bit more clear.0304

So some function f, and we are taking points in R2, the vector in 2-space, a point in 2-space, we are doing something to that vector, we are spitting out a number. Let us say the domain is this.0314

Basically, all of the points in here, those are the arguments that go into the actual function. Those are the points that we operate on, if you will. That is all.0324

Just a notion that you should be aware of. It is going to come up a lot, again. All of the definitions and the theorems.0334

Okay, now let us go ahead an move on to our partial derivatives.0340

Again, we can take a second and third, fourth, fifth derivatives of functions of several variables, but now of course we are dealing with partial derivatives in the sense that we are holding one variable constant and we are differentiating with respect to the others.0345

So, let us talk about notation first. This is very, very important.0360

As you noticed from the last lesson, the notation for partial differentiation tends to become a little bit more involved because again you have more variables involved, that is what is going.0365

It is very important to be accustomed to the notation and the scientific literature and the different books that you are going to read and the different teachers that you are going to have, they tend to use notations form a lot of different sources.0373

We want to make sure that if you have not seen a particular notation, at least you can recognize was is being said.0387

We said we have... let me just write here, notation... so, we will let f be a function from R2 to R, so in this case we are going to start a function of 2 variables.0393

So, an f(x,y) if we happen to call the variables x and y, but we do not have to it could be x1, x1, the variables themselves do not actually matter.0413

It is the notion underneath. The partial derivative of f, with respect to the first variable x is notated like that, that little modified d, df/dx.0420

We also do d1, so d1 means take the partial derivative with respect to the first variable, the first variable being x.0433

They are written in order in the argument, that is it.0440

Again, you also have seen it as dx. Well, df/dy the partial derivative with respect to y, that is one notation, and we often write it as d2, so we are taking the partial derivative with respect to the second variable.0444

This capital D notation, that is actually very, very common in multi-variable calculus. You will this a lot also, but often times just to sort of keep the mathematics on the paper reasonably sane.0461

When you have all these df/dx's df/dy's floating around, it gets really crazy -- so, this d notation, this capital D notation - very, very, convenient.0472

Again, you will also see it as dy, specifically.0483

Now, if we want to take the derivative with respect to x of the first partial, in other words, d/dx of df/dx, it is like this.0488

It is going to be d/dx of df/dx, so the partial derivative with respect to x of the partial derivative with respect to x.0501

It is notated as d2f/dx2 completely analogous to single variable calculus.0516

The capital notation is something like this, just D1/D1f, in other words you have taken the first partial derivative, now take the partial derivative with respect to the first variable of what you just did.0523

This is also... you see it this way, d2f. This means take the partial derivative with respect to x, then take the partial derivative with respect to x again, or the first variable.0539

Now, let us do d/dy of df/dy, that is going to be the same as d2f, dy2.0553

Again, you have got d2/d2f and this is often written -- sometimes -- squared f.0565

I personally do not like this notation myself, this d12f, d22f.0575

I like to see everything that I am doing, so this tells me that, it takes the first, I take the derivative with respect to the second variable, and then of that thing, I take the derivative with respect to the second variable.0578

These are just differential operators. They tell me what to differentiate. That is all.0590

Now, let us go ahead and do, d/dx of df/dy.0596

Now, I have taken the partial derivative with respect to y, the second variable, now I am going to go ahead and take the partial derivative of that with respect to the first variable.0610

That becomes... actually, let me just do this one in reverse. This one I am going to write the capital D notation first, because I think it is a little bit better, and then I will write this modified D notation here.0623

So, d1d1, that means I have taken the derivative with respect to the first variable x, and now I differentiate that with respect to the second variable.0637

That is going to be the same as d/dy of df/dx, and that is d2f dy dx.0649

Clearly you can see that this notation is going to start to get really, really cumbersome very, very quickly.0664

I mean it is very beautiful aesthetically, and it is very nice to look at, and it is really important that when you do your math you at least like what you see, but it can be a little daunting.0669

Let us go ahead and specifically say, this is what we are doing first. We are moving from right to left.0682

This means you have a function f, do D1 to it, then do D2 on it. That is first, and this is second, order is very important here.0688

Now, of course we have D1, D2 of f, which means we have done D2 first, then done D1, and this is going to be d/dx of df/dy = d2f dx dy.0698

Wow, that is a lot.0717

Alright, so let us just do an example. That is the best way to make sense of this.0722

Again, you are reasonably comfortable with differentiation of basic functions. With multi-variable calculus just be a little bit more careful.0727

You have to remember which variable you are differentiating with respect to so just go a little bit slower. That is all you have to do with Math in order to be correct.0736

Just go slow and be careful. So, example 1.0742

We will let f(x,y) = x3y2, so d1 the derivative with respect to x is equal 3x2y2, in other words I am treating y as a constant and just differentiating with respect to x.0751

D2 = x3y, this time I ended up holding x constant and differentiating with respect to the second variable, y.0772

Now, I am going to do d2 of d1, in other words I have done d1, now I am going to differentiate this thing with respect to y, the second variable.0791

This is going to give me, 2 × 3 is going to give me 6x2y.0801

Now, I am going to do D1 of D2, which I just did. I just did D2 that is x3y, then do D1 of that, which means I am going to differentiate this with respect to x.0809

I get 6x2y. We will stop and take a look at this.0823

D2D1, I did D1 first then I did D2, I got 6x2y.0828

here I did D2 first, then I got D1, then I did 6x2y and they ended up being the same. This is not a coincidence.0832

So, we will put MV, which means... take note of this... these ended up being equal.0843

Not a coincidence. Now, let us go ahead and write down the theorem that allows us to always that this will be the case.0863

Theorem. Let f be a function from R2 to R, defined on an open set.0874

Now, we are going to be using... I just introduced the term open set, f, R2, this is often how you are going to see theorems in mathematics.0887

This is very precisely stated theorem, but it is nothing that we do not understand and know. It is just we want something that is formal and that is precise.0896

Defined on an open set, and assume D1 D2, D2 D1, exists. In other words assume that the partial derivatives actually exist, and are continuous.0906

These clearly exist, and these are clearly continuous functions. There is no problem here. Then, D1 D2 of f = D2 D1 of f.0935

Notice up here just real quickly that I just wrote, D1 D2, D2 D1, but I did not put the f.0952

Again this is sort of a personal thing that I do. I tend to sort of minimize my notation simply because I do not like a lot of things floating around on a piece of paper.0960

The idea is that you know what you are talking about. You know that you are dealing with f, you know that D1 is the first partial derivative, you know that D2 is the second partial derivative.0969

You do not have to be that explicit. You can modify your notation depending on how much you know, how much you are comfortable with.0978

You have that freedom, do not feel that you are constrained to always write this, this, this, this. Unless you have the kind of teacher that actually demands that you write all the things out, all the x's, all the parentheses, all the y's be there.0984

Please, feel free to take some liberties with this. You are the one doing the math, you are the one that needs to be comfortable with this.0997

For me, I tend not to write the f. I know that this is not confusing at all.1004

Now, let us consider a function from R3 to R. Now let us move from 2-space to 3-space, a function of 2 variables, to a function of 3 variables.1010

Consider the function from R3 to R, or f(x,y,z).1021

That is it. Now, if we take f(y,z), so now we have D1, we will have D2, and we will have D3, partial with respect to x, partial with respect to y, partial with respect to z.1035

Now we can sort of mix and match. Notice I can have... I can do D1, D2, D3, I can do D3, D2, D1... D2, D1, D3... D2, D3, D1... so all kinds of mixed partials are possible now.1050

Repeated applications of the theorem, which we will do an example of in just a minute, demonstrate that mixed partials are equal, mixed partial as of course 1,2,3... so no worries there... are equal regardless of order, as long as the variables with respect to which we differentiate are the same.1065

Let me just show you what that means. In other words, if I take... so D1, D2, D3 of f.1144

I have taken D3 first, then I have done D2 to that, then I have taken D1 of that.1153

Well, if I do it in different order, do... let us say, D3, D2, D1 of f... if I do D2 first, then D2, then D3... or if I do D2, D3, D1... again we are working from right to left, in other words I do D1 first, then 3 then 2.1159

As it turns out, all of these mixed partials are the same. It does not matter which order you actually differentiate in, as long as the 1,2,3... 1,2,3... 1,2,3... as long as that is the same.1178

You can actually do it in any order. This is really, really extraordinary. I mean there is no reason to think that if you take the partial derivative of a function of several variables in... you do the mixed partial derivatives, there is no reason to believe that they should be equal and yet there it is. Really, really fantastic.1190

Okay, let us just do an example, and so it is nice to see these things sort of fall out.1209

We will let f(x,y,z) = x2y2z3.1219

Now, we are just going to do a bunch of partials. Let us do, let us start with D3, in other words z.1230

So, D3 is going to equal 3x2y2z2.1240

Now we will do D2 of D3, in other words we are going to differentiate with respect to y, this thing up here. D2 of D3.1247

That is going to equal 6x2yz2, and now we are going to do D1 of D2,D3. In other words we are going to differentiate with respect to x, this thing right here.1257

We end up getting 12xyz2.1272

So, we will do that, now let us go ahead and od it in a different order.1279

This time let us take D1, well D1 which is just Dx is just going to be 2xy2z3.1285

Now let us do D2 of D1, which is... I am going to take the derivative with respect to y of this one, of D1.1296

That is going to equal 4xyz3.1303

Now, I am going to take the derivative with respect to z of the D2 D1 that I just got.1310

I end up with 12xyz2.1317

Wow, what do you know. They are the same. Alright, now let us try another order.1322

Now let us do D1 first, we did D1 already, that is 2xy2z3.1329

Now I am going to do, instead of D2, I will do D3 next. So I will do D3 D1.1338

I am going to take the... derivative with respect to z of D1, so that is going to be 6xy2z2, I think.1344

Now, I will do D2 of the D3 D1, and I get... 12xyz2.1357

Is it the same? Yes. 12xyz2, 2xyz2, 2xyz2. I put the partial derivatives in different orders and yet it ended up being exactly the same.1368

This is very, very, very deep and extraordinary. I am just going to write that. I am just going to write "pretty amazing".1380

Now, let us see what else we can do here.1392

Regarding notation, there is just one caution that I am going to throw out there.1400

So, let us write caution: do not confuse the following.1405

Much of the problem with higher mathematics is notation. Notation, it is just one of those things. We have to have some way of representing what it is that we are actually doing.1418

As things become more complicated... well, things get more complicated, so we just want to make sure we know how we are operating.1427

What kind of math we are actually doing. Do not confuse the following.1436

If you see d/dx2 of f, that is the same as d2fdx2.1441

This is D1(D1) of f. In other words take the derivative with respect to the first variable, then take the derivative again with respect to the same variable.1455

It is not the same as this one. df/dx2, which is D1 of f squared.1470

This one says take the derivative, take the first partial derivative, or the derivative with respect to the first variable and square it as a number, or a variable, or a function.1486

This one notice, it is the d/dx, the differential operator that is squared.1499

Here, f is actually inside, here f is outside. This one right here, this right here, not the same. This one is also written this way.1504

Remember we said often times we will write the square, D12f, so D12f is not the same as D1f2.1514

This one says differentiate with respect to f, then differentiate again with respect to the same variable.1524

This one says differentiate with respect to f, then square that number or function.1530

Two very, very different things. Another reason why I actually prefer that notation.1533

This one is reasonably clear, this one can be a bit confusing.1538

Okay, let us just do another example here. So, example 3.1543

We will let f(x,y) = sin(x2+y), and we will do... well, let us just run through them.1551

Let us run through all of them. We have D1.1569

The derivative with respect to x. We have cos(x2+y) × 2x, which we will write again... I mean you leave it like this it is not a problem but it is traditional to sort of bring all the functions forward and leave the trigonometric function to the end.1575

So, 2x × cos(x2+y).1599

D2 = cos(x2+y) × 1, which equals... because this is the derivative of the inside, right? chain rule... cos(x2+y).1607

Now, let us do this for the heck of it, let us do D12 which is the same as D1, D1.1625

So, we have D1 already, now we are going to differentiate this with respect to x. So this is going to be... I tend to pull my constants out, so I am going to pull this out, and it is going to be a power rule. x × cos(x2+y).1636

So it is going to be this × the derivative of that, and end up being -x × sin(x2+y) × 2x + that × the derivative of that, which is this × the derivative of this which is just 1.1652

Again, I have pulled the 2 out. So, it is cos(x2+y), so again, the idea is just be really, really careful and go really slowly.1674

There is no hurry, the idea is to be right, not to finish quickly.1681

When we put this together, we end up with, so the 2, and the 2... this is a × not a minus sign.1687

You end up with -4x2 × sin(x2+y) + cos(x2+y). That is the derivative of the derivative.1696

Derivative with respect to x of the derivative with respect to x. D12, D1 D1. I really love this capital notation, and I love not writing the f's.1710

Now, let us go ahead and do D22.1722

So, D22, which is D2 of D2, when we do that, we end up with -sin(x2+y) × 1.1726

So, equals -sin(x2+y). I hope you guys are checking this.1742

Now let us do D1 D2. When we do D1 D2, we end up with, in other words, we have taken D2, now we are going to take D1 of D2.1750

-sin(x2+y) × 2x is going to be -2xsin(x2+y).1761

Now, we will do D2 of D1, in other words we have done D1 first, now we are going to do D2 of that.1783

We get -2x × sin(x2+y) × 1, which equals -2xsin(x2+y).1789

Well, what do you know, this ends up being the same as that. This mixed partial is the same as that mixed partial.1803

The order is irrelevant, but it is 1,2... 2,1.1810

That is it. So mixed partial derivatives of higher order are the same for functions of several variables provided... so the order does not matter... provided that all of the particular variables with respect to which you are differentiating are there.1815

So 2,3,1... 3,2,1... 1,2,3... 1,3,2... all of those will actually end up being equal, and I personally think that is absolutely extraordinary.1833

Thank you for joining us here at educator.com, we will see you next time for some more Multivariable Calculus. Take care, bye-bye.1842

OR

### Start Learning Now

Our free lessons will get you started (Adobe Flash® required).