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Raffi Hovasapian

Raffi Hovasapian

Divergence & Curl in 3-Space

Slide Duration:

Table of Contents

I. 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
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

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
II. 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
III. 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
Notation and Properties for Gradient
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
IV. 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
Norm of the Gradient f(p)
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
V. 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
VI. 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
VII. 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
VIII. 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
IX. 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
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Lecture Comments (4)

1 answer

Last reply by: Professor Hovasapian
Wed May 11, 2016 3:35 AM

Post by Tram T on May 7, 2016

In example 2 last minute, for j^ term, i think it should be -yz^4 + x^2y.

1 answer

Last reply by: Professor Hovasapian
Sun Jan 20, 2013 10:19 PM

Post by mateusz marciniak on January 19, 2013

professor in example 1 why did you not distribute the (-) sign to e^x yz

Divergence & Curl in 3-Space

Find the divergence of the vector field F = (2x + y,x + y + z,x − 2y).
  • The divergence of a vector field F(x,y,z) = (f1(x,y,z),f2(x,y,z),f3(x,y,z)) is div(F) = [(df1)/dx] + [(df2)/dy] + [(df3)/dz]
  • For f1 = 2x + y, f2 = x + y + z and f3 = x − 2y we have [(df1)/dx] = 2, [(df2)/dy] = 1 and [(df3)/dz] = 0.
Hence div(F) = 2 + 1 + 0 = 3
Find the divergence of the vector field F = (cos(x),sin(x),tan(x + y + z)).
  • The divergence of a vector field F(x,y,z) = (f1(x,y,z),f2(x,y,z),f3(x,y,z)) is div(F) = [(df1)/dx] + [(df2)/dy] + [(df3)/dz]
  • For f1 = cos(x), f2 = sin(x) and f3 = tan(x + y + z) we have [(df1)/dx] = − sin(x), [(df2)/dy] = 0 and [(df3)/dz] = sec2(x + y + z).
Hence div(F) = − sin(x) + sec2(x + y + z)
Find the curl of the vector field F = ( − y2 + z,x − y + z, − x2 + z).
  • The curl of a vector field F(x,y,z) = (f1(x,y,z),f2(x,y,z),f3(x,y,z)) is curl(F) = ( [(df3)/dy] − [(df2)/dz],[(df1)/dz] − [(df3)/dx],[(df2)/dx] − [(df1)/dy] )
  • For f1 = − y2 + z,, f2 = x − y + z and f3 = − x2 + z we have curl(F) = ( − 2 − 1,0 − 1,1 − 1 ).
Simplifying yields div(F) = ( − 3, − 1,0)
Find the curl of the vector field F = (e − xyz,exyz,exy).
  • The curl of a vector field F(x,y,z) = (f1(x,y,z),f2(x,y,z),f3(x,y,z)) is curl(F) = ( [(df3)/dy] − [(df2)/dz],[(df1)/dz] − [(df3)/dx],[(df2)/dx] − [(df1)/dy] )
For f1 = e − xyz, f2 = exyz and f3 = exy we have curl(F) = ( xexy − xyexyz, − xye − xyz − yexy,yzexyz + xze − xyz ).
Find ∇×F and ∇×F of the vector field F = ( 1 − [(x2)/4],1 − [(y2)/4],z2 ).
  • Note that ∇ is the operator assigning ( [d/dx],[d/dy],[d/dz] ), so that ∇×F corresponds to the divergence and ∇×F corresponds to the curl of the vector field F.
  • Then ∇×F = ( [d/dx],[d/dy],[d/dz] ) ×( 1 − [(x2)/4],1 − [(y2)/4],z2 ) = [d/dx]( 1 − [(x2)/4] ) + [d/dy]( 1 − [(y2)/4] ) + [d/dz]( z2 ) = − [x/2] − [y/2] + 2z
And ∇×F = (
i
j
k
[d/dx]
[d/dy]
[d/dz]
1 − [(x2)/4]
1 − [(y2)/4]
z2
) = (0 − 0)i − (0 − 0)j + (0 − 0)k or (0,0,0)
Find ∇×F and ∇×F of the vector field F = ( √{xy} ,√{xz} ,√{xy} ).
  • Note that ∇ is the operator assigning ( [d/dx],[d/dy],[d/dz] ), so that ∇×F corresponds to the divergence and ∇×F corresponds to the curl of the vector field F.
  • Then ∇×F = ( [d/dx],[d/dy],[d/dz] ) ×( √{xy} ,√{xz} ,√{xy} ) = [d/dx]( √{xy} ) + [d/dy]( √{xz} ) + [d/dz]( √{xy} ) = [y/(2√{xy} )] + 0 + 0 = [y/(2√{xy} )]
And ∇×F = (
i
j
k
[d/dx]
[d/dy]
[d/dz]
√{xy}
√{xz}
√{xy}
) = ( [x/(2√{xy} )] − [x/(2√{xz} )] )i − ( [y/(2√{xy} )] − 0 )j + ( [z/(2√{xy} )] − [x/(2√{xz} )] )k or [1/(2√x )]( [x/(√y )] − [x/(√z )], − √y ,[z/(√y )] − [x/(√z )] )
Find ∇×F and ∇×F of the vector field F = ( [1/yz],[1/xz],[1/xy] ).
  • Note that ∇ is the operator assigning ( [d/dx],[d/dy],[d/dz] ), so that ∇×F corresponds to the divergence and ∇×F corresponds to the curl of the vector field F.
  • Then ∇×F = ( [d/dx],[d/dy],[d/dz] ) ×( [1/yz],[1/xz],[1/xy] ) = [d/dx]( [1/yz] ) + [d/dy]( [1/xz] ) + [d/dz]( [1/xy] ) = 0 + 0 + 0 = 0
And ∇×F = (
i
j
k
[d/dx]
[d/dy]
[d/dz]
[1/xy]
[1/xz]
[1/xy]
) = ( − [1/(xy2)] + [1/(xz2)] )i − ( − [1/(x2y)] − 0 )j + ( − [1/(x2z)] + [1/(xy2)] )k or [1/x]( − [1/(y2)] + [1/(z2)],[1/xy], − [1/xz] + [1/(y2)] )
Find ∇×F and ∇×F of the vector field F = ( xcos(y)ez,ycos(x)ez,z ).
  • Note that ∇ is the operator assigning ( [d/dx],[d/dy],[d/dz] ), so that ∇×F corresponds to the divergence and ∇×F corresponds to the curl of the vector field F.
  • Then ∇×F = ( [d/dx],[d/dy],[d/dz] ) ×( xcos(y)ez,ycos(x)ez,z ) = [d/dx]( xcos(y)ez ) + [d/dy]( ycos(x)ez ) + [d/dz]( z ) = cos(y)ez + cos(x)ez +
And ∇×F = (
i
j
k
[d/dx]
[d/dy]
[d/dz]
xcos(y)ez
ycos(x)ez
z
) = ( 0 + ycos(x)ez )i − ( 0 − xcos(y)ez )j + ( − ysin(x)ez − xsin(y)ez )k or ez( ycos(x),xcos(y), − ysin(x) − xsin(y) )
Let F = (x + y + z,x − y − z, − x + y − z) be a vector field.
i) Find ∇×F
We compute ∇×F = (
i
j
k
[d/dx]
[d/dy]
[d/dz]
x + y + z
x − y − z
− x + y − z
) = ( 1 + 1 )i − ( − 1 − 1 )j + ( 1 − 1 )k to obtain ( 2,2,0 )
Let F = (x + y + z,x − y − z, − x + y − z) be a vector field.
ii) Find ∇×(∇×F)
  • Since ∇ is an operator, we follow our usual order of operations. From our previous problem ∇×F = ( 2,2,0 ).
Then ∇×(∇×F) = ∇×( 2,2,0 ) = (
i
j
k
[d/dx]
[d/dy]
[d/dz]
2
2
0
) = 0i − 0j + 0k to obtain ( 0,0,0 )
Let F = (x + y + z,x − y − z, − x + y − z) be a vector field.
iii) Find ∇×(∇×F)
  • Since ∇ is an operator, we follow our usual order of operations. From our previous problem ∇×F = ( 2,2,0 ).
Then ∇×(∇×F) = ∇×( 2,2,0 ) = ( [d/dx],[d/dy],[d/dz] ) ×( 2,2,0 ) = 0 + 0 + 0 = 0
Let F = (x2,y2,z2) be a vector field and ∇f = (2x,2y,2z) be the gradient of F.
i) Find ∇×F
We compute ∇×( x2,y2,z2 ) = ( [d/dx],[d/dy],[d/dz] ) ×( x2,y2,z2 ) to obtain 2x + 2y + 2z
Let F = (x2,y2,z2) be a vector field and ∇f = (2x,2y,2z) be the gradient of F.
ii) Find ∇2f
  • Note that ∇2f = ∇×(∇f) which is equivalent to taking the divergence of the gradient of F.
Then ∇×(∇f) = ∇×(2x,2y,2z) = ( [d/dx],[d/dy],[d/dz] ) ×(2x,2y,2z) = 2 + 2 + 2 = 6
Let F = (x2,y2,z2) be a vector field and ∇f = (2x,2y,2z) be the gradient of F.
iii) Find ∇×[ ( ∇2f )F ]
  • Note that ∇2f = 6 which is a number. Then ( ∇2f )F is just a multiplication which yields 6(x2,y2,z2) = (6x2,6y2,6z2).
Then ∇×[ ( ∇2f )F ] = (
i
j
k
[d/dx]
[d/dy]
[d/dz]
6x2
6y2
6z2
) = ( 0 − 0 )i − ( 0 − 0 )j + ( 0 − 0 )k or ( 0,0,0 )

*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.

Answer

Divergence & Curl in 3-Space

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
  • Divergence and Curl in 3-Space 0:26
    • Introduction to Divergence and Curl in 3-Space
    • Define: Divergence of F
    • Define: Curl of F
    • The Del Operator
    • Symbolically: Div(F)
    • Symbolically: Curl(F)
    • Example 1
    • Example 2

Transcription: Divergence & Curl in 3-Space

Hello and welcome back to educator.com and multi variable calculus.0000

Today's lesson we are going to be talking about divergence and curl in 3-space.0004

So we have talked about divergence and curl before, we did it for 2-space when we discussed Green's Theorem.0008

Now we are going to talk about it in 3-space. We are going to introduce some new symbolism that will take care of all cases all at once.0016

Let us just go ahead and jump right on in. Okay.0024

So, we will let f(x,y,z) = f1(x,y,z), f2(x,y,z), and f3(x,y,z).0028

This is just a highly explicit way of representing a vector field. A vector field, these are the coordinate functions, the coordinate functions are functions of x,y,z themselves. I have just written everything out.0051

So, let this be a vector field on an open set s in R3, in 3-space, 3-dimensional space.0064

Okay. That is... and I know you know what this is but I am just going to repeat it for the sake of being complete.0084

For each point in s, there is a vector given by f pointing in some direction away from that point... pointing in some direction... starting at that point. That is it.0095

So, if I take some 3-space, so in this particular case let us just say that our open set s happens to be all of 3-dimensional space.0140

If I pick a point at random, well if I pick a point and I put that point in for f, I am going to get a vector going this way, maybe for this point I have a vector this way, maybe for this point a vector this way, that way, that way, that way, that way, could be any number of things. That is a vector field in 3-space. That is all it is.0150

Okay, let us define the divergence of f.0171

The divergence of f, which is symbolized as div(f) = df1 dx + df2 dy + df3 dz, or in terms of capital D notation, D1f1 + D2f2, + D3f3. Okay.0180

It is a scalar, it is a number. It is a scalar. A number at a particular point, Df dx + Df2 dy + Df3 dz at a particular point you put that particular point into this thing and it is just going to spit out a number.0218

Okay. That is the divergence in 3-space. It is just the analog of the divergence in 2-space.0246

So the curl is the... it is the analog of the curl in 2-space that we discussed previously, but it is a little bit more notationally complicated.0253

So, let us go ahead and write it out, and then we will go ahead and give you a symbolic way of representing it.0262

So, curl of f... define the other definition, the curl of the vector field... is equal to... this one I am going to do the capital D notation first, and then I will do the regular partial derivative notation.0267

It is going to be D2f3 - D3f2, D3f1 - D1f3, D1f2 - D2f1. Notice this is not a scalar. This is a vector. The curl of a vector field gives you another vector field.0282

In terms of partial derivative notation, this looks like this. So, D2 of f3, this is Df3 dy - Df2 dz.0316

This is Df1 dz - Df3 dx.0335

This is Df2 dx - Df1 dy.0346

I hope to heaven I have got all of those correct. Okay. That is the curl. It is a vector.0356

So, when you are given a vector field f, when you take the divergence of it, you want to put a number at a given point.0367

When you take the vector field f, the curl of that vector field, what you end up with is another vector field. It is a vector at a given point.0374

Okay. Now we are going to introduce something called the Del operator.0386

The del operator is a symbolic way of simplifying the calculations. That is really what it is. And making things just look more elegant. The del operator, okay, it is usually an upside down triangle, or you can just write del.0392

So, the definition of the operator is the following. For the time being, the notion of an operator, an operator is just something that tells you to do something to something else.0413

In other words, we talk about the differential operator d dx. So, for example you knew what this is, d dx. If I apply this differential operator to some function f, d dx of f, well, that is just df dx.0423

That just means take the derivative of it. The integral operator. The integral operator says integrate on the function f, you are going to end up getting something else, so that is all an operator is.0440

It is a fancy term that says do something to this function. Well, the del operator is the same thing.0450

It is an operator that says do this to a given function, except it has multiple parts. These are individual operators. The differential, the integral operators. The del operator actually is written symbolically in the form of a vector.0456

So, but other than that you treat it exactly the same way as you do anything else. It says do something to something.0473

As it turns out, the del operator is a differential operator. It is a partial differential operator. You will see what we mean in just a minute.0480

So, let us go ahead and write out the symbol and then do some examples and of course everything will make sense.0487

Let us see, so this, or del, is equivalent to d dx, d dy, d dz, or D1, D2, D3, notice there is no function here because it is an operator.0493

I have to choose a function and then say do this to it. That is the whole idea. We write it as a vector because it is going to operate as a vector on another vector. That is the whole idea.0518

So, now, let us go ahead and write what we mean by divergence and curl symbolically, using this del operator notation.0534

Okay. So, symbolically, and again, this is all symbolic. Symbolically, the divergence of f = the del operator dotted with the vector field, f.0544

Well, the del operator dotted with the vector field f, is equal to... well, the del operator is D1, D2, D3, that is my symbolic operator for del, dotted with well... f is f1, f2, f3.0568

Well, I know what a dot product is. It is just this × this, this × this, this × this, added together.0585

Except I am multiplying these two numbers, this is a symbolic vector operation. It says do d1 to f1, in other words, take the derivative of f1 with respect to x. Take the derivative of f2 with respect to y, take the derivative of f3 with respect to z. This is a symbolic notion.0591

We are symbolizing using the idea of a vector, that is what an operator is. So, this is equal to... so, it is d1f1 + d2f2 + d3f3, except this is not multiplication, this d1f1... this is a differential operator.0614

It says take the derivative of f1 with respect to the first variable, this says take the derivative of f2 with respect to the second variable, this says take the derivative of f3 with respect to the 3rd variable. I hope this makes sense.0637

Now the curl of f. Okay. This is going to be kind of interesting. Let me go to the next page.0651

So, the curl of f. Now you can memorize this any way you want to. If you want to just go back to what I initially wrote on the first page of this lesson, when I defined divergence and curl, you are more than welcome to remember curl in that way if you want to remember the indices -- 23 32, 13 31, 12 21.0657

That is fine, or you can remember it this way, symbolically.0677

So, the curl of f, it is defined as del cross f, the del operator crossed with the vector field. Well we know what a cross product is. We have been dealing with it symbolically. It is the symbolic determinant i,j,k, and now del cross f... it is like this is a vector, this is a vector.0682

Well, the first vector is in the second row, so we will just write D1 D2 D3, and f is just f1 f2 f3.0705

We will symbolically take the determinant of that. When you do that, you end up with the following.0720

You end up with (D2f3 - D3f2)i - (D1f3 - D3f1)j + (D1f2 - D2f1)k. i... j... k... this is the first component function, second component function, third component function.0726

What you end up with is exactly what we had earlier. You get D2f3 - D3f2, this - turns this into a negative, turns this into a positive... what you end up with is D3f1 - D1f3 and you get D1f2 - D2f1.0784

This is the vector representation, this is the i,j,k, representation. This is the curl.0813

So, all I have done is I have taken this... I have created this thing called the del operator, given it a symbol like a vector, and I have been able to define the divergence and curl in terms of the two vector operations that I have.0819

The divergence of the vector field f is equal to del · f, and then curl of the vector field f is equal to del cross f.0835

This is the symbolic way of keeping things straight. So, let us just do some examples and I think it will make sense here. So, Example 1.0843

We will let our vector field f = the first component function is x2, the second component function is xy, and our third component function is going to be ex,y,z.0859

Okay. So, the divergence of f, divergence is always going to be easier because you are just sort of taking partial derivatives 1 by 1. So, the divergence, the partial of this with respect to x is going to be 2x, + the partial of this with respect to y, so this is going to be cos(xy) × x, so it is going to be + x × cos(xy), and the partial of this with respect to z is just going to be ecy.0875

There you go. That is the divergence. It is a scalar, not a vector.0906

Now, let us go ahead and form the curl of f. The curl of f, well, we said that the curl of f = del cross f, so let us go ahead and form our symbolic determinant here... i, j, k.0915

We have d1, d2, d3, or if you want, let us go ahead this time for this particular example, let us use our partial differential and our d dx, d dy, it does not really matter.0934

So, let us do i, j, k, so we have d dx, d dy, an d dz, that is the del vector operator, and then we have f, which is x2, sin(xy), and exyz. We want to form this determinant.0950

Okay. So, let us see what we get. When we expand along the first row, we get the derivative of this with respect to y - the derivative of this with respect to z, so I end up getting the derivative of this with respect to y is exz, ex × z - 0 × i - ... because remember it is + - +, alternating sign.0978

The derivative of this with respect to x - the derivative of this with respect to z.1008

So the derivative of this with respect to x is exyz is e2 × yz - 0, and that is going to be j.1014

Of course our last one k is going to be the derivative of this with respect to x - the derivative of this with respect to y.1025

So, the derivative of this with respect to x is going to be y × cos(xy) - 0 × k, so is the... so I get this, this, and this... and so I end up with ex × z is my first component function of my curl, ex × yz is the second component function of my curl, and y × cos(xy) is the third component of my curl vector. There we go.1032

This is a scalar divergence... curl is a vector at a given point.1071

If I take all of the points, it gives me a vector field. That is it.1078

Okay. Let us do another example. So, example 2.1085

f(x,y,z) = x2yz, xy3z, and xyz4. Okay, well, let us see what we have got.1095

The divergence of f, we said, is equal to del · f, well del · f is d1f1 + d2f2 + d3f3, but we are not multiplying the d and the f, this means take the derivative of f.1122

Well, the derivative of this with respect to x is going to be 2xyz + this one, the derivative with respect to y is going to be + 3xy2z, and the derivative with respect to z of this one is going to be +4xyz3. This is our divergence of the particular vector field.1141

Okay. So, now let us go ahead and do the curl of the vector field. The curl of this vector field, well it is equal to del cross f, and del cross f, well, is symbolic.1169

It is going to be i, j, k... let us do capital D notation here... D1, D2, D3, and we have x2yz, we have xy3z, and we have xyz4.1187

Okay. So, now let us go ahead and expand along the first row. It is going to be the derivative with respect to y of this - the derivative with respect to z of this. The derivative with respect to y of this is xz4.1208

The derivative with respect to z of this is xy3. This is the i component... - , now we go to the next one.1227

The derivative with respect to x of this - the derivative with respect to z of this, derivative with respect to the first variable which is x, derivative with respect to the third variable which is z. That is what is going on here.1241

This × this, this × this, except it is not times, it is symbolic. It means operate on this.1253

Okay. So, the derivative with respect to x of this is yz4 - the derivative with respect to z of this, x2y. This is the j component.1262

Okay, we are almost done. Now, the derivative with respect to x of this - the derivative with respect to y of this.1273

So, the derivative with respect to x of this is going to be y3z - the derivative with respect to y of this is going to be x2z.1282

Again, I hope that you are confirming this for me. There are lots of x's, y's, z's, i's, j's, k's, floating around. y3z, there you go.1294

I will go ahead and actually leave it in this form. That is the first coordinate function of the curl, that is the second coordinate function of the curl, and that is the third coordinate function of the curl.1304

Actually, you know what, let me go ahead and write it out. It is not a problem.1319

So, I will go ahead and erase these stray lines here.1320

So, we have got xz4 - xy3, that is the first component function, we have yz4 -x2y, and then we have y3z - x2z, notice the divergence is a scalar, the curl is a vector.1329

It has three component functions... an x, a y and a z. At a given point (x,y,z), there is some vector pointing in some direction and away from that point. That is the whole idea.1359

Again, it is all based on this notion of what we call an operator. It is just a symbolic way of telling you what to do to a given function.1370

It is a unifying scheme, so you had this thing. You can take the divergence and curl of a vector field. We want to be able to express that in terms that we know.1380

Well, if we gave this... we called it a del operator, we have it a symbolic representation s D1, D2, D3, as a symbolic vector... If we do del · f, we get divergence of f, if we get del cross f, we get the curl of f. That is it. It is just a unifying scheme.1390

Okay, thank you for joining us here at educator.com for divergence and curl. We will see for a discussion of the divergence theorem in 3-space. Take care, bye-bye.1410

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