Table of contents
Lecture 1: The chain rule ......................................................................................................................... 7
Differentials ......................................................................................................................................... 7
The differential of a function............................................................................................................... 7
Application........................................................................................................................................... 7
The chain rule ...................................................................................................................................... 7
The chain rule (general version) .......................................................................................................... 9
Implicit differentiation ........................................................................................................................ 9
Lecture 2: Directional derivates ............................................................................................................ 12
Directional derivatives....................................................................................................................... 12
The gradient vector ........................................................................................................................... 12
Function of three variables ............................................................................................................... 12
Maximizing the directional derivative ............................................................................................... 13
Tangent planes to level surfaces ....................................................................................................... 13
Lecture 3: Local maximum and minimum values .................................................................................. 14
Local maximum and minimum values ............................................................................................... 14
Critical points ..................................................................................................................................... 14
The second derivates test ................................................................................................................. 15
Examples........................................................................................................................................ 15
Lecture 4: Absolute maximum and minimum values ............................................................................ 17
Closed and bounded sets .................................................................................................................. 17
The extreme value theorem .............................................................................................................. 17
Absolute maximum and minimum values ......................................................................................... 17
Lecture 5: Double integrals over rectangles.......................................................................................... 21
Riemann sums ................................................................................................................................... 21
The midpoint rule .......................................................................................................................... 21
Iterated integrals ............................................................................................................................... 22
Double integrals over rectangles....................................................................................................... 22
Average value ................................................................................................................................ 23
Lecture 6: Double integrals over general regions ................................................................................. 24
Double integrals over general regions .............................................................................................. 24
Regions of type I ................................................................................................................................ 24
Regions of type II ............................................................................................................................... 24
More general regions ........................................................................................................................ 25
Double integrals over general regions .............................................................................................. 27
, Area of a region ................................................................................................................................. 30
Example ......................................................................................................................................... 30
Lecture 7: Double integrals in polar coordinates .................................................................................. 31
Polar coordinates .............................................................................................................................. 31
Double integrals in polar coordinates ............................................................................................... 31
Example 1 ...................................................................................................................................... 32
Example 2 ...................................................................................................................................... 33
Lecture 8: Applications of double integrals........................................................................................... 36
Density and mass............................................................................................................................... 36
Center of mass................................................................................................................................... 36
Density and mass............................................................................................................................... 36
Moments ........................................................................................................................................... 37
Center of mass................................................................................................................................... 37
Example 1 ...................................................................................................................................... 37
Example 2 ...................................................................................................................................... 39
Moment of inertia ............................................................................................................................. 39
Lecture 9: Triple integrals ...................................................................................................................... 40
Triple integrals ................................................................................................................................... 40
Triple integrals over rectangular boxes ............................................................................................. 40
Fubini’s theorem for triple integrals ................................................................................................. 41
Example 1 ...................................................................................................................................... 41
Triple integrals over general regions ................................................................................................. 41
Triple integrals over a region of type 1 ............................................................................................. 42
Triple integrals over a region of type 2 ............................................................................................. 42
Triple integrals over a region of type 3 ............................................................................................. 42
Example 2 ...................................................................................................................................... 42
Triple integrals ................................................................................................................................... 43
Example 3 ...................................................................................................................................... 43
Applications of triple integrals .......................................................................................................... 44
Center of mass................................................................................................................................... 44
Example ......................................................................................................................................... 44
Moment of inertia ............................................................................................................................. 45
Example ......................................................................................................................................... 45
Lecture 10: Triple integrals in cylindrical coordinates .......................................................................... 46
Cylindrical coordinates ...................................................................................................................... 46
Volume element in cylindrical coordinates ....................................................................................... 46
, Triple integrals in cylindrical coordinates.......................................................................................... 46
Example ......................................................................................................................................... 47
Applied project: Roller derby ............................................................................................................ 48
Lecture 11: Triple integrals in spherical coordinates ............................................................................ 49
Spherical coordinates ........................................................................................................................ 49
Volume element in spherical coordinates......................................................................................... 49
Triple integrals in spherical coordinates ........................................................................................... 49
Example ......................................................................................................................................... 50
Applied project: Roller derby ............................................................................................................ 52
Lecture 12: Change of variables in multiple integrals ........................................................................... 53
Change of variables ........................................................................................................................... 53
One-to-one transformations ............................................................................................................. 53
Example ......................................................................................................................................... 54
The Jacobian of a transformation...................................................................................................... 55
Change of variables in a double integrals ......................................................................................... 55
Example ......................................................................................................................................... 56
Polar coordinates .............................................................................................................................. 56
Triple integrals ................................................................................................................................... 57
Cylindrical coordinates ...................................................................................................................... 57
Spherical coordinates ........................................................................................................................ 58
Lecture 13: Vector fields ....................................................................................................................... 59
Example ......................................................................................................................................... 59
More examples .................................................................................................................................. 60
Examples............................................................................................................................................ 60
Gradient (vector) fields ..................................................................................................................... 60
Example ......................................................................................................................................... 61
Conservative vector fields ................................................................................................................. 61
Lecture 14: Line integrals and the arc length of a curve ....................................................................... 63
Parametrization of a curve ................................................................................................................ 63
Example ......................................................................................................................................... 64
The arc length of a curve ................................................................................................................... 64
Examples........................................................................................................................................ 64
The arc length function ..................................................................................................................... 64
The line integral of a scalar function ................................................................................................. 65
Example ......................................................................................................................................... 65
Application: physical interpretation .................................................................................................. 65
, Example ......................................................................................................................................... 66
The line integral of a vector field ...................................................................................................... 67
Example ......................................................................................................................................... 67
Notation............................................................................................................................................. 68
Lecture 15: The fundamental theorem for line integrals ...................................................................... 69
Line integrals ..................................................................................................................................... 69
The fundamental theorem for line integrals ..................................................................................... 70
Independence of path ....................................................................................................................... 70
Closed path ........................................................................................................................................ 70
Open and connected regions in ℝ2 .................................................................................................. 70
Conservative vector fields ................................................................................................................. 71
Simple curves..................................................................................................................................... 71
Simply-connected regions ................................................................................................................. 71
Conservative vector fields ................................................................................................................. 71
Lecture 16: Green’s theorem ................................................................................................................ 72
Example ......................................................................................................................................... 72
The orientation of a plane curve ....................................................................................................... 72
Green’s theorem ............................................................................................................................... 73
Example ......................................................................................................................................... 73
Application: calculating areas............................................................................................................ 74
Example ......................................................................................................................................... 74
Extended proof of Green’s theorem ................................................................................................. 75
Example ......................................................................................................................................... 75
Example ......................................................................................................................................... 76
Lecture 17: Curl and divergence............................................................................................................ 77
The curl of a vector field.................................................................................................................... 77
The vector differential operator ∇ .................................................................................................... 77
Example ......................................................................................................................................... 77
The curl of a conservative vector field .............................................................................................. 78
Example ......................................................................................................................................... 78
Example ......................................................................................................................................... 78
The divergence of a vector field ........................................................................................................ 79
Example ......................................................................................................................................... 80
Example ......................................................................................................................................... 80
The Laplace operator......................................................................................................................... 80
Vector forms of Green’s theorem ..................................................................................................... 81
Lecture 1: The chain rule ......................................................................................................................... 7
Differentials ......................................................................................................................................... 7
The differential of a function............................................................................................................... 7
Application........................................................................................................................................... 7
The chain rule ...................................................................................................................................... 7
The chain rule (general version) .......................................................................................................... 9
Implicit differentiation ........................................................................................................................ 9
Lecture 2: Directional derivates ............................................................................................................ 12
Directional derivatives....................................................................................................................... 12
The gradient vector ........................................................................................................................... 12
Function of three variables ............................................................................................................... 12
Maximizing the directional derivative ............................................................................................... 13
Tangent planes to level surfaces ....................................................................................................... 13
Lecture 3: Local maximum and minimum values .................................................................................. 14
Local maximum and minimum values ............................................................................................... 14
Critical points ..................................................................................................................................... 14
The second derivates test ................................................................................................................. 15
Examples........................................................................................................................................ 15
Lecture 4: Absolute maximum and minimum values ............................................................................ 17
Closed and bounded sets .................................................................................................................. 17
The extreme value theorem .............................................................................................................. 17
Absolute maximum and minimum values ......................................................................................... 17
Lecture 5: Double integrals over rectangles.......................................................................................... 21
Riemann sums ................................................................................................................................... 21
The midpoint rule .......................................................................................................................... 21
Iterated integrals ............................................................................................................................... 22
Double integrals over rectangles....................................................................................................... 22
Average value ................................................................................................................................ 23
Lecture 6: Double integrals over general regions ................................................................................. 24
Double integrals over general regions .............................................................................................. 24
Regions of type I ................................................................................................................................ 24
Regions of type II ............................................................................................................................... 24
More general regions ........................................................................................................................ 25
Double integrals over general regions .............................................................................................. 27
, Area of a region ................................................................................................................................. 30
Example ......................................................................................................................................... 30
Lecture 7: Double integrals in polar coordinates .................................................................................. 31
Polar coordinates .............................................................................................................................. 31
Double integrals in polar coordinates ............................................................................................... 31
Example 1 ...................................................................................................................................... 32
Example 2 ...................................................................................................................................... 33
Lecture 8: Applications of double integrals........................................................................................... 36
Density and mass............................................................................................................................... 36
Center of mass................................................................................................................................... 36
Density and mass............................................................................................................................... 36
Moments ........................................................................................................................................... 37
Center of mass................................................................................................................................... 37
Example 1 ...................................................................................................................................... 37
Example 2 ...................................................................................................................................... 39
Moment of inertia ............................................................................................................................. 39
Lecture 9: Triple integrals ...................................................................................................................... 40
Triple integrals ................................................................................................................................... 40
Triple integrals over rectangular boxes ............................................................................................. 40
Fubini’s theorem for triple integrals ................................................................................................. 41
Example 1 ...................................................................................................................................... 41
Triple integrals over general regions ................................................................................................. 41
Triple integrals over a region of type 1 ............................................................................................. 42
Triple integrals over a region of type 2 ............................................................................................. 42
Triple integrals over a region of type 3 ............................................................................................. 42
Example 2 ...................................................................................................................................... 42
Triple integrals ................................................................................................................................... 43
Example 3 ...................................................................................................................................... 43
Applications of triple integrals .......................................................................................................... 44
Center of mass................................................................................................................................... 44
Example ......................................................................................................................................... 44
Moment of inertia ............................................................................................................................. 45
Example ......................................................................................................................................... 45
Lecture 10: Triple integrals in cylindrical coordinates .......................................................................... 46
Cylindrical coordinates ...................................................................................................................... 46
Volume element in cylindrical coordinates ....................................................................................... 46
, Triple integrals in cylindrical coordinates.......................................................................................... 46
Example ......................................................................................................................................... 47
Applied project: Roller derby ............................................................................................................ 48
Lecture 11: Triple integrals in spherical coordinates ............................................................................ 49
Spherical coordinates ........................................................................................................................ 49
Volume element in spherical coordinates......................................................................................... 49
Triple integrals in spherical coordinates ........................................................................................... 49
Example ......................................................................................................................................... 50
Applied project: Roller derby ............................................................................................................ 52
Lecture 12: Change of variables in multiple integrals ........................................................................... 53
Change of variables ........................................................................................................................... 53
One-to-one transformations ............................................................................................................. 53
Example ......................................................................................................................................... 54
The Jacobian of a transformation...................................................................................................... 55
Change of variables in a double integrals ......................................................................................... 55
Example ......................................................................................................................................... 56
Polar coordinates .............................................................................................................................. 56
Triple integrals ................................................................................................................................... 57
Cylindrical coordinates ...................................................................................................................... 57
Spherical coordinates ........................................................................................................................ 58
Lecture 13: Vector fields ....................................................................................................................... 59
Example ......................................................................................................................................... 59
More examples .................................................................................................................................. 60
Examples............................................................................................................................................ 60
Gradient (vector) fields ..................................................................................................................... 60
Example ......................................................................................................................................... 61
Conservative vector fields ................................................................................................................. 61
Lecture 14: Line integrals and the arc length of a curve ....................................................................... 63
Parametrization of a curve ................................................................................................................ 63
Example ......................................................................................................................................... 64
The arc length of a curve ................................................................................................................... 64
Examples........................................................................................................................................ 64
The arc length function ..................................................................................................................... 64
The line integral of a scalar function ................................................................................................. 65
Example ......................................................................................................................................... 65
Application: physical interpretation .................................................................................................. 65
, Example ......................................................................................................................................... 66
The line integral of a vector field ...................................................................................................... 67
Example ......................................................................................................................................... 67
Notation............................................................................................................................................. 68
Lecture 15: The fundamental theorem for line integrals ...................................................................... 69
Line integrals ..................................................................................................................................... 69
The fundamental theorem for line integrals ..................................................................................... 70
Independence of path ....................................................................................................................... 70
Closed path ........................................................................................................................................ 70
Open and connected regions in ℝ2 .................................................................................................. 70
Conservative vector fields ................................................................................................................. 71
Simple curves..................................................................................................................................... 71
Simply-connected regions ................................................................................................................. 71
Conservative vector fields ................................................................................................................. 71
Lecture 16: Green’s theorem ................................................................................................................ 72
Example ......................................................................................................................................... 72
The orientation of a plane curve ....................................................................................................... 72
Green’s theorem ............................................................................................................................... 73
Example ......................................................................................................................................... 73
Application: calculating areas............................................................................................................ 74
Example ......................................................................................................................................... 74
Extended proof of Green’s theorem ................................................................................................. 75
Example ......................................................................................................................................... 75
Example ......................................................................................................................................... 76
Lecture 17: Curl and divergence............................................................................................................ 77
The curl of a vector field.................................................................................................................... 77
The vector differential operator ∇ .................................................................................................... 77
Example ......................................................................................................................................... 77
The curl of a conservative vector field .............................................................................................. 78
Example ......................................................................................................................................... 78
Example ......................................................................................................................................... 78
The divergence of a vector field ........................................................................................................ 79
Example ......................................................................................................................................... 80
Example ......................................................................................................................................... 80
The Laplace operator......................................................................................................................... 80
Vector forms of Green’s theorem ..................................................................................................... 81
