Classnotes - MA1101
Functions of Several Variables
Arindama Singh
Department of Mathematics
Indian Institute of Technology Madras
,Contents
1 Differential Calculus 4
1.1 Regions in the plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Level curves and surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4 Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.5 Increment Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.6 Chain Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.7 Directional Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.8 Normal to Level Curve and Tangent Planes . . . . . . . . . . . . . . . . . . . . . 24
1.9 Taylor’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.10 Extreme Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.11 Lagrange Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
1.12 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2 Multiple Integrals 40
2.1 Volume of a solid of revolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.2 The Cylindrical Shell Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.3 Approximating Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.4 Riemann Sum in Polar coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.5 Triple Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.6 Triple Integral in Cylindrical coordinates . . . . . . . . . . . . . . . . . . . . . . . 58
2.7 Triple Integral in Spherical coordinates . . . . . . . . . . . . . . . . . . . . . . . 60
2.8 Change of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
2.9 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3 Vector Integrals 74
3.1 Line Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.2 Line Integral of Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.3 Conservative Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.4 Green’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.5 Curl and Divergence of a vector field . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.6 Surface Area of solids of Revolution . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.7 Surface area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.8 Integrating over a surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
2
, 3.9 Surface Integral of a Vector Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3.10 Stokes’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
3.11 Gauss’ Divergence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
3.12 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
Bibliography 120
Appendix A One Variable Summary 121
A.1 Graphs of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
A.2 Concepts and Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
A.3 Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Index 138
3
, Chapter 1
Differential Calculus
1.1 Regions in the plane
Let D be a subset of the plane R2 and let (a, b) ∈ R2 be any point.
An -disk around (a, b) is the set of all points (x, y) ∈ R2 whose distance from (a, b) is less than .
(a, b) is an interior point of D iff some -disk around (a, b) is contained in D.
(a, b) ∈ D is an isolated point of D iff (a, b) is the only point of D that is contained in some -disk
around (a, b).
(a, b) is a boundary point of D iff every -disk around (a, b) contains points from D and points
not from D.
R is an open subset of R2 iff all points of D are its interior points.
D is a closed subset of R2 iff it contains all its boundary points.
D = D∪ the set of boundary points of D; It is the closure of D.
D is a bounded subset of R2 iff D is contained in some -disk. (around some point)
An interior point A boundary point
4
Functions of Several Variables
Arindama Singh
Department of Mathematics
Indian Institute of Technology Madras
,Contents
1 Differential Calculus 4
1.1 Regions in the plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Level curves and surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4 Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.5 Increment Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.6 Chain Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.7 Directional Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.8 Normal to Level Curve and Tangent Planes . . . . . . . . . . . . . . . . . . . . . 24
1.9 Taylor’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.10 Extreme Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.11 Lagrange Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
1.12 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2 Multiple Integrals 40
2.1 Volume of a solid of revolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.2 The Cylindrical Shell Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.3 Approximating Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.4 Riemann Sum in Polar coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.5 Triple Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.6 Triple Integral in Cylindrical coordinates . . . . . . . . . . . . . . . . . . . . . . . 58
2.7 Triple Integral in Spherical coordinates . . . . . . . . . . . . . . . . . . . . . . . 60
2.8 Change of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
2.9 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3 Vector Integrals 74
3.1 Line Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.2 Line Integral of Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.3 Conservative Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.4 Green’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.5 Curl and Divergence of a vector field . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.6 Surface Area of solids of Revolution . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.7 Surface area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.8 Integrating over a surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
2
, 3.9 Surface Integral of a Vector Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3.10 Stokes’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
3.11 Gauss’ Divergence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
3.12 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
Bibliography 120
Appendix A One Variable Summary 121
A.1 Graphs of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
A.2 Concepts and Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
A.3 Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Index 138
3
, Chapter 1
Differential Calculus
1.1 Regions in the plane
Let D be a subset of the plane R2 and let (a, b) ∈ R2 be any point.
An -disk around (a, b) is the set of all points (x, y) ∈ R2 whose distance from (a, b) is less than .
(a, b) is an interior point of D iff some -disk around (a, b) is contained in D.
(a, b) ∈ D is an isolated point of D iff (a, b) is the only point of D that is contained in some -disk
around (a, b).
(a, b) is a boundary point of D iff every -disk around (a, b) contains points from D and points
not from D.
R is an open subset of R2 iff all points of D are its interior points.
D is a closed subset of R2 iff it contains all its boundary points.
D = D∪ the set of boundary points of D; It is the closure of D.
D is a bounded subset of R2 iff D is contained in some -disk. (around some point)
An interior point A boundary point
4
