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MATH 215:
Calculus IV
Vincent Bouchard
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Contents
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Front Matter
How to use these notes
Supplementary material and references
1
A preview of vector calculus
A preview of vector calculus
2
One-forms and vector fields
One-forms and vector fields
Exact one-forms and conservative vector fields
Changes of variables
The pullback of a one-form
3
Integrating one-forms: line integrals
Integrating a one-form over an interval in
R
Parametric curves in
R
n
Line integrals
Fundamental Theorem of line integrals
Applications of line integrals
Poincare's lemma for one-forms
4
k
-forms
Differential forms revisited: an algebraic approach
Multiplying
k
-forms: the wedge product
Differentiating
k
-forms: the exterior derivative
The exterior derivative and vector calculus
Physical interpretation of grad, curl, div
Exact and closed
k
-forms
The pullback of a
k
-form
Hodge star
5
Integrating two-forms: surface integrals
Integrating zero-forms and one-forms
Orientation of a region in
R
2
Integrating a two-form over a region in
R
2
Parametric surfaces in
R
n
Orientation of parametric surfaces in
R
3
Surface integrals
Green's theorem
Stokes' theorem
Applications of surface integrals
6
Beyond one- and two-forms
Generalized Stokes' theorem
Divergence theorem in
R
3
Divergence theorem in
R
n
Applications of the divergence theorem
Integral theorems: when to use what
7
Unoriented line and surface integrals
Unoriented line integrals
Unoriented surface integrals
Applications of unoriented line and surface integrals
Back Matter
A
List of results
B
List of definitions
C
List of examples
D
List of exercises
Authored in PreTeXt
đź”—
Chapter
5
Integrating two-forms: surface integrals
đź”—
We study how two-forms can be integrated along surfaces, which leads us to introduce the concept of surface integrals (also called “flux integrals”).
5.1
Integrating zero-forms and one-forms
5.2
Orientation of a region in
R
2
5.3
Integrating a two-form over a region in
R
2
5.4
Parametric surfaces in
R
n
5.5
Orientation of parametric surfaces in
R
3
5.6
Surface integrals
5.7
Green's theorem
5.8
Stokes' theorem
5.9
Applications of surface integrals