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MATH 315:
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
🔗
MATH 315:
Calculus IV
Vincent Bouchard
University of Alberta
March 5, 2024
🔗
These are notes for MATH 315 offered at the University of Alberta, which is the fourth calculus course in the calculus sequence. MATH 315 is devoted to vector calculus.
How to use these notes
Supplementary material and references