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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
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 1D
Parametric curves in 2D and 3D
Line integrals
Fundamental Theorem of line integrals
Applications of line integrals
Poincare's lemma for one-forms
4
\(n\)
-forms
Multiplying one-forms: the wedge product
\(n\)
-forms
Differentiating
\(n\)
-forms: the exterior derivative
The exterior derivative and vector calculus
\(d^2=0\)
Exact and closed
\(n\)
-forms
Poincare's lemma for two-forms
Hodge star
5
Integrating two-forms: surface integrals
Integrating a two-form over a region in 2D
Parametric surfaces in 3D, part I
Parametric surfaces in 3D, part II
Orientation of surfaces
The pullback of a two-form along a surface
Surface integrals
Applications of surface integrals
6
Stokes' Theorem
Integrating a three-form over a region in 3D
Stokes' Theorem
Green's Theorem and Stokes' Theorem
The Divergence Theorem
Applications of Stokes' Theorem I
Applications of Stokes' Theorem II
7
Unoriented line and surface integrals
Unoriented line integrals
Unoriented surface integrals
Applications of unoriented line and surface integrals
Authored in PreTeXt
Chapter
4
\(n\)
-forms
4.1
Multiplying one-forms: the wedge product
4.2
\(n\)
-forms
4.3
Differentiating
\(n\)
-forms: the exterior derivative
4.4
The exterior derivative and vector calculus
4.5
\(d^2=0\)
4.6
Exact and closed
\(n\)
-forms
4.7
Poincare's lemma for two-forms
4.8
Hodge star