Appendix A List of results
1 A preview of vector calculus
2 One-forms and vector fields
Lemma 2.2.10 Exact one-forms in \(\mathbb{R}^2\) are closed
Lemma 2.2.11 Screening test for conservative vector fields in \(\mathbb{R}^2\)
Lemma 2.2.15 Exact one-forms in \(\mathbb{R}^3\) are closed
Lemma 2.2.16 Screening test for conservative vector fields in \(\mathbb{R}^3\)
Lemma 2.4.4 The pullback of \(dx \)
Lemma 2.4.5 The pullback of a one-form
3 Integrating one-forms: line integrals
Lemma 3.1.5 Integrals of one-forms over intervals are invariant under orientation-preserving reparametrizations
Lemma 3.1.6 Integrals of one-forms over intervals pick a sign under orientation-reversing reparametrizations
Lemma 3.2.6 Parametric curves are oriented
Lemma 3.2.9 Reparametrizations of a curve
Lemma 3.2.11 Orientation-preserving reparametrizations
Lemma 3.3.5 Line integrals are invariant under orientation-preserving reparametrizations
Lemma 3.3.7 Line integrals in terms of vector fields
Theorem 3.4.1 The Fundamental Theorem of line integrals
Corollary 3.4.2 The line integrals of an exact form along two curves that start and end at the same points are equal
Corollary 3.4.3 The line integral of an exact one-form along a closed curve vanishes
Theorem 3.4.5 The Fundamental Theorem of line integrals for vector fields
Theorem 3.6.1 Poincare's lemma, version I
Theorem 3.6.3 Equivalent formulations of exactness on \(\mathbb{R}^n\)
Theorem 3.6.4 Poincare's lemma, version II
4 \(k\)-forms
Lemma 4.1.6 Antisymmetry of basic \(k\)-forms
Lemma 4.2.7 The wedge product of two one-forms is the cross-product of the associated vector fields
Lemma 4.2.8 The wedge product of a one-form and a two-form is the dot product of the associated vector fields
Lemma 4.3.2 The exterior derivative in \(\mathbb{R}^3\)
Lemma 4.3.6 The graded product rule for the exterior derivative
Lemma 4.3.9 \(d^2=0\)
Lemma 4.4.9 Vector calculus identities, part 1
Lemma 4.4.10 Vector calculus identities, part 2
Lemma 4.4.11 Vector calculus identities, part 3
Lemma 4.4.12 Vector calculus identities, part 4
Lemma 4.6.3 Exact \(k\)-forms are closed
Theorem 4.6.4 Poincare's lemma for \(k\)-forms, version 1
Theorem 4.6.5 Poincare's lemma for \(k\)-forms, version II
Lemma 4.7.1 The pullback of a \(k\)-form
Lemma 4.7.4 The pullback commutes with the exterior derivative
Lemma 4.7.7 The pullback of a top form in \(\mathbb{R}^n\) in terms of the Jacobian determinant
Lemma 4.7.9 An explicit formula for the pullback of a basic one-form
Lemma 4.7.11 The pullback commutes with the wedge product
Corollary 4.7.12 An explicit formula for the pullback of a basic \(k\)-form
Lemma 4.8.8 The Laplace-Beltrami operator and the Laplacian of a function
Lemma 4.8.9 The Laplace-Beltrami operator and the Laplacian of a vector field
Lemma 4.8.10 Vector calculus identities, part 5
5 Integrating two-forms: surface integrals
Theorem 5.1.7 The Fundamental Theorem of Calculus
Theorem 5.1.8 The Fundamental Theorem of line integrals
Lemma 5.3.7 Integrals of two-forms over regions in \(\mathbb{R}^2\) are invariant under orientation-preserving reparametrizations
Lemma 5.5.3 Parametric surfaces are oriented
Lemma 5.5.7 Orientation-preserving reparametrizations
Lemma 5.6.3 Surface integrals are invariant under orientation-preserving reparametrizations
Lemma 5.6.4 The pullback of a two-form along a parametric surface in terms of vector fields
Theorem 5.7.1 Green's theorem
Theorem 5.8.1 Stokes' theorem
Corollary 5.8.2 The surface integrals of an exact two-form along two surfaces that share the same oriented boundary are equal
Corollary 5.8.3 The surface integral of an exact two-form along a closed surface vanishes
Theorem 5.8.11 Stokes' theorem for vector fields
6 Beyond one- and two-forms
Theorem 6.1.1 The generalized Stokes' theorem
Lemma 6.2.5 Integrals of three-forms over regions in \(\mathbb{R}^3\) are oriented and reparametrization-invariant
Theorem 6.2.6 The divergence theorem in \(\mathbb{R}^3\)
Lemma 6.3.1 Rewriting the left-hand-side
Lemma 6.3.2 Rewriting the right-hand-side
Theorem 6.3.3 Divergence theorem in \(\mathbb{R}^n\)
Lemma 6.4.1 Green's first identity
Lemma 6.4.2 Green's second identity
7 Unoriented line and surface integrals