Appendix B List of definitions
1 A preview of vector calculus
2 One-forms and vector fields
Definition 2.1.1 One-forms
Definition 2.1.2 Vector fields
Definition 2.2.1 Differential of a function
Definition 2.2.5 Exact one-forms
Definition 2.2.6 Conservative vector fields
Definition 2.2.9 Closed one-forms in \(\mathbb{R}^2\)
Definition 2.2.14 Closed one-forms in \(\mathbb{R}^3\)
Definition 2.3.3 The pullback of a function on \(\mathbb{R}\)
Definition 2.3.4 The pullback of a one-form on \(\mathbb{R}\)
Definition 2.4.1 The pullback of a function
3 Integrating one-forms: line integrals
Definition 3.1.1 The integral of a one-form over \([a,b]\)
Definition 3.1.3 The orientation of an interval
Definition 3.1.4 The oriented integral of a one-form
Definition 3.2.1 Parametric curves
Definition 3.2.2 Closed parametric curves
Definition 3.2.4 The tangent vector to a parametric curve
Definition 3.2.5 Orientation of a curve
Definition 3.3.2 (Oriented) line integrals
4 \(k\)-forms
Definition 4.1.1 The basic one-forms
Definition 4.1.3 Basic two-forms
Definition 4.1.4 Basic three-forms
Definition 4.1.5 Basic \(k\)-forms
Definition 4.2.1 The wedge product
Definition 4.3.1 The exterior derivative of a \(k\)-form
Definition 4.4.1 The gradient of a function
Definition 4.4.2 The curl of a vector field
Definition 4.4.3 The divergence of a vector field
Definition 4.6.1 Exact and closed \(k\)-forms
Definition 4.7.5 The Jacobian
Definition 4.7.6 Top form
Definition 4.7.8 The pullback of a basic one-form with respect to a linear map
Definition 4.7.10 The pullback of a basic \(k\)-form with respect to a linear map
Definition 4.8.7 The codifferential and the Laplace-Beltrami operator
5 Integrating two-forms: surface integrals
Definition 5.1.1 Oriented points
Definition 5.1.2 Integral of a zero-form over an oriented point
Definition 5.1.4 The orientation of an interval
Definition 5.1.5 The integral of a one-form over an oriented interval \([a,b]_{\pm}\)
Definition 5.1.6 (Oriented) line integrals
Definition 5.2.1 Orientation of \(\mathbb{R}^n\)
Definition 5.2.2 Canonical orientation of \(\mathbb{R}^n\)
Definition 5.2.6 Regions in \(\mathbb{R}^2\)
Definition 5.2.8 Orientation of a closed bounded region in \(\mathbb{R}^2\)
Definition 5.2.9 Induced orientation on the boundary of a region in \(\mathbb{R}^2\)
Definition 5.3.1 Integral of a two-form over an oriented closed bounded region in \(\mathbb{R}^2\)
Definition 5.3.6 Orientation-preserving reparametrizations of regions in \(\mathbb{R}^2\)
Definition 5.4.1 Parametric surfaces in \(\mathbb{R}^n\)
Definition 5.4.2 Closed parametric surfaces
Definition 5.4.8 Tangent planes to a parametric surface
Definition 5.4.9 Normal vectors to a parametric surface in \(\mathbb{R}^3\)
Definition 5.5.1 Orientable surfaces and orientation
Definition 5.5.5 Induced orientation on the boundary of a parametric surface
Definition 5.6.1 Surface integrals
Definition 5.9.1 The flux of a vector field across a surface
6 Beyond one- and two-forms
Definition 6.2.1 Orientation of a region in \(\mathbb{R}^3\) and induced orientation on the boundary
Definition 6.2.3 Integral of a three-form over a closed bounded region in \(\mathbb{R}^3\)
7 Unoriented line and surface integrals
Definition 7.1.1 Unoriented line integrals
Definition 7.1.4 Arc length of a curve
Definition 7.2.1 Unoriented surface integrals
Definition 7.2.3 Surface area of a parametric surface in \(\mathbb{R}^3\)