Appendix C List of examples
1 A preview of vector calculus
2 One-forms and vector fields
Example 2.1.4 A one-form and its associated vector field
Example 2.2.3 The differential and gradient of a function
Example 2.2.7 An exact one-form and its associated conservative vector field
Example 2.2.8 The gravitational force field is conservative
Example 2.2.12 Exact one-forms are closed
Example 2.2.13 Closed one-forms are not necessarily exact
Example 2.3.2 An example of a change of variables
Example 2.3.5 Change of variables as pullback
3 Integrating one-forms: line integrals
Example 3.1.2 An example of an integral of a one-form over an interval
Example 3.2.3 Parametrizing the unit circle
Example 3.2.8 Parametrizing the unit circle counterclockwise
Example 3.2.12 Two parametrizations of the unit circle
Example 3.2.14 Parametrizing a triangle
Example 3.3.1 Pulling back along a circle
Example 3.3.3 An example of a line integral
Example 3.3.6 How line integrals change under reparametrizations
Example 3.4.4 An example of a line integral of an exact one-form
Example 3.5.1 Work done by a (non-conservative) force field
Example 3.6.2 Closed forms are exact
Example 3.6.5 An example of a closed one-form that is not exact
4 \(k\)-forms
Example 4.2.2 The wedge product of two one-forms
Example 4.2.3 The wedge product of a one-form and a two-form
Example 4.2.4 The wedge product of a zero-form and a \(k\)-form
Example 4.3.3 The exterior derivative of a zero-form on \(\mathbb{R}^3\)
Example 4.3.4 The exterior derivative of a one-form on \(\mathbb{R}^3\)
Example 4.3.5 The exterior derivative of a two-form on \(\mathbb{R}^3\)
Example 4.3.7 The exterior derivative of the wedge product of two one-forms
Example 4.4.7 Maxwell's equations
Example 4.5.1 The direction of steepest slope
Example 4.5.4 The curl of the velocity field of a moving fluid
Example 4.5.5 An irrotational velocity field
Example 4.5.6 Another irrotational velocity field
Example 4.5.7 The divergence of the velocity field of an expanding fluid
Example 4.5.8 An imcompressible velocity field
Example 4.5.9 Another incompressible velocity field
Example 4.6.2 Exact and closed one-forms in \(\mathbb{R}^3\)
Example 4.7.2 The pullback of a two-form
Example 4.7.3 The pullback of a three-form
Example 4.8.2 The action of the Hodge star in \(\mathbb{R}\)
Example 4.8.3 The action of the Hodge star in \(\mathbb{R}^2\)
Example 4.8.4 The action of the Hodge star in \(\mathbb{R}^3\)
Example 4.8.5 An example of the Hodge star action in \(\mathbb{R}^3\)
Example 4.8.6 Maxwell's equations using differential forms (optional)
5 Integrating two-forms: surface integrals
Example 5.1.3 Integral of a zero-form at points
Example 5.2.3 Orientation of \(\mathbb{R}\) and choice of positive or negative direction
Example 5.2.4 Orientation of \(\mathbb{R}^2\) and choice of counterclockwise or clockwise rotation
Example 5.2.5 Orientation of \(\mathbb{R}^3\) and choice of right-handed or left-handed twirl
Example 5.2.10 Closed disk in \(\mathbb{R}^2\)
Example 5.2.11 Closed square in \(\mathbb{R}^2\)
Example 5.2.12 Annulus in \(\mathbb{R}^2\)
Example 5.3.3 Integral of a two-form over a rectangular region with canonical orientation
Example 5.3.4 Integral of a two-form over an \(x\)-supported (or type I) region with canonical orientation
Example 5.3.8 Area of a disk
Example 5.4.4 The graph of a function in \(\mathbb{R}^3\)
Example 5.4.5 The sphere
Example 5.4.6 The cylinder
Example 5.4.7 Grid curves on the sphere
Example 5.5.6 Upper half-sphere
Example 5.6.2 An example of a surface integral
Example 5.6.7 An example of a surface integral of a vector field
Example 5.7.4 Using Green's theorem to calculate line integrals
Example 5.7.5 Area of an ellipse
Example 5.8.5 Using Stokes' theorem to evaluate a surface integral by transforming it into a line integral
Example 5.8.7 Using Stokes' theorem to evaluate a surface integral by using a simpler surface
Example 5.8.8 Using Stokes' theorem to evaluate a line integral by transforming it into a surface integral
Example 5.9.2 The electric flux and net charge of a point source
6 Beyond one- and two-forms
Example 6.2.2 Solid region bounded by a sphere in \(\mathbb{R}^3\)
Example 6.2.4 Integral of a three-form over a recursively supported region
Example 6.2.8 Using the divergence theorem to evaluate the flux of a vector field over a closed surface in \(\mathbb{R}^3\)
7 Unoriented line and surface integrals
Example 7.1.3 An example of an unoriented line integral
Example 7.1.5 Calculating the arc length of a parametric curve
Example 7.2.2 An example of an unoriented surface integral
Example 7.2.4 Calculating the surface area of a parametric surface
Example 7.3.1 Finding the centre of mass of a wire in \(\mathbb{R}^2\)
Example 7.3.2 Finding the centre of mass of a sheet in \(\mathbb{R}^3\)