Appendix B Definitions
0.1 Sets and functions
Definition 0.1.1 Sets
Definition 0.1.2 Set equality
Definition 0.1.3 Set inclusion (subsets)
Definition 0.1.5 Set-builder notation
Definition 0.1.8 Basic set operations
Definition 0.1.9 Common mathematical sets
Definition 0.1.10 Cartesian product
Definition 0.1.13 Functions
Definition 0.1.17 Function equality
Definition 0.1.18 Image of a set
Definition 0.1.19 Injective, surjective, bijective
Definition 0.1.21 Function composition
Definition 0.1.22 Identity and inverse functions
0.2 Logic
Definition 0.2.1 Logical operators
1.1 Double integrals over rectangles
Definition 1.1.1 Rectangles, partitions, pointed partitions
Definition 1.1.2 Riemann sums in two variables
Definition 1.1.3 Double integral over a rectangle
Definition 1.1.7 Volume between graph and \(xy\)-plane (rectangular base)
1.2 Iterated integrals and Fubini's theorem
Definition 1.2.1 Doubly iterated integral over rectangle
1.3 Double integrals over general regions
Definition 1.3.2 Double integral over bounded region
Definition 1.3.4 Volume below graph and over planar region
Definition 1.3.5 Elementary planar region
1.4 Area of planar regions and average value
Definition 1.4.1 Area of planar region
Definition 1.4.4 Average valueAverage value over planar region
1.5 Triple integrals
Definition 1.5.1 Triple integral over a box
Definition 1.5.3 Triple integral over bounded region
Definition 1.5.5 Volume of solid region
Definition 1.5.6 Elementary solid region
Definition 1.5.16 Average value over solid region
1.6 Substitution: general
Definition 1.6.1 Transformations
Definition 1.6.2 Determinant and Jacobian
Definition 1.6.4 Invertible linear transformations
1.9 Applications of multiple integrals
Definition 1.9.1 Center of mass
Definition 1.9.4 Random variable
Definition 1.9.8 Expected value
2.1 Line integrals of scalar functions
Definition 2.1.1 Curve parametrization
Definition 2.1.4 Line integral of scalar function
2.2 Line integrals of vector fields
Definition 2.2.1 Vector fields
Definition 2.2.3 Integral of a vector field along a curve
Definition 2.2.8 Line integrals with respect to \(dx, dy, dz\)
Definition 2.2.10 Work and flow
Definition 2.2.12 Flux
2.3 Path independence, conservative fields, potential functions
Definition 2.3.1 Gradient vector fields and potential functions
Definition 2.3.7 Conservative vector field
Definition 2.3.10 Open connected sets
Definition 2.3.12 Curl of a vector field
Definition 2.3.17 Simply connected sets
2.4 Green's theorem in the plane
Definition 2.4.3 Scalar curl
Definition 2.4.9 Divergence
2.5 Surfaces and their area
Definition 2.5.1 Surface parametrization
Definition 2.5.9 Parametrization tangent vectors
Definition 2.5.10 Smooth surface parametrization
Definition 2.5.11 Tangent plane to smooth surface
Definition 2.5.13 Surface area
2.6 Surface integrals
Definition 2.6.1 Surface integral of scalar function
Definition 2.6.7 Surface integral for piecewise smooth surfaces
Definition 2.6.9 Surface orientation
Definition 2.6.13 Surface integral of vector field
2.7 Stokes's theorem
Definition 2.7.1 Induced orientation
Definition 2.7.2 Orientation of piecewise smooth surface