Appendix C Theory and procedures
0.1 Sets and functions
Theorem 0.1.23 Invertible is equivalent to bijective
1.1 Double integrals over rectangles
Theorem 1.1.5 Integrable over a rectangle
Procedure 1.1.6 Limit definition of double integral (equal partition)
1.2 Iterated integrals and Fubini's theorem
Theorem 1.2.2 Fubini's theorem over rectangles
1.3 Double integrals over general regions
Theorem 1.3.6 Fubini's theorem over elementary planar regions
Theorem 1.3.9 Properties of double integrals
Theorem 1.3.11 Double integral of even/odd functions
1.5 Triple integrals
Interpretation 1.5.2 Riemann sums as approximations
Interpretation 1.5.4 Interpreting integrals
Theorem 1.5.7 Fubini's theorem over elementary solid regions
Theorem 1.5.9 Fubini's theorem expanded
Corollary 1.5.10 Fubini's theorem for boxes
Theorem 1.5.15 Properties of triple integrals
1.6 Substitution: general
Theorem 1.6.6 Substitution
Procedure 1.6.8 Substitution as change of variable
1.7 Substitution: polar and cylindrical coordinates
Theorem 1.7.3 Polar change of variables
Theorem 1.7.9 Cylindrical change of variables
Procedure 1.7.10 Integrating using cylindrical change of variables
1.8 Substitution: spherical coordinates
Theorem 1.8.2 Spherical change of variables
Procedure 1.8.3 Integrating using spherical change of variables
2.1 Line integrals of scalar functions
Theorem 2.1.6 Scalar line integrals
Theorem 2.1.8 Interpretations
Theorem 2.1.10 Piecewise smooth line integral
2.2 Line integrals of vector fields
Procedure 2.2.2 Visualizing vector fields
2.3 Path independence, conservative fields, potential functions
Theorem 2.3.3 Fundamental theorem of line integrals
Corollary 2.3.4 Integral of gradient field over closed curve
Theorem 2.3.8 Loop property of conservative fields
Theorem 2.3.9 Gradient implies conservative
Theorem 2.3.11 Gradient iff conservative (connected domain)
Theorem 2.3.13 Gradient implies zero curl
Corollary 2.3.14 Gradient implies zero curl (\(n=2\))
Theorem 2.3.18 Gradient iff zero curl (connected and simply connected domain)
Theorem 2.3.19 Logical housekeeping
2.4 Green's theorem in the plane
Theorem 2.4.1 Green's theorem (circulation or tangential form)
Corollary 2.4.4 Area via line integral
Theorem 2.4.6 Scalar curl interpretation
Corollary 2.4.8 Green's theorem (flux or normal form)
Theorem 2.4.11 Divergence interpretation
2.5 Surfaces and their area
Procedure 2.5.2 Plane parametrization
2.6 Surface integrals
Interpretation 2.6.5 Surface integral interpretation
Theorem 2.6.15 Surface integral of vector field
2.7 Stokes's theorem
Theorem 2.7.3 Stokes's theorem
Corollary 2.7.6 Two surfaces, same boundary
Theorem 2.7.9 Curl interpretation
2.8 Divergence theorem
Theorem 2.8.2 Divergence (or Gauss's) theorem
Corollary 2.8.4 Zero divergence implies zero flux
Corollary 2.8.5 Identical flux
Theorem 2.8.6 \(\diver\curl\boldF=0\)
Corollary 2.8.8 Curl field has zero flux
Theorem 2.8.10 Divergence interpretation