MATH2020: Advanced Calculus II
Proper multiple integrals, rigorous coordinate changes, and vector integral theorems with their actual proof domains.
Double integrals and iterated integration
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Chapter 1
Multiple integrals
Multiple integrals
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Double integrals and iterated integration
Build double integrals from area sums and choose an integration order from the geometry of a region.
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Change of variables and Jacobians
Transform an integration region and its area scale without losing multiplicity or orientation.
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Triple integrals and three-dimensional coordinates
Set up volume integrals in Cartesian, cylindrical and spherical coordinates.
Chapter 2
Integration foundations and applications
Integration foundations and applications
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Double-integral depth: Darboux construction and a complete continuous Fubini proof
Define the rectangular integral and prove the order theorem for continuous functions before applying it.
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Integral depth: coordinate coverage, slicing and improper failures
Use Jacobians with global domain control, select integration order geometrically, and diagnose an iterated-integral counterexample.
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Integral applications: mass, centroid, inertia and surface area
Derive physical integral models, preserve units and use symmetry without replacing weighted quantities by geometric averages.
Chapter 3
The Jordan and change-of-variables proof chain
The Jordan and change-of-variables proof chain
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Change of variables foundations: Jordan covers and two-sided cell distortion
Derive finite-cover boundary estimates and uniform two-sided volumes for small transformed cubes.
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C1 change of variables: a finite-dimensional Jordan/Riemann proof
Prove the nonlinear determinant formula for compact Jordan regions and bounded Riemann-integrable functions, with coverage and boundary hypotheses explicit.
Chapter 4
Line, surface and vector integral theorems
Line, surface and vector integral theorems
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Line integrals and conservative fields
Distinguish scalar arc-length integrals from oriented work and test path independence.
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Green's theorem and orientation
Relate circulation and flux around a planar boundary to integrals over the enclosed region.
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Surface integrals and flux
Parametrize surfaces, distinguish surface density from oriented flux, and choose normal directions.
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Stokes' theorem and the divergence theorem
Relate circulation to curl and closed-surface flux to divergence, with compatible orientation.
Chapter 5
Proofs, orientation and domain obstructions
Proofs, orientation and domain obstructions
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Vector fields, gradient, divergence and curl
Build the differential operators and test conservation with explicit regularity and domain hypotheses.
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Proofs of Green, parametrized Stokes and sliced divergence
Establish integral identities on explicit regular classes instead of replacing boundary arguments with pictures.
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Orientation, cavities and singular-field synthesis
Choose integral theorems by geometry and exclude singularities before converting a boundary integral.