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COURSE 08 / COURSE

Mathematical Methods for Continuum Physics

Acquire the language that fields and fluids require.

OUTCOMEUse vector calculus, partial derivatives, linear algebra, and differential equations to express local conservation laws.

PREREQUISITECourses 01–07, or equivalent calculus-based physics.

Mathematical Methods for Continuum Physics: original course diagram — Course visual: a compact prompt for the physical ideas developed below.

12 CHAPTERS

Course map

01Fields and multivariable functionsRepresent a physical quantity that varies across space and time.→02Partial derivatives and total changeDistinguish holding a coordinate fixed from following a moving system.→03Gradient and directional changeRead the geometry of scalar fields.→04Divergence and fluxMeasure sources, sinks, and net flow through a surface.→05Curl and circulationMeasure local rotation and connect it to path integrals.→06Integral theoremsMove between local and global statements with Green, Gauss, and Stokes.→07Material derivativeFollow a parcel through an Eulerian velocity field.→08Ordinary differential equationsSolve first- and second-order evolution laws.→09Partial differential equationsClassify transport, diffusion, and wave behavior.→10Boundary and initial conditionsMake a differential equation into a physical problem.→11Scaling and nondimensionalizationFind dominant terms before reaching for a solver.→12Methods studio: derive a conservation lawTranslate a control-volume balance into a local differential equation.→