## 18.089 Summer 2013

Meetings:
June 10-June 14: MTWThF 12:30–3:00 (E17-129)
June 17-June 28: MTWThF 12:30–2:30 (E17-129)
July 1-July 12: MTWThF 12:30–2:30 (4-153)
July 15-July 19: MTWThF 12:30–2:30 (4-159)

Instructors:
Ailsa Keating (ailsa@math.mit.edu)
John Lesieutre (johnl@math.mit.edu)

Office hours: Tu Th 11-12am, 4th floor common room, building 2

Homeworks
Homework 0 (not due); solutions by Gary Tam.
Homework 1 due Friday, June 21 in class. Solutions.
Homework 2 due Friday, June 28 in class. Solutions.
Homework 3 due Friday, July 5 in class. Solutions: part i, part ii.
Homework 4 due Friday, July 12 in class. Solutions.
Homework 5 due Friday, July 19 in class.

Mid-terms
06/14 Mid-term 1: due Tuesday, June 18th in class (postpone one day due to Navy commitments). Solutions.
07/05 Mid-term 2: due Monday, July 8. Solutions.
07/18 Mid-term 3: due Monday, July 22.

Lecture notes
06/10 Lecture 1: differentiation from first principles; some basic examples; linearity, product and chain rule, quotient rule; higher derivatives; graphing functions; equation of a tangent line
06/11 Lecture 2: implicit differentiation, max/min and related rates problems, exponentials and logarithms, trigonometric functions and their inverses, derivatives thereof
06/12 Lecture 3: indefinite integration: basics, guessing and adjusting, recognising the chain rule and substitution, substitutions with arcsin and arctan, partial fractions; definite integration: definition, fundamental theorem of calculus
06/13 Lecture 4: definite integrals; areas bounded by curves; arclengths; volumes of solids of revolution: little discs and little cylinders methods; surface area of solids of revolution; more integration tools: some trigonometric substitutions and integration by parts
06/14 Lecture 5: integration by parts (continued); indefinite limits and l'Hopital's rule; indefinite integrals; comparison test and limit comparison test; infinite series: examples, comparison tests
06/17 Lecture 6: vectors: addition, multiplication by a scalar, dot product, cross product & geometric interpretations
06/18 Lecture 7: matrices as linear transformations, composition, determinants & inverses; lines in 2D space, lines and planes in 3D space: parametric and implicit equations
06/19 Lecture 8: more on infinite series: integral test, alternating series test, root and ratio tests; powers series: examples, radius of convergence, basic operations (incl. differentiating and integrating), Taylor series
06/20 Lecture 9: curves in 2D and 3D space: parametric and implicit equations, velocity, speed & tangent vector, acceleration, curvature
06/21 Lecture 10: normal and tangent vector to a curve; tangential and normal components of acceleration; 2D polar coodinated; going from cartesian to polar equations and back; arclength and area for polar curves
06/24 Lecture 11: area inside a curve with polar coordinates; functions of two variables: examples, domain, graph; partial derivatives; implicit differentiation; equation of tangent plane at a point
06/25 Lecture 12: differentiability for functions of two variables; tangent plane as approximation; total derivative; gradient, directional derivatives
06/26 Lecture 13: exercises with gradient, directional derivatives and tangent planes; gradient as normal to level curves; generalisations to functions of more than two variables
06/27 Lecture 14: total derivative and generalisation of the chain rule for functions of several variables, applications; critical points; generalisation of second derivative test to functions of two variables.
07/01 Lecture 15: double integrals: finding volume, writing down bounds, switching order of integration, center of mass etc.
07/02 Lecture 16: polar coordinates: integration in polar coordinates, changing coordinates for double integrals. triple integrals: setting up bounds and evaluating triple integrals, applications to mechanics. Here are some extra worked examples, and notes from 18.02.
07/03 Lecture 17: triple integrals in spherical and cylindrical coordinates. Change of variables for double integrals. extra worked examples, more, and notes from 18.02.
07/05 Lecture 18: double integrals for surface area; line integrals, path independence extra worked examples, vector fields (18.02), line integrals (18.02).
07/08 Lecture 20: line integrals, take II, gradient fields, finding potential function, simply connected regions. problems I, problems II, solutions I, solutions II
07/09 Lecture 21: Green's theorem, applications (planimeter) examples.
07/10 Lecture 22: tricky Lagrange multipliers; Green's theorem for regions that aren't simply connected. (18.02 notes)
07/11 Lecture 23: flux in 2D, normal form of Green's theorem, vector fields in space
07/12 Lecture 25: line integrals, fundamental theorem of calculus, conservative fields in 3D. Start of flux across surfaces.
07/15 Lecture 26: more flux across surfaces, the divergence theorem.
07/16 Lecture 27: more divergence theorem, parametrized surfaces and flux, start of Stokes' theorem.
07/17 Lecture 28 parametrized surfaces, more Stokes, recap of the four main integral theorems (FTC, Green, divergence, Stokes).
07/18 Lecture 29: intro PDEs: examples (heat, wave, Laplace), separation of variables (heat), continuity eqn for fluids. Better notes on the heat eqn.

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