**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

**Links**

Stellar (grades)

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.

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.

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.