Notes

Math 18 (multivariable calculus)

Introduction
Planes, domain and range, level sets, contour curves
Domain and range, partial derivatives
Dot product, tangent planes
Maximizing and minimizing functions
Limits and continuity, polar coordinates
Polar coordinates, differentiability
Equality of second partials, classifying critical points
Chain rule, implicit differentiation
Directional derivatives, gradient
Critically analyzing functions, sketching special graphs, how to study for a college math test
Review 1: Partial derivatives, classifying critical points, directional derivative, tangent planes, optimization
Review 2: Optimization, limits, chain rule
Lagrange multipliers, functions of three variables
Iterated integrals, volumes over rectangular regions
Volumes over general regions
Double integrals in polar coordinates
Decomposition of regions, density, triple integrals
Triple integrals
Cylindrical coordinates and a 'brilliant polar trick'
Spherical coordinates
Review problems for midterm 2, center of mass in 1D, 2D, 3D
List of integrals to know, volume above the xy-plane, center of mass in 2D and 3D
Review: A sheet to take notes on classmates' review examples
Center of mass in 3D, change of variables, (u,v)-substitution
Change of variables, the Jacobian, Cartesian to polar proof of r dr dθ
Change of variables, vector fields
Vector fields, gradient fields, equipotential curves, preview of the rest of the course
Path integrals, circulation, flux
Circulation, flux, conservative vector fields
Conservative vector fields, Green's theorem for circulation and flux
Proof of Green's theorem, curl, divergence
Preview of the rest of the course; vector fields, curl and divergence in 3D
Surface integrals of functions and of vector fields
Stokes' theorem, examples and proof
Divergence theorem (only one page)
Review sheet for final exam, or 'how to ensure your success on a college math final'
Review 1: Tangent planes, optimizing, directional derivative, gradient, max/min on bounded domain, Lagrange multipliers, limits
Review 2: Center of mass in Cartesian and cylindrical coordinates, double and triple integrals, spherical and cylindrical coordinates
Review 3: Line integrals, conservative vector fields; Green's, Stokes' and divergence theorem, and figuring out which one to use

Math 9 (introductory calculus)

I had the idea to hand out my lecture notes a few weeks into the semester, so the early ones are a bit rough.

Exponential functions, growth and decay
Inverse functions, logarithms, change of base
Inverse trig functions, inverse functions, logarithm example
Average speed, instantaneous speed, secant and tangent lines
Limits, limit laws, sandwich theorem, rational functions
ε-δ definitions, finite limits, limit laws, asymptotes
Continuity, continuous functions, limits, removable singularities, intermediate value theorem (IVT)
IVT, slope and derivative at a point
Differentiable functions, IVT, Darboux's theorem, differentiation rules
Differentiation rules (constants, power, exponential, product, quotient, trig)
Differentiating trig functions, applying derivatives to motion (ballistics)
Chain rule, repeated chain rule
Parametric equations, implicit differentiation
Review: Derivative definition, limits, asymptotes, continuous functions, IVT, taking derivatives, motion
Derivatives of inverse functions, logarithmic differentiation
Proof of power rule, what is e, derivatives of inverse trig, differentials
Related rates
Finding maxima and minima, extreme value theorem
Mean value theorem, Rolle's theorem
Using derivatives to test maxima and minima; more related rates
Concavity and curve sketching, second derivative test
Examples of optimization (word problems)
Curve sketching, related rates with optimization, applied optimization
l'Hopital's rule
l'Hopital's rule, finding antiderivatives
Review: Inverse trig, logarithms, chain rule, implicit differentiation, related rates, extreme values, concavity, curve sketching, applied optimization
Antiderivatives and motion, the integral sign
Estimating with finite sums, sigma notation
Definite integrals and areas under curves, average value
Fundamental theorem of calculus
The end of the course is missing. TBA.

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