# Introductory Course Topic Lists

## Math 20 Series

This series covers differential calculus, integral calculus, and power series in one variable. It can be started at any point in the sequence for those with sufficient background.

### Math 19

1. Definition and properties of functions, exponentials, logarithms, trigonometric functions and their inverses.
2. Definition of limits, computing limits, squeeze theorem, continuity, infinite limits and limits at infinity.
3. Definition and intuition for the derivative.
4. Computing derivatives, product rule, quotient rule, chain rule.
5. Derivatives of inverse functions, implicit differentiation (logarithmic differentiation if time/interest).
6. Linear approximation.
7. Max/min problems, optimization, applications of derivatives.
8. L’Hopital’s rule.
9. Related rates.
10. Curve sketching using limits and derivatives.

### Math 20

1. The definite integral as area under the curve.
2. Antiderivatives and indefinite integrals, Fundamental Theorem of Calculus.
3. Integration techniques: basic integration, substitution, integration by parts, using tables of integrals (partial fractions and/or trig substitution if time/interest).
4. Applications of integration: area between curves, volumes of solids.
5. Introduction to differential equations: initial value problems, slope fields, separable equations.
6. Applications of differential equations and modeling, logistic equation.
7. Parametric equations, arc length of parametric curves.

### Math 21

1. Review of limits and integration techniques.
2. Improper integrals: unbounded interval (type 1) and bounded interval with asymptote (type 2); comparison tests for convergence.
3. Sequences, geometric series, general series.
4. Convergence tests for series: divergence test, integral test, comparison test, limit comparison test, ratio test, alternating series test, absolute convergence test.
5. Power series, radius/interval of convergence, differentiation/integration of power series.
6. Taylor polynomials, Taylor series (in particular, the series and radius of convergence for eˣ, sin x, cos x, arctan x, ln x).
7. Applications of Taylor series, error in Taylor polynomial approximations.

## Math 50 Series

This series provides the necessary mathematical background for majors in all disciplines, especially the Natural Sciences, Mathematics, Data Science, Economics, and Engineering.

### Math 51

1. Vectors, dot product, length, planes in space.
2. Span, subspaces, basis & dimension of subspaces, projection to subspaces.
3. Orthogonal basis of planes, linear regression, multivariable functions, level sets and graphs, contour plots, partial derivatives (and tangent planes).
5. Linear functions, matrices, derivative matrix, linear transformations, matrix algebra.
6. Applications of matrix algebra (Markov Chains, gambling), Chain Rule for multivariable functions, matrix inverses, multivariable Newton’s method (latter on HW, not exams).
7. Linear independence, Gram-Schmidt process, transpose, orthogonal matrices, symmetric matrices, quadratic forms, linear systems (column and null spaces).
8. QR/LU-decompositions, eigenvalues and eigenvectors (only compute in 2-dim’l & triangular cases), Spectral Theorem, applications to quadratic forms (definiteness) & matrix powers.
9. Higher-order partial derivatives, Hessian matrix, multivariable Second Derivative Test for local extrema via eigenvalues.
10. (not for HW or exams) Singular Value Decomposition via Spectral Theorem & eigenvalues.

### Math 52

1. Double and triple integrals over various regions.
2. Applications of double and triple integrals.
3. Double integrals via polar coordinates, and to compute areas and some volumes.
4. Cylindrical and spherical coordinates, associated integration formulas.
5. Determinants and cross products: properties, calculations, geometric meaning (and orientation). Change of Variables formula for multiple integrals.
6. Vector fields and associated derivative operators: grad, curl, div.
7. Parametric curves and line integrals, path independence and Fundamental Theorem.
8. Green’s theorems and applications; relation to flux, circulation, and area (planimeter).
9. Parametric surfaces, surface area & other integrals. Tangents, normal, and orientation.
10. Divergence Theorem, Stokes’ Theorem, and applications; conservative vector fields revisited.

### Math 53

1. First-order ODE’s, initial value problems, slope fields.
2. Stationary values and stability, phase line, second-order linear ODE’s, linear ODE systems and eigenvectors.
3. Complex eigenvalues and phase portraits for x⁰ = Ax, matrix exponential, inhomogeneous linear ODE, variation of parameters and method of undetermined coefficients.
4. Rewriting ODE system as first-order, existence/uniqueness theorem, chaos and bifurcation.
5. Linearization, non-linear dynamics
6. Euler’s methods, Runge-Kutta, stiff ODE’s, power series.
7. Introduction to PDE’s, separation of variables
8. Fourier series in trigonometric and exponential form, using Fourier methods to solve initial- boundary value problems on some 2-dimensional regions.
9. Fourier transform, uncertainty principle, convolution, application to PDE’s on a line: heat and wave equations.
10. (not for HW or exams) Applications of Fourier transform beyond PDE’s (e.g., medical imaging, bell curve in probability, or signals).