OM011
- Honors Beginning Algebra (Directed study with discussion: 10
units)
For students who previously have had little or
no formal exposure to algebra. Primary topics include: the elementary
structure and language of real numbers, understanding and
manipulating algebraic expressions including polynomials and rational
expressions, solving linear and second-degree equations,
understanding inequalities and systems of equations. Emphasis is
placed on word problems and graphing.
M012 - Honors
Intermediate Algebra (Directed study with discussion: 10
units)
For
students with previous exposure to algebra but not sufficient mastery
for OM013 Precalculus with Trigonometry. This course reviews and
extends the topics of beginning algebra: linear equations and
inequalities, absolute value, quadratic inequalities, roots and
exponents, and systems of equations. Prerequisite: OM011:
Honors Beginning Algebra
OM015
- Honors Geometry (Directed study with discussion: 10 units)
This
course combines the traditional deductive approach to geometry in the
tradition of Euclid with the contemporary computational and
discovery approaches. Primary topics include: logic, congruence
of polygons, inequalities, similarity, properties of circles, area
of plane figures, surface area and volume of solids, basic
trigonometry, coordinate geometry, and transformational geometry. Prerequisites: OM011 Beginning Algebra required; OM012
Intermediate Algebra strongly recommended
M013
- Honors Precalculus with Trigonometry (Directed study with
discussion: 10 units)
For
students who have had substantial previous exposure to algebra. The
course builds on and deepens all the topics from OM011 Beginning
Algebra and OM012 Intermediate Algebra. Functions are studied, in
particular, their composition and inverses. Other topics include: the
algebra of exponential and logarithmic functions, techniques of
graphing and matrices, mathematical induction, sequences and series,
and analytic geometry. Approximately one third of the course focuses
on trigonometry and its applications. Prerequisite: OM012
Intermediate Algebra
M040
- AP Calculus A (Directed study with discussion: 5 units)
An
introduction to the differential calculus. Topics: limits and
continuity, derivatives and differentiability, applications of the
derivative, curve sketching, related rates, and implicit
differentiation. Prerequisite: OM013 Honors Precalculus with
Trigonometry
M041
- AP Calculus B (Directed study with discussion: 5 units)
An
introduction to the integral calculus. Topics: parametric equations,
Riemann sums, indefinite and definite integrals, techniques of
integration and applications of the integral. Calculus A and Calculus
B together prepare students to take the Calculus AB AP exam. Prerequisite: OM040 AP Calculus A
M042
- AP Calculus C (Directed study with discussion: 5
units)
Further study of differential and integral
calculus. Topics: a more rigorous development of limits and
derivatives, advanced techniques and applications of integration,
power series, calculus for parametric equations and polar
coordinates, simple differential equations. Together with Calculus A
and Calculus B, this course prepares students to take the Calculus BC
AP exam. Prerequisite: OM041 AP Calculus B
University Level
UM51A - Linear Algebra (Directed
study: 7.5 units)
An introductory course in linear algebra.
Topics: linear spaces, transformations, matrices, eigenvalues,
eigenvectors, and linear operators. Prerequisite: UM52A
Multivariable Differential Calculus and consent of the instructor
UM52A - Multivariable Differential
Calculus (Directed study: 7.5 units)
Differential
calculus for functions of two or more variables. Topics: vectors and
vector-valued functions in 2-space and 3-space, tangent and normal
vectors, curvature, functions of two or more variables, partial
derivatives and differentiability, directional derivatives and
gradients, maxima and minima, optimization using Lagrange
multipliers. Prerequisite: OM042 Calculus BC with AP sCore 4 or 5
and consent of the instructor
UM52B - Multivariable Integral
Calculus (Directed study: 7.5 units)
Integral calculus
for functions of two or more variables. Topics: double and triple
integrals, change of variables and the Jacobian, vector fields, line
integrals, independence of path and the fundamental theorem of line
integrals, Green's theorem, divergence theorem, and Stokes' theorem. Prerequisite: UM52A Multivariable Differential Calculus and
consent of the instructor
UM53A - Differential Equations (Directed study: 7.5 units)
Basic techniques and
methods for solving ordinary differential equations. Topics: linear,
separable, and exact equations, existence and uniqueness theorems,
difference equations, basic theory of higher order equations,
variation of parameters, undetermined coefficients, series solutions,
Laplace transform, systems of equations. Prerequisites: UM51A
Linear Algebra and UM52A Multivariable Differential Calculus and
consent of the instructor
UM106 - Complex Analysis (Directed
study: 7.5 units)
Theory of differentiation and integration of
complex functions. Topics: algebra of complex numbers, complex
functions, multi-valued functions, exponentials, logarithms,
analyticity, integrals, power series, Laurent series, residues,
isolated singularities, poles and zeros. Prerequisites: UM51A
Linear Algebra and UM115 Real Analysis and consent of the instructor
UM109 - Modern Algebra (Directed
study: 7.5 units)
Theory of abstract algebra, with particular
emphasis on applications involving symmetry. Topics: groups, rings,
fields, matrix and crystallographic groups, and constructibility. Prerequisites: UM51A Linear Algebra and UM52B Multivariable
Integral Calculus and consent of the instructor
UM115 - Real Analysis (Directed
study: 7.5 units)
Theory of functions of a real variable.
Topics: sequences, series, limits, continuity, differentiation,
integration, and basic point-set topology. Prerequisite: UM52B
Multivariable Integral Calculus and UM51A Linear Algebra and
consent of the instructor
UM131 - Partial Differential Equations (Directed study: 7.5 units)
Theory of differential
equations involving functions of more than one variable. Topics:
first order equations, classification of second order equations,
initial-boundary value problems for heat equation, wave and related
equations, separation of variables, eigenvalue problems, Fourier
series, existence and uniqueness questions. Prerequisites: UM52B
Multivariable Integral Calculus, UM53A Differential Equations, UM106
Complex Analysis and consent of the instructor
UM152 - Number Theory (Directed
study: 7.5 units)
Introduction to number theory and its
applications. Topics: Euclid's algorithm, divisibility, prime
numbers, congruence of numbers, theorems of Fermat, Euler, Wilson,
Lagrange; residues of power, quadratic residues, introduction to
binary quadratic forms. Prerequisite: OM013 Precalculus with
Trigonometry and consent of the instructor
UM157 - Introduction to Logic (Directed study: 7.5 units)
A standard introduction to sentential
and first-order logic. Topics: semantics and syntax of sentential
logic, truth tables, inference rules, proofs, counterexamples,
quantification, symbolizing English sentences, consistency proofs and
independence. Prerequisite: OM013 Precalculus with Trigonometry
and consent of the instructor |
