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Johns Hopkins University
Department of Mathematics
404 Krieger Hall
3400 N. Charles Street
Baltimore, MD 21218
410-516-7397 Phone
410-516-5549 Fax
|
Undergraduate
Courses
|
110.105
|
Introduction to Calculus
|
|
110.106-107
|
Calculus I and II (Biological and Social Sciences)
|
|
110.108-109
|
Calculus I and II (Physical Sciences and Engineering)
|
|
110.113 |
Honors
Single Variable Calculus |
|
110.201 |
Linear Algebra
|
|
110.202 |
Calculus III--Calculus of Several Variables
|
|
110.211 |
Honors Multivariable Calculus |
|
110.212 |
Honors Linear Algebra
|
|
110.225 |
Putnam Problem Solving |
|
110.302 |
Differential Equations with Applications
|
|
110.304
|
Elementary Number Theory
|
|
110.311 |
Methods of Complex Analysis |
|
110.328 |
Non-Euclidean Geometry
|
|
110.401 |
Advanced Algebra I
|
|
110.402 |
Advanced Algebra II
|
|
110.405 |
Real Analysis I |
|
110.406 |
Real Analysis II |
|
110.407-408 |
Geometry and Relativity |
|
110.413 |
Introduction to Topology
|
|
110.415 |
Honors Analysis I |
|
110.416 |
Honors Analysis II |
|
110.417 |
Partial Differential Equations for
Applications
|
|
110.421 |
Dynamical Systems |
|
110.423
|
Lie Groups for Undergraduates
|
|
110.427 |
Introduction of the Calculus of Variations
|
|
110.429 |
Mathematics of Quantum Mechanics |
|
110.431 |
Introduction to Knot Theory |
|
110.439 |
Introduction to Differential Geometry
|
|
110.443 |
Fourier Analysis
|
|
110.462 |
Prime Numbers and Riemann's Zeta Function
|
|
110.599 |
Independent Study, Undergraduate
|
Course Descriptions
|
110.105 |
(Q)
Introduction to Calculus |
| |
This course
is a pre-calculus course and provides students with all the
background necessary for the study of calculus. Includes a review of
algebra, trigonometry, exponential and logarithmic functions,
coordinates and graphs. Each of these tools is introduced in its
cultural and historical context. The concept of the rate of change of a
function will be introduced. Not open to students who have studied
Calculus in high school. |
| 4
credits |
Syllabus: |
110.105 |
|
110.106
110.107 |
(Q)
Calculus I, II (Biological and Social Sciences) |
| |
Differential and integral Calculus. Includes analytic geometry,
functions, limits, integrals and derivatives, introduction to
differential equations, functions of several variables, linear systems,
applications for systems of linear differential equations, probability
distributions.
Applications to the biological and social sciences
will be discussed, and the courses are designed to meet the needs of
students in these disciplines.
|
| 4
credits |
Syllabus: |
110.106,
110.107 |
|
110.108
110.109 |
(Q)
Calculus I, II (Physical Sciences and Engineering) |
| |
Differential and integral Calculus. Includes analytic geometry,
functions, limits, integrals and derivatives, polar coordinates,
parametric equations, Taylor's theorem and applications,
infinite sequences and series. Applications to the physical
sciences and engineering will be discussed, and the courses are
designed to meet the needs of students in these disciplines.
|
| 4
credits |
Syllabi: |
110.108,
110.109 |
|
110.113 |
(Q)
Honors Single Variable Calculus |
| |
The honors alternative to the Calculus sequences
110.106-107 or
110.108-109. Meets the general
requirement for both Calculus I and Calculus II in one semester (counts
as only one course). A highly theoretical treatment of one
variable differential and integral calculus based on our modern understanding of
the real number system as explained by Cantor, Dedekind, and Weierstrass. Previous background in Calculus is not
assumed. Content includes differential calculus (derivatives,
differentiation, chain rule, optimization, related rates, etc), the
theory of integration, the Fundamental Theorems of Calculus, with
applications of integration, and Taylor series. |
|
Prerequisites: |
A
strong ability to learn mathematics quickly and on a higher
level than that of the regular Calculus sequences.
|
| 4
credits |
Syllabus: |
110.113 |
|
110.201 |
(Q)
Linear Algebra |
| |
Vector spaces, matrices, and linear transformations. Solutions of
systems of linear equations. Eigenvalues, eigenvectors, and
diagonalization of matrices. Applications to differential equations. |
|
Prerequisites: |
Calculus II
or
110.113 or
a 5 on the Advanced Placement BC exam. |
| 4
credits |
Syllabus: |
110.201 |
|
110.202 |
(Q)
Calculus III--Calculus of Several Variables |
| |
Calculus of functions of more than one variable: partial
derivatives, and applications; multiple integrals, line and
surface integrals; Green's Theorem, Stokes' Theorem, and Gauss'
Divergence Theorem. |
|
Prerequisites: |
Calculus II
or
110.113 or
a 5 on the Advanced Placement BC exam. |
| 4
credits |
Syllabus: |
110.202 |
|
110.211 |
(Q)
Honors Multivariable Calculus |
| |
This course includes the material in 110.202
Calculus III but with a strong emphasis on theory and proofs.
Recommended only for mathematics majors or mathematically able
students majoring in physical science or engineering. |
|
Prerequisites: |
B+
or better in
Linear Algebra,
or Honors
Linear Algebra. Either of these courses may be taken
as a co-requisite. |
| 4
credits |
Syllabus: |
110.211 |
|
110.212 |
(Q)
Honors Linear Algebra |
| |
This course includes the material in
110.201
Linear Algebra with a strong emphasis on theory and proofs.
Recommended only for mathematics majors or mathematically able
students majoring in physical science, engineering. |
|
Prerequisites: |
B+
or better in
Calculus II,
or 5 on the Advanced Placement BC exam. or
110.113. |
| 4
credits |
Syllabus: |
110.212 |
|
110.302 |
(Q,E)
Differential Equations with Applications |
| |
This is an applied course in ordinary differential equations,
which is primarily for students in the biological, physical and
social sciences, and engineering. Techniques for solving
ordinary differential equations are studied. Topics covered
include first order differential equations, second order linear
differential equations, applications to electric circuits,
oscillation of solutions, power series solutions, systems of
linear differential equations, autonomous systems, Laplace
transforms and linear differential equations, mathematical
models (e.g., in the sciences or economics). |
|
Prerequisites: |
Calculus II
or 110.113. |
| 4
credits |
Syllabus: |
110.302 |
|
110.304 |
(Q)
Elementary Number Theory |
| |
This course provides
some historical background and examples of topics of current
research interest in number theory. Includes concrete
examples of some of the abstract concepts studied in
110.401-402. Topics include primes and prime
factorization, congruences, Euler's function, quadratic
reciprocity, primitive roots, solutions to polynomial
congruences (Chevalley's theorem), Diophantine equations
including the Pythagorean and Pell equations, Gaussian integers,
and Dirichlet's theorem on primes. |
|
Prerequisites: |
Calculus II
and Linear
Algebra. |
| 4
credits |
Syllabus: |
110.304 |
| 110.311 |
(Q)
Methods of Complex Analysis |
| |
This course is an
introduction to the theory of functions of one complex variable.
Its emphasis is on techniques and applications, and it serves as
a basis for more advanced courses. Functions of a complex
variable and their derivatives; power series and Laurent
expansions; Cauchy integral theorem and formula; calculus of
residues and contour integrals; harmonic functions. |
|
Prerequisites: |
Calculus III. |
| 4
credits |
|
|
|
110.328 |
(Q)
Non-Euclidean Geometry |
| |
For 2,000 years, Euclidean geometry was the
geometry. In the 19th century, new, equally consistent but
very different geometries were discovered. This course
will delve into these geometries on an elementary but
mathematically rigorous level. |
|
Prerequisites: |
High
school geometry. |
| 4
credits |
|
|
| 110.401
|
(Q)
Advanced Algebra I |
| |
An introduction to the basic notions of modern algebra. Elements of
group theory: groups, subgroups, normal subgroups, quotients,
homomorphisms. Generators and relations, free groups, products,
commutative (Abelian) groups, finite groups. Groups acting on sets, the
Sylow theorems. Definition and examples of rings and ideals.
Introduction to field theory. Linear algebra over a field. Field
extensions, constructible polygons, non-trisectability. |
|
Prerequisites: |
Linear Algebra |
| 4
credits |
Syllabus: |
110.401 |
| 110.402
|
(Q)
Advanced Algebra II |
| |
This is a continuation of
110.401. Theory of fields (continued).
Splitting field of a polynomial, algebraic closure of a field. Galois
theory: correspondence between subgroups and subfields. Solvability of
polynomial equations by radicals. Modules over a ring. Principal ideal
domains, structure of finitely generated modules over them.
|
|
Prerequisites: |
110.401 |
| 4
credits |
Syllabus: |
110.402 |
| 110.405 |
(Q)
Real Analysis I |
| |
This course is designed to give a firm grounding in the basic tools of
analysis. It is recommended as preparation (but may not be a
prerequisite) for other advanced analysis courses. Real and complex
number systems, topology of metric spaces, limits, continuity, infinite
sequences and series, differentiation, Riemann-Stieltjes integration.
|
|
Prerequisites: |
Calculus III,
Linear Algebra. |
| 4
credits |
Syllabus: |
110.405 |
| 110.406 |
(Q)
Real Analysis II |
| |
This course
continues
110.405, with an
emphasis on the fundamental notions of modern analysis.
Sequences and series of functions, Fourier series,
equicontinuity and the Arzela-Ascoli theorem, the Stone-Weierstrass
theorem. Functions of several variables, the inverse and
implicit function theorems, introduction to the Lebesgue
integral. |
|
Prerequisites: |
110.405, or
110.415. |
| 4
credits |
Syllabus: |
110.406 |
|
110.407
110.408 |
(Q,N)
Geometry and Relativity |
| |
Special relativity: Lorentz transformation, Minkowski spacetime, mass,
energy-momentum, stress-energy tensor, electrodynamics. Introduction to
differential geometry: theory of surfaces, first and second fundamental
forms, curvature. Gauss's theorema egregium, differentiable
manifolds, connections and covariant differentiation, geodesics,
differential forms, Stokes theorem. Gravitation as a geometric theory:
Lorentz metrics, Riemann curvature tensor, tidal forces and geodesic
deviation, gravitational redshift, Einstein field equation, the
Schwarzschild solution, perihelion precession, the deflection of
light, black holes, cosmology. |
|
Prerequisites: |
Calculus II,
Linear Algebra,
General Physics II. |
| 4
credits |
|
|
|
110.413
|
(Q)
Introduction to Topology |
| |
The basic concepts of point-set topology: topological spaces,
connectedness, compactness, quotient spaces, metric spaces,
function spaces. An introduction to algebraic topology: covering
spaces, the fundamental group, and other topics as time permits.
|
|
Prerequisites: |
Calculus III. |
| 4
credits |
Syllabus: |
110.413 |
|
110.415
|
(Q)
Honors Analysis I |
| |
This highly theoretical sequence in analysis is
reserved for mathematics majors and/or the most mathematically able students. The sequence covers the real number
system, metric spaces, basic functional analysis, the Lebesgue integral,
and other topics. |
|
Prerequisites: |
Calculus III
and Linear
Algebra. |
| 4
credits |
|
|
|
110.416 |
(Q)
Honors Analysis II |
| |
This course continues
110.415
Honors Analysis I, with an emphasis on the fundamental notions of modern
analysis. Topic here include functions of bounded variation, Riemann-Stieltjes
integration, Riesz representation theorem, along with measures,
measurable functions, and the lebesgue integral, properties of Lp-
spaces, and Fourier series. |
|
Prerequisites: |
110.415. |
| 4
credits |
|
|
|
110.417
|
(Q,E) Partial
Differential Equations for Applications |
| |
Characteristics. classification of second order equations, well-posed
problems. separation of variables and expansions of solutions. The wave
equation: Cauchy problem, Poisson's solution, energy inequalities,
domains of influence and dependence. Laplace's equation: Poisson's
formula, maximum principles, Green's functions, potential theory
Dirichlet and Neumann problems, eigenvalue problems. The heat equation:
fundamental solutions, maximum principles. |
|
Prerequisites:
Recommended:
|
Calculus III.
110.405
or 110.415. |
| 4
credits |
|
|
|
110.421
|
(Q) Dynamical Systems |
| |
A basic introduction to the
general theory of dynamical systems from a mathematical standpoint, this
course studies the properties of continuous and discrete dynamical
systems, in the form of ordinary differential and difference equations
and iterated maps. Topics include contracting and expanding maps,
interval and circle maps, toral flows, billiards, limit sets and
recurrence, topological transitivity, bifurcation theory and chaos. Applications include
classical mechanics and optics, inverse and implicit functions theorems,
the existence and uniqueness of general ODEs, stable and center
manifolds, and structural stability. |
|
Prerequisites: |
Calculus III,
Linear Algebra,
and
Differential Equations. |
| 4
credits |
|
|
|
110.423 |
(Q) Lie Groups for
Undergraduates |
| |
This course is an introduction to Lie Groups and their representations
at the upper undergraduate level. It will cover basic Lie Groups such as
SU (2), U(n) , the Euclidean Motion Group and Lorentz Group. This course
is useful for students who want a working knowledge of group
representations. Some aspects of the role of symmetry groups in particle
physics such as some of the formal aspects of the electroweak and the
strong interactions will also be discussed. |
|
Prerequisites: |
Calculus III.
Prior knowledge of group theory (e.g.
110.401)
would be helpful. |
| 4
credits |
|
|
|
110.427
|
(Q) Introduction to the
Calculus of Variations |
| |
The calculus of variations is concerned with finding optimal solutions
(shapes, functions, etc.) where optimality is measured by minimizing a
functional (usually an integral involving the unknown functions)
possibly with constraints. Applications include mostly
one-dimensional (often geometric) problems: brachistochrone, geodesics,
minimum surface area of revolution, isoperimetric problem, curvature
flows, and some differential geometry of curves and surfaces.
|
|
Prerequisites: |
Calculus III.
|
| 4
credits |
|
|
|
110.429
|
(Q) Mathematics of
Quantum Mechanics |
| |
The basis of quantum mechanics is the Schrodinger equation. The focus of
this course will be on one dimensional Schrodinger equations. Topics
include eigenvalue problems, bound states, scattering states, tunneling,
uncertainty principle, dynmaics, semi-classical limit. The ideas will be
illustrated through many examples. |
|
Prerequisites: |
Differential
Equations or the permission of the instructor. |
| 4
credits |
|
|
|
110.431
|
(Q) Introduction to Knot Theory |
| |
The theory of knots and
links is a facet of modern topology. The course will be mostly
self-contained, but a good working knowledge of groups will be helpful.
Topics include braids, knots and links, the fundamental group of a knot
or link complement, spanning surfaces, and low dimensional homology
groups. |
|
Prerequisites: |
Calculus III.
|
| 4
credits |
|
|
|
110.439
|
(Q) Introduction to Differential Geometry |
| |
Theory of curves and surfaces in Euclidean space: Frenet equations,
fundamental forms, curvatures of a surface, theorems of Gauss and
Mainardi-Codazzi, curves on a surface; introduction to tensor analysis
and Riemannian geometry; theorema egregium; elementary global theorems.
|
|
Prerequisites: |
Calculus III,
Linear Algebra. |
| 4
credits |
|
|
|
110.443
|
(Q,E) Fourier
Analysis and Generalized Functions |
| |
An introduction to the
Fourier transform and the construction of fundamental solutions of
linear partial differential equations. Homogeneous distributions on the
real line: the Dirac delta function, the Heaviside step function.
Operations with distributions: convolution, differentiation, Fourier
transform. Construction of fundamental solutions of the wave, heat,
Laplace and Schrödinger equations. Singularities of fundamental
solutions and their physical interpretations (e.g., wave fronts).
Fourier analysis of singularities, oscillatory integrals, method of
stationary phase. |
|
Prerequisites:
Recommended: |
Calculus III,
Linear Algebra.
110.405
or 110.415. |
| 4
credits |
|
|
|
110.462
|
(Q) Prime Numbers and Riemann's Zeta Function |
| |
This course is devoted to such questions as: How many prime numbers are
there less than N? How are they spaced apart? Although prime numbers at
first sight have nothing to do with complex numbers, the answers to
these questions due to Gauss, Riemann, Hadamard) involve complex
analysis and in particular the Riemann zeta function, which controls the
the distribution of primes. This course builds on 110.311 and is an
introduction to Analytic Number Theory for undergraduates.
|
|
Prerequisites: |
110.311. |
| 4
credits |
|
|
|
110.599 |
Independent
Study, Undergraduate |
| |
Topics vary based on an
agreement between the student and the Instructor. |
|
credits vary |
| |
