# Physics and Mathematics, BS

### Mathematics deals with abstraction, logic, and quantitative reasoning. Because it has applications to nearly every branch of science and engineering, it’s essential for mathematicians to think about how their work infiltrates other branches of learning. Advances in physics — for example, those in electromagnetism and thermodynamics — often resonate deeply with mathematics.

At the School of Engineering, the BS in Applied Physics and Mathematics program serves as a means to bridge these 2 disciplines. The dual major allows you to gain a foothold in separate but substantial fields. In addition to learning the fundamentals of physics and math, our students pursue a specialized course of study that a minor in either field just can’t match.

But we also want to make sure your skills transfer over to the real world. That’s why we provide internship opportunities at major financial, insurance, and technology firms in the New York area.

Students with experience in both mathematics and physics enjoy diverse and interesting careers. Our graduates have the freedom to explore such stimulating fields as chemistry, biology, medicine, and engineering. They’re also qualified for positions in software design, economics, aerospace engineering, law, and business.

## Curriculum

OverviewYou must complete 128 credits, as defined below, to graduate from the School of Engineering with a Bachelor of Science in Physics and Mathematics. Please note that the curriculum that follows applies to students who began classes in the fall of 2014 or later. If you entered the School of Engineering prior to that date, please review the curriculum and typical course schedule for students entering prior to fall 2014.

Physics Requirements (33 credits)

- 3 Credits Mechanics PH-UY1013
- This course is the first of a three-semester lecture sequence in general physics for science and engineering students. Motion of particles and systems of particles. One-dimensional motion. Vectors and two-dimensional motions. Forces and acceleration. Conservation of energy and momentum. Rotations. The free and driven harmonic oscillator. Gravitation. (This class meets four hours per week for lectures and recitation.)

Prerequisites: MA-UY 1024 or an approved equivalent. Corequisites: MA-UY 1124 or approved equivalent, and EX-UY 1 - 3 Credits Electricity, Magnetism, & Fluids PH-UY2023
- This is the second course of a three-semester lecture sequence in general physics for science and engineering students. Fluids at rest and in motion. An introduction to electric and magnetic forces and fields. Electric charge density. Electric fields from simple charge distributions. Electric potential. Capacitance.

Magnetic forces. Magnetic field from a current loop. Inductance. Magnetism

in matter. Current and resistance. (This class meets four hours per week for lectures and recitation.)

Prerequisites: PH-UY 1013 and MA-UY 1124 or an approved equivalent. Co-requisite: PH-UY 2121 General Physics Laboratory I, and EX-UY 1 - 3 Credits Waves, Optics, & Thermodynamics PH-UY2033
- This is the third course of a three-semester lecture sequence in general physics for science and engineering students. Water, sound and electromagnetic waves. Reflection, scattering and absorption. Standing waves and spectra. Superposition, diffraction and beats. Geometrical optics. Introduction to thermodynamics; temperature, heat, and entropy. (This class meets four hours per week for lectures and recitation.)

Prerequisites: PH-UY 2121 and PH-UY 2023. Co-requisites: PH-UY 2131, and EX-UY 1. - 4 Credits Analytical Mechanics PH-UY2104
- The course covers statics by virtual work and potential energy methods. Stability of equilibrium. Particle dynamics, harmonic oscillator and planetary motion. Rigid body dynamics in two and three dimensions. Lagrangian mechanics. Dynamics of oscillating systems.

Prerequisite: PH-UY 2023; Co-requisite: MA-UY 2034 - 1 Credits General Physics Laboratory I PH-UY2121
- PH-UY 2121 General Physics Laboratory I (0.5:1:0:1). An introductory level experimental course. Fundamental laboratory experiments in classical mechanics and electrostatics. Stresses basic experimental techniques, error analysis, and written presentation of experiment results. Experiments require progressively more detailed and sophisticated analysis. This laboratory class meets for three hours on alternate weeks.

Prerequisites: PH-UY 1013 and MA-UY 1124 or equivalent. Co-requisite: PH-UY 2023. - 1 Credits General Physics Laboratory II PH-UY2131
- PH 2131 General Physics Laboratory II (0.5:1:0:1). The second part of the introductory physics laboratory program. Fundamental laboratory experiments in E&M, waves, optics, and thermodynamics. Stresses experimental models and design, error and data analysis. This laboratory class meets for three hours on alternate weeks.

Prerequisites: PH-UY 2121 and PH-UY 2023. Corequisite: PH-UY 2033 - 4 Credits Introduction to Modern and Solid State Physics PH-UY2344
- Special theory of relativity, Michelson Morley experiment. Planck’s quantum hypothesis, photoelectric effect, Compton effect, Rutherford scattering, Bohr’s atom, DeBroglie wavelength, electron diffraction, wave function, uncertainty principle, Schrodinger equation. Application to: square well potential, one electron atom. Atomic nucleus, fission and fusion. Energy bands in a periodic lattice, Kronig Penney model, valence, conduction bands, impurity states, electron mobility. Semiconductor properties. Introduction to superconductivity; electron pairs, energy gap, Josephson effect.

Prerequisites: PH-UY 2023 and MA-UY 2034; Co-requisite: PH-UY 2033. - 2 Credits Junior Physics Laboratory PH-UY3002
- An intermediate level laboratory course providing in depth exposure to a selection of classic physics experiments. Students' experimental skill set is expanded and data analysis and communication skills developed.

Prerequisites: PH-UY 2131 and PH-UY 2033; Co-requisites: PH-UY 2344 and MA-UY 2224. - 4 Credits Electricity and Magnetism PH-UY3234
- The course covers properties of the electrostatic, magnetostatic and electromagnetic field in vacuum and in material media. Maxwell’s equations with applications to elementary problems.

Prerequisites: PH-UY 2033 and MA-UY 2114. - 4 Credits Thermodynamics and Statistical Physics PH-UY4124
- The course covers fundamental laws of macroscopic thermodynamics, heat, internal energy and entropy. Topics include an introduction to statistical physics, and applications of Maxwell, Fermi-Dirac and Bose-Einstein distributions.

Prerequisites: PH-UY 2344, MA-UY 2114, and MA-UY 2224. - 4 Credits Introduction to the Quantum Theory PH-UY4364
- The course introduces quantitative introduction to the quantum theory, which describes understanding light, electrons, atoms, nuclei and solid matter. Superposition principle, expectation values, momentum operator and wave function, duality, current vector, Hermitian operators, angular momentum, solution of the radial equation, electron in a magnetic field, perturbation theory, WKB approximation, identical particles. Applications include alpha decay, electrons in a periodic lattice, hydrogen spectrum, helium atom, neutron-proton scattering, and quark model of baryons.

Prerequisites: PH-UY 2344, MA-UY 2114, and MA-UY 2224.

Math Requirements (29 credits)

- 4 Credits Calculus I for Engineers MA-UY1024
- This course covers: Library of Functions, functions of one variable. Limits, derivatives of functions defined by graphs, tables and formulas, differentiation rules for power, polynomial, exponential and logarithmic functions, derivatives of trigonometric functions, the product and quotient rules, the chain rule, applications of the chain rule, maxima and minima, optimization. The definite integral, the Fundamental Theorem of Calculus and interpretations, theorems about definite integrals, anti-derivatives. MA-UY 1324 is for students who wish to take MA-UY 1024 but need more review of precalculus. MA-UY 1324 covers the same material as MA-UY 1024 but with more contact hours per week, incorporating a full discussion of the required precalculus topics.

Prerequisite: Placement Exam or MA-UY 912 or MA-UY 914 (with a grade of B or better). Corequisite: EX-UY 1 - 4 Credits Calculus II for Engineers MA-UY1124
- This course covers techniques of integration, introduction to ordinary differential equations, improper integrals, numerical methods of integration, applications of integration, sequences, series, power series, approximations of functions via Taylor polynomials, Taylor series, functions of two variables, graphs of functions of two variables, contour diagrams, linear functions, functions of three variables. MA-UY 1424 is for students who wish to take MA-UY 1124 but need more review of precalculus. MA-UY 1424 covers the same material as MA-UY 1124 but with more contact hours per week, incorporating a full discussion of the required precalculus topics.

Prerequisites: MA-UY 1022 (with a grade of B or better) or MA-UY 1024 or MA-UY 1324 (with a grade of B or better).

Corequisite: EX-UY 1. - 4 Credits Linear Algebra and Differential Equations MA-UY2034
- MA-UY 2034 is an introduction to ordinary differential equations and linear algebra. The course develops the techniques for the analytic and numeric solutions of ordinary differential equations (and systems) that are widely used in modern engineering and science. Linear algebra is used as a tool for solving systems of linear equations as well as for understanding the structure of solutions to linear (systems) of differential equations. Topics covered include the fundamental concepts of linear algebra such as Gaussian elimination, matrix theory, linear transformations, vector spaces, subspaces, basis, eigenvectors, eigenvalues and the diagonalization of matrices, as well as the techniques for the analytic and numeric solutions of ordinary differential equations (and systems) that commonly appear in modern engineering and science.

Prerequisite: MA-UY 1124, MA-UY 1424 or MA-UY 1132. Note: Not open to students who have taken MA-UY 3044 or MA-UY 3054 or MA-UY 3083 or MA-UY 4204. - 4 Credits Calculus Iii: Multi-dimensional Calculus MA-UY2114
- Vectors in the plane and space. Partial derivatives with applications, especially Lagrange multipliers. Double and triple integrals. Spherical and cylindrical coordinates. Surface and line integrals. Divergence, gradient, and curl. Theorems of Gauss and Stokes.

Prerequisite: MA-UY 1124 or MA-UY 1424 or MA-UY 1132. - 4 Credits Data Analysis MA-UY2224
- An introductory course to probability and statistics. It affords the student some acquaintance with both probability and statistics in a single term. Topics in Probability include mathematical treatment of chance; combinatorics; binomial, Poisson, and Gaussian distributions; the Central Limit Theorem and the normal approximation. Topics in Statistics include sampling distributions of sample mean and sample variance; normal, t-, and Chi-square distributions; confidence intervals; testing of hypotheses; least squares regression model. Applications to scientific, industrial, and financial data are integrated into the course.NOTE: Not open to students who have taken MA-UY 2233 or MA-UY 3012 or MA-UY 3022.

Prerequisite: MA-UY 1124, MA-UY1424, or MA-UY 1132 - 3 Credits Advanced Linear Algebra and Complex Variables MA-UY3113
- This course provides a deeper understanding of topics introduced in MA-UY 2012 and MA-UY 2034 and continues the development of those topics, while also covering functions of a Complex Variable. Topics covered include: The Gram-Schmidt process, inner product spaces and applications, singular value decomposition, LU decomposition. Derivatives and Cauchy-Riemann equations, integrals and Cauchy integral theorem. Power and Laurent Series, residue theory.

Prerequisites: (MA-UY 2122 or MA-UY 2114) AND (MA-UY 2012 or MA-UY 2034). Note: Not open to students who have taken MA-UY 1533, MA-UY 3112 or MA-UY 4433. - 3 Credits Applied Partial Differential Equations MA-UY4413
- Modeling of physical processes. Classification of equations. Formulation and treatment of boundary- and initial-value problems. Green’s functions. Maximum principle. Separation of variables. Fourier series and integrals. Quasilinear first-order equations and characteristics. D’Alembert solution of wave equation. Conservation laws and shock waves.

Prerequisite: MA-UY 2114 and (MA-UY 3083 or MA-UY 4204) - MA-UY4423 Please refer to the bulletin for more information

Physics Electives (7 Credits)

Select 7 credits from the list of undergraduate applied physics elective courses. Graduate courses may be substituted with advisor’s approval.

Math Electives (10 Credits)

Select 10 credits from the list of undergraduate math elective courses. Graduate courses may be substituted with advisor’s approval.

Free Electives, Independent Study and Projects (14 Credits)

14 credits are reserved for free electives and independent study courses, of which 8 credits are reserved for a 6 credit applied physics project plus a 2 credit senior physics seminar or a 4 credit math project/thesis and an extra 4 credit math elective.

Electives in the Humanities and Social Sciences (16 Credits)

You are required to take 16 credits in the humanities and social sciences requiring EXPOS-UA1 and EXPOS-UA2 as prerequisites. To gain some breadth and depth of knowledge, it is required that you take courses in at least two disciplines and at least one course at an advanced level.

Other Required Courses (19 Credits)

- 1 Credits Engineering and Technology Forum EG-UY1001
- In this course the notion of invention, innovation and entrepreneurship (i2e) is introduced to the students’ educational experience. Students will be exposed to elements of a research-intensive institution and diverse research performed by leading engineers, scientists, inventors and entrepreneurs.
- 4 Credits General Chemistry for Engineers CM-UY1004
- This is a one-semester introductory course in general chemistry. It covers chemical equations, stoichiometry, thermodynamics, gases, atomic and molecular structure, periodic table, chemical bonding, states of matter, chemical equilibrium, organic, inorganic and polymeric materials and electrochemistry.

Corequisite: EX-UY 1 - EXPOS-UA1 Please refer to the bulletin for more information
- EXPOS-UA2 Please refer to the bulletin for more information
- 4 Credits Introduction to Programming & Problem Solving CS-UY1114
- This course introduces problem solving and computer programming and is for undergraduate Computer Science and Computer Engineering majors who have limited prior experience in programming in any language. The course covers fundamentals of computer programming and its underlying principles using the Python programming language. Concepts and methods introduced in the

course are illustrated by examples from various disciplines. ABET competencies: a,b,c, e, f, g, k

Corequisite: EX-UY 1

And either of these courses can be taken:

- 2 Credits Physics: the Genesis of Technology PH-UY1002
- This course introduces contemporary topics in physics, along with readings and discussions of topics with technological implications.

Prerequisite: Only first-year students are permitted to enroll in this introductory level course. - 2 Credits The Art of Mathematics MA-UY1002
- This is an introductory course about Mathematics. Areas of Mathematics. History of Mathematics. Mathematical Methods. Great Mathematicians. Famous open and solved mathematical problems. The Study of Mathematics. Mathematical Software.

Prerequisite: Only first-year students are permitted to enroll in this introductory level course.

## Sample Course Schedule

OverviewThis schedule lists specific courses you might take during your 4 years at the School of Engineering; it does not indicate required courses. It applies to students who begin classes in the fall of 2009 and onwards. If you entered the School of Engineering prior to that date, please review the curriculum and typical course schedule for students entering prior to fall 2009.

First Year

#### Fall Semester

**Choose one:**

- 2 Credits Physics: the Genesis of Technology PH-UY1002
- This course introduces contemporary topics in physics, along with readings and discussions of topics with technological implications.

Prerequisite: Only first-year students are permitted to enroll in this introductory level course. - 2 Credits The Art of Mathematics MA-UY1002
- This is an introductory course about Mathematics. Areas of Mathematics. History of Mathematics. Mathematical Methods. Great Mathematicians. Famous open and solved mathematical problems. The Study of Mathematics. Mathematical Software.

Prerequisite: Only first-year students are permitted to enroll in this introductory level course.

**Take all:**

- 4 Credits Calculus I for Engineers MA-UY1024
- This course covers: Library of Functions, functions of one variable. Limits, derivatives of functions defined by graphs, tables and formulas, differentiation rules for power, polynomial, exponential and logarithmic functions, derivatives of trigonometric functions, the product and quotient rules, the chain rule, applications of the chain rule, maxima and minima, optimization. The definite integral, the Fundamental Theorem of Calculus and interpretations, theorems about definite integrals, anti-derivatives. MA-UY 1324 is for students who wish to take MA-UY 1024 but need more review of precalculus. MA-UY 1324 covers the same material as MA-UY 1024 but with more contact hours per week, incorporating a full discussion of the required precalculus topics.

Prerequisite: Placement Exam or MA-UY 912 or MA-UY 914 (with a grade of B or better). Corequisite: EX-UY 1 - 4 Credits General Chemistry for Engineers CM-UY1004
- This is a one-semester introductory course in general chemistry. It covers chemical equations, stoichiometry, thermodynamics, gases, atomic and molecular structure, periodic table, chemical bonding, states of matter, chemical equilibrium, organic, inorganic and polymeric materials and electrochemistry.

Corequisite: EX-UY 1 - EXPOS-UA1 Please refer to the bulletin for more information
- 1 Credits Engineering and Technology Forum EG-UY1001
- In this course the notion of invention, innovation and entrepreneurship (i2e) is introduced to the students’ educational experience. Students will be exposed to elements of a research-intensive institution and diverse research performed by leading engineers, scientists, inventors and entrepreneurs.

#### Spring Semester

- 3 Credits Mechanics PH-UY1013
- This course is the first of a three-semester lecture sequence in general physics for science and engineering students. Motion of particles and systems of particles. One-dimensional motion. Vectors and two-dimensional motions. Forces and acceleration. Conservation of energy and momentum. Rotations. The free and driven harmonic oscillator. Gravitation. (This class meets four hours per week for lectures and recitation.)

Prerequisites: MA-UY 1024 or an approved equivalent. Corequisites: MA-UY 1124 or approved equivalent, and EX-UY 1 - 4 Credits Calculus II for Engineers MA-UY1124
- This course covers techniques of integration, introduction to ordinary differential equations, improper integrals, numerical methods of integration, applications of integration, sequences, series, power series, approximations of functions via Taylor polynomials, Taylor series, functions of two variables, graphs of functions of two variables, contour diagrams, linear functions, functions of three variables. MA-UY 1424 is for students who wish to take MA-UY 1124 but need more review of precalculus. MA-UY 1424 covers the same material as MA-UY 1124 but with more contact hours per week, incorporating a full discussion of the required precalculus topics.

Prerequisites: MA-UY 1022 (with a grade of B or better) or MA-UY 1024 or MA-UY 1324 (with a grade of B or better).

Corequisite: EX-UY 1. - 4 Credits Introduction to Programming & Problem Solving CS-UY1114
- This course introduces problem solving and computer programming and is for undergraduate Computer Science and Computer Engineering majors who have limited prior experience in programming in any language. The course covers fundamentals of computer programming and its underlying principles using the Python programming language. Concepts and methods introduced in the

course are illustrated by examples from various disciplines. ABET competencies: a,b,c, e, f, g, k

Corequisite: EX-UY 1 - EXPOS-UA2 Please refer to the bulletin for more information

Second Year

#### Fall Semester

- 3 Credits Electricity, Magnetism, & Fluids PH-UY2023
- This is the second course of a three-semester lecture sequence in general physics for science and engineering students. Fluids at rest and in motion. An introduction to electric and magnetic forces and fields. Electric charge density. Electric fields from simple charge distributions. Electric potential. Capacitance.

Magnetic forces. Magnetic field from a current loop. Inductance. Magnetism

in matter. Current and resistance. (This class meets four hours per week for lectures and recitation.)

Prerequisites: PH-UY 1013 and MA-UY 1124 or an approved equivalent. Co-requisite: PH-UY 2121 General Physics Laboratory I, and EX-UY 1 - 1 Credits General Physics Laboratory I PH-UY2121
- PH-UY 2121 General Physics Laboratory I (0.5:1:0:1). An introductory level experimental course. Fundamental laboratory experiments in classical mechanics and electrostatics. Stresses basic experimental techniques, error analysis, and written presentation of experiment results. Experiments require progressively more detailed and sophisticated analysis. This laboratory class meets for three hours on alternate weeks.

Prerequisites: PH-UY 1013 and MA-UY 1124 or equivalent. Co-requisite: PH-UY 2023. - 4 Credits Analytical Mechanics PH-UY2104
- The course covers statics by virtual work and potential energy methods. Stability of equilibrium. Particle dynamics, harmonic oscillator and planetary motion. Rigid body dynamics in two and three dimensions. Lagrangian mechanics. Dynamics of oscillating systems.

Prerequisite: PH-UY 2023; Co-requisite: MA-UY 2034 - 4 Credits Linear Algebra and Differential Equations MA-UY2034
- MA-UY 2034 is an introduction to ordinary differential equations and linear algebra. The course develops the techniques for the analytic and numeric solutions of ordinary differential equations (and systems) that are widely used in modern engineering and science. Linear algebra is used as a tool for solving systems of linear equations as well as for understanding the structure of solutions to linear (systems) of differential equations. Topics covered include the fundamental concepts of linear algebra such as Gaussian elimination, matrix theory, linear transformations, vector spaces, subspaces, basis, eigenvectors, eigenvalues and the diagonalization of matrices, as well as the techniques for the analytic and numeric solutions of ordinary differential equations (and systems) that commonly appear in modern engineering and science.

Prerequisite: MA-UY 1124, MA-UY 1424 or MA-UY 1132. Note: Not open to students who have taken MA-UY 3044 or MA-UY 3054 or MA-UY 3083 or MA-UY 4204.

HuSS Elective 1, Credits: 4.00*

#### Spring Semester

- 3 Credits Waves, Optics, & Thermodynamics PH-UY2033
- This is the third course of a three-semester lecture sequence in general physics for science and engineering students. Water, sound and electromagnetic waves. Reflection, scattering and absorption. Standing waves and spectra. Superposition, diffraction and beats. Geometrical optics. Introduction to thermodynamics; temperature, heat, and entropy. (This class meets four hours per week for lectures and recitation.)

Prerequisites: PH-UY 2121 and PH-UY 2023. Co-requisites: PH-UY 2131, and EX-UY 1. - 1 Credits General Physics Laboratory II PH-UY2131
- PH 2131 General Physics Laboratory II (0.5:1:0:1). The second part of the introductory physics laboratory program. Fundamental laboratory experiments in E&M, waves, optics, and thermodynamics. Stresses experimental models and design, error and data analysis. This laboratory class meets for three hours on alternate weeks.

Prerequisites: PH-UY 2121 and PH-UY 2023. Corequisite: PH-UY 2033 - 4 Credits Calculus Iii: Multi-dimensional Calculus MA-UY2114
- Vectors in the plane and space. Partial derivatives with applications, especially Lagrange multipliers. Double and triple integrals. Spherical and cylindrical coordinates. Surface and line integrals. Divergence, gradient, and curl. Theorems of Gauss and Stokes.

Prerequisite: MA-UY 1124 or MA-UY 1424 or MA-UY 1132.

MA ***3 Math Elective, Credits: 3.00

HuSS Elective 2, Credits: 4.00*

**To gain some breadth and depth of knowledge, take courses in at least two disciplines and at least one course at an advanced level.*

Third Year

#### Fall Semester

- 4 Credits Introduction to Modern and Solid State Physics PH-UY2344
- Special theory of relativity, Michelson Morley experiment. Planck’s quantum hypothesis, photoelectric effect, Compton effect, Rutherford scattering, Bohr’s atom, DeBroglie wavelength, electron diffraction, wave function, uncertainty principle, Schrodinger equation. Application to: square well potential, one electron atom. Atomic nucleus, fission and fusion. Energy bands in a periodic lattice, Kronig Penney model, valence, conduction bands, impurity states, electron mobility. Semiconductor properties. Introduction to superconductivity; electron pairs, energy gap, Josephson effect.

Prerequisites: PH-UY 2023 and MA-UY 2034; Co-requisite: PH-UY 2033. - 4 Credits Data Analysis MA-UY2224
- An introductory course to probability and statistics. It affords the student some acquaintance with both probability and statistics in a single term. Topics in Probability include mathematical treatment of chance; combinatorics; binomial, Poisson, and Gaussian distributions; the Central Limit Theorem and the normal approximation. Topics in Statistics include sampling distributions of sample mean and sample variance; normal, t-, and Chi-square distributions; confidence intervals; testing of hypotheses; least squares regression model. Applications to scientific, industrial, and financial data are integrated into the course.NOTE: Not open to students who have taken MA-UY 2233 or MA-UY 3012 or MA-UY 3022.

Prerequisite: MA-UY 1124, MA-UY1424, or MA-UY 1132 - Chemical Laboratory Safety CM-GY5040
- This course discusses problems of health and safety in chemical laboratories, including how to work safely with dangerous chemicals. This course must be completed by graduate and undergraduate chemistry students before they begin laboratory research.

PH ***4 Physics Elective, Credits: 4.00

HuSS Elective 3, Credits: 4.00*

#### Spring Semester

- 2 Credits Junior Physics Laboratory PH-UY3002
- An intermediate level laboratory course providing in depth exposure to a selection of classic physics experiments. Students' experimental skill set is expanded and data analysis and communication skills developed.

Prerequisites: PH-UY 2131 and PH-UY 2033; Co-requisites: PH-UY 2344 and MA-UY 2224. - 4 Credits Electricity and Magnetism PH-UY3234
- The course covers properties of the electrostatic, magnetostatic and electromagnetic field in vacuum and in material media. Maxwell’s equations with applications to elementary problems.

Prerequisites: PH-UY 2033 and MA-UY 2114. - 3 Credits Advanced Linear Algebra and Complex Variables MA-UY3113
- This course provides a deeper understanding of topics introduced in MA-UY 2012 and MA-UY 2034 and continues the development of those topics, while also covering functions of a Complex Variable. Topics covered include: The Gram-Schmidt process, inner product spaces and applications, singular value decomposition, LU decomposition. Derivatives and Cauchy-Riemann equations, integrals and Cauchy integral theorem. Power and Laurent Series, residue theory.

Prerequisites: (MA-UY 2122 or MA-UY 2114) AND (MA-UY 2012 or MA-UY 2034). Note: Not open to students who have taken MA-UY 1533, MA-UY 3112 or MA-UY 4433.

MA ***3 Math Elective, Credits: 4.00

HuSS Elective 4, Credits: 4.00*

**To gain some breadth and depth of knowledge, take courses in at least two disciplines and at least one course at an advanced level.*

Fourth Year

#### Fall Semester

Take all:

- 4 Credits Introduction to the Quantum Theory PH-UY4364
- The course introduces quantitative introduction to the quantum theory, which describes understanding light, electrons, atoms, nuclei and solid matter. Superposition principle, expectation values, momentum operator and wave function, duality, current vector, Hermitian operators, angular momentum, solution of the radial equation, electron in a magnetic field, perturbation theory, WKB approximation, identical particles. Applications include alpha decay, electrons in a periodic lattice, hydrogen spectrum, helium atom, neutron-proton scattering, and quark model of baryons.

Prerequisites: PH-UY 2344, MA-UY 2114, and MA-UY 2224. - 3 Credits Applied Partial Differential Equations MA-UY4413
- Modeling of physical processes. Classification of equations. Formulation and treatment of boundary- and initial-value problems. Green’s functions. Maximum principle. Separation of variables. Fourier series and integrals. Quasilinear first-order equations and characteristics. D’Alembert solution of wave equation. Conservation laws and shock waves.

Prerequisite: MA-UY 2114 and (MA-UY 3083 or MA-UY 4204)

PH ***3 Physics Elective, Credits: 3.00

XYZ ***3 Free Elective, Credits: 3.00

and either take both:

- 2 Credits Introduction to Senior Project in Physics PH-UY4902
- A qualified senior physics student or group of students work with a faculty member (and possibly graduate students) on an advanced problem in physics. In this introductory phase the student(s) and adviser select a suitable theoretical or experimental problem in the subject area and use various resources to solve it.
- 2 Credits Senior Seminar in Physics PH-UY4912
- Senior physics students, in consultation with the instructor, study and prepare presentations on several current research topics in the general area of interdisciplinary physics. Students’ performance is based on the mastery of the material chosen and also on the quality of the presentation made to the instructor and the seminar members.

or take:

MA ***4 Math Elective, Credits: 4.00

#### Spring Semester

Take all:

- 4 Credits Thermodynamics and Statistical Physics PH-UY4124
- The course covers fundamental laws of macroscopic thermodynamics, heat, internal energy and entropy. Topics include an introduction to statistical physics, and applications of Maxwell, Fermi-Dirac and Bose-Einstein distributions.

Prerequisites: PH-UY 2344, MA-UY 2114, and MA-UY 2224. - MA-UY4423 Please refer to the bulletin for more information

MA ***3 Math Elective, Credits: 3.00

XYZ ***3 Free Elective, Credits: 3.00

Choose one:

- 4 Credits Senior Project in Physics PH-UY4904
- In the project’s concluding phase, senior physics students or group of students work with a faculty member (and possibly graduate students) to solve an advanced problem in interdisciplinary physics. The conclusion of the project is a written report and an oral presentation made to the supervising faculty.
- 4 Credits Project in Mathematics I MA-UY3914
- In this course, students read, study, and investigate selected topics in mathematics. Problems are discussed and presented by participating students.

Prerequisite: approval of departmental adviser. Note: This course is repeatable for credit, but does not allow multiple enrollment in the same term.