Math and Music Studio

New Classes





 

 


 
 
Math & Music Studio's Fall 2016 Course Offerings

1.  NEW --
Elementary Differential Geometry

Differential geometry is the application of methods of calculus to curves and surfaces, and it studies the effects of small local differences on various kinds of multidimensional shapes.  This subject is typically taught at the university level where math, science and engineering students take the course during their junior or senior year.  Differential geometry was developed in the 19th century by C.F. Gauss for the study of surfaces and was later extended to higher dimensions by Bernhard Riemann.  Einstein learned differential geometry and used it to characterize gravity as the result of the curved geometry of warped space and time.

In today's world, physicists use differential geometry to study Einstein's theory of general relativity, electromagnetic fields, stability of matter, atomic physics (including the theory of quarks), Stephen Hawking's black hole theories and more.  Additionally, biophysicists use differential geometry to study the structure of protein molecules, and engineers and applied mathematicians use it to study crystallography, computerized vision, robotics and kinematic modeling.

Topics presented in this two-semester course will include differential forms, Frenet-Serret formulas, frame fields, covariant derivatives, connection forms, integration of forms, Gaussian curvature, Gauss-Bonnet theorem, Riemannian geometry, geodesics, Poincare-Hopf theorem, minimal surfaces, and classification of compact surfaces.
Prerequisite:  Either Multivariable Calculus and Differential Equations, or Complex Analysis, along with some knowledge of Linear Algebra

2.  Classical Euclidean Geometry

The primary focus of this two-semester course in geometry is on the basic theorems of Euclid.  Topics will include parallel lines, transversals, triangles, congruence, perpendicular bisectors, parallelogram theorems, trapezoids, similarity, right triangles, trigonometry, Pythagorean theorem, inscribed angle theorems, polygons, and area and perimeter.  Students will present their own proofs of the major and related theorems in class each week.
Prerequisite:  Algebra 1

3.  Algebra 2 with Trigonometry

Algebra 2 is considered by many as a gateway course that predicts student graduation from college, and it's typically taught in the tenth grade.  This course builds on the foundation laid in algebra 1 and geometry, and it stresses student mastery of the analysis and graphing of polynomials, and logarithmic, exponential and trigonometric functions.  Topics covered in this two-semester course include linear equations, matrices, determinants, polynomials, rational functions, imaginary and complex numbers, quadratic equations, inequalities, trigonometry, vectors, polar coordinates, conic sections, circles, parabolas, ellipses and hyperbolas.
Prerequisite:  Algebra 1 and Geometry

4.  Multivariable Calculus and Differential Equations

This introductory course will focus on the unifying concept of a dynamical system.  Examples of dynamical systems include things that change over time such as wind, weather, ocean currents, planetary motion, pendulums and mass spring systems.  Topics covered in this two-semester course are first order linear differential equations, vector spaces, vector fields, harmonic oscillator equations, eigenvalues, techniques of Euler and Lagrange for solving second order linear differential equations, partial derivatives, Jacobian matrices, gradient, extrema, higher dimension integration, Fubini's theorem, line integrals, Green's theorem, Stokes' theorem and differential forms.  Students will be using two textbooks for this course.
Prerequisite:  Differential and Integral Calculus

5.  Music Theory and Composition

This course is targeted for musically gifted young people who are interested in delving deeper into aspects of music theory and improving their appreciation and mastery of music.  Ear training is strongly emphasized and students will learn to instantly recognize various intervals, chords and modes.  Each week students will be challenged to create original music pieces based on specific criteria, and will receive positive and constructive feedback from Mr. Rosasco -- a hit songwriter, producer, arranger, keyboardist and film composer with four Grammy nominations to his credit.  Additionally, the class will study composition, melody, structure, harmony, rhythm patterns, chord progressions, and a variety of scales and modes.  Please note that mutual respect for each person's work will set the tone.
Prerequisite:  Some experience playing a musical instrument

6.  Differential and Integral Calculus

Calculus is the mathematical study of change and its two major branches, differential and integral, are related to each other by the fundamental theorem of calculus.  In today's world, calculus has broad applications in science, mathematics, engineering, economics and more.  Topics studied in this course include limits, derivatives, related rates, mean value theorem, optimization, integrals, fundamental theorem of calculus, volume, surface area, infinite sequences and series, power series, and Taylor series.  This two-semester course is the equivalent of taking both AB and BC Calculus.
Prerequisite:  Algebra 2 with Trigonometry or Precalculus

7.  Introduction to Group Theory

This innovative course is uniquely designed for young, gifted students who possess a keen interest in math and desire for discovery.  Finite groups found in abstract algebra are the central focus, and students will use Sodoku-like Cayley tables and diagrams as methods to visually display group structures.  The class will also study designs and patterns, and the mathematics of symmetry which lie beneath.  Topics will include cosets, Lagrange's theorem, isomorphisms, permutation groups, direct sums of groups, and factor groups.
Prerequisite:  A good working knowledge of arithmetic

8.  Group Theory 2

Students will have the opportunity to build on the foundational concepts learned in Introduction to Group Theory in this second year course.  Over the past few decades symmetry groups have been found to be the mathematical basis for particle physics.  The primary areas of study in this course will focus on non-commutative groups, such as dihedral and matrix groups (also known as symmetry groups).  Students will also study Cayley diagrams, normal subgroups, factor groups, permutation groups and quotient groups.
Prerequisite:  A good working knowledge of arithmetic, as well as some exposure to Group Theory