Mathematics

Prerequisites: Mathematics score of 550 on the SAT exam, taken within the past year. Recommended: MATH S0065. Algebra review, graphs and functions, polynomial functions, rational functions, conic sections, systems of equations in two variables, exponential and logarithmic functions, trigonometric functions and trigonometric identities, applications of trigonometry, sequences, series, and limits.

Prerequisites: high school mathematics through trigonometry or MATH S1003, or the equivalent. Functions, limits, derivatives, introduction to integrals.

Prerequisites: high school mathematics through trigonometry or MATH S1003, or the equivalent. Functions, limits, derivatives, introduction to integrals.

Prerequisites: MATH S1101 Calculus I, or the equivalent. Methods of integration, applications of the integral, Taylor's theorem, infinite series.

Prerequisites: MATH S1102, or the equivalent. Columbia College students who aim at an economics major AND have at least the grade of B in Calculus I may take Calculus III directly after Calculus I. However, all students majoring in engineering, science, or mathematics should follow Calculus I with Calculus II. Vectors in dimensions 2 and 3, complex numbers and the complex exponential function with applications to differential equations, Cramer's rule, vector-valued functions of one variable, scalar-valued functions of several variables, partial derivatives, gradients, surfaces, optimization, the method of Lagrange multipliers.

Prerequisites: MATH S1102, or the equivalent. Columbia College students who aim at an economics major AND have at least the grade of B in Calculus I may take Calculus III directly after Calculus I. However, all students majoring in engineering, science, or mathematics should follow Calculus I with Calculus II. Vectors in dimensions 2 and 3, complex numbers and the complex exponential function with applications to differential equations, Cramer's rule, vector-valued functions of one variable, scalar-valued functions of several variables, partial derivatives, gradients, surfaces, optimization, the method of Lagrange multipliers.

Calculus I
Material includes the study of limits, derivatives, and applications of derivatives, as well as an introduction to integrals. Success in this course requires intense commitment to study and collaboration both in and out of class, along with active participation during class, and the ability to follow both written and verbal directions.   
The overarching objective of the course is for students to develop conceptual fluency and computational competence working with limits, derivatives, and their applications.   Students will enhance their quantitative reasoning, particularly through interpreting/analyzing charts and graphs in the study of limits and derivatives.  Problem solving skills will be augmented through applied word problems, particularly relating to optimization, related rates, and graphing.  Writing justifications/explanations for answers on all graded work will expand students' ability to clearly communicate mathematical ideas in writing. While collaborative projects will advance their ability to do so orally and visually.

Calculus I
Material includes the study of limits, derivatives, and applications of derivatives, as well as an introduction to integrals. Success in this course requires intense commitment to study and collaboration both in and out of class, along with active participation during class, and the ability to follow both written and verbal directions.   
The overarching objective of the course is for students to develop conceptual fluency and computational competence working with limits, derivatives, and their applications.   Students will enhance their quantitative reasoning, particularly through interpreting/analyzing charts and graphs in the study of limits and derivatives.  Problem solving skills will be augmented through applied word problems, particularly relating to optimization, related rates, and graphing.  Writing justifications/explanations for answers on all graded work will expand students' ability to clearly communicate mathematical ideas in writing. While collaborative projects will advance their ability to do so orally and visually.

Prerequisite for Calc III is Calc I.  
For exam placement, see dept site: https://math.barnard.edu/placement-calculus-classes

• To develop a conceptual understanding of the fundamental principles of multivariable differential calculus and explore their practical relevance to other branches of natural and social sciences.
• Students will enhance their critical thinking by using 3D graphical tools and engaging with three dimensional representations of curves, surfaces and solids.
• Writing justifications/explanations for answers on all graded work will expand students' ability to clearly communicate mathematical ideas in writing. 
• Problem solving skills will be augmented through applications in physics and real world problems involving the dynamics of movement, and optimization problems involving multiple inputs (parameters)

• The sequence of topics covered is as follows.

  • ​​​​1.  Vectors. Equations of lines and planes. Cylinders and quadric surfaces. 

  • 2. Vector functions. Space curves. Arc length and curvature. Applications.

  • 3. Functions of several variables. Partial derivatives. Gradient. Optimization. Lagrange multipliers.

  •  

Prerequisite for Calc III is Calc I.  
For exam placement, see dept site: https://math.barnard.edu/placement-calculus-classes

• To develop a conceptual understanding of the fundamental principles of multivariable differential calculus and explore their practical relevance to other branches of natural and social sciences.
• Students will enhance their critical thinking by using 3D graphical tools and engaging with three dimensional representations of curves, surfaces and solids.
• Writing justifications/explanations for answers on all graded work will expand students' ability to clearly communicate mathematical ideas in writing. 
• Problem solving skills will be augmented through applications in physics and real world problems involving the dynamics of movement, and optimization problems involving multiple inputs (parameters)

• The sequence of topics covered is as follows.

  • ​​​​1.  Vectors. Equations of lines and planes. Cylinders and quadric surfaces. 

  • 2. Vector functions. Space curves. Arc length and curvature. Applications.

  • 3. Functions of several variables. Partial derivatives. Gradient. Optimization. Lagrange multipliers.

  •  

Prerequisites: MATH S1201 Calculus III, or the equivalent. Matrices, vector spaces, linear transformation, Eigenvalues and Eigenvectors, canonical forms, applications.

Prerequisites: MATH S1201 Calculus III, or the equivalent. Matrices, vector spaces, linear transformation, Eigenvalues and Eigenvectors, canonical forms, applications.

Linear algebra with a focus on probability and statistics. The course covers the standard linear algebra topics: systems of linear equations, matrices, determinants, vector spaces, bases, dimension, eigenvalues and eigenvectors, the Spectral Theorem and singular value decompositions. It also teaches applications of linear algebra to probability, statistics and dynamical systems giving a background sufficient for higher level courses in probability and statistics. The topics covered in the probability theory part include conditional probability, discrete and continuous random variables, probability distributions and the limit theorems, as well as Markov chains, curve fitting, regression, and pattern analysis. The course contains applications to life sciences, chemistry, and environmental life sciences. No a priori background in the life sciences is assumed.

This course is best suited for students who wish to focus on applications and practical approaches to problem solving. It is recommended to students majoring in engineering, technology, life sciences, social sciences, and economics.

Math majors, joint majors, and math concentrators must take MATH UN2010 Linear Algebra or MATH UN1207 Honors Math A, which focus on linear algebra concepts and foundations that are needed for upper-level math courses. MATH UN2015 (Linear Algebra and Probability) does NOT replace MATH UN2010 (Linear Algebra) as prerequisite requirements of math courses. Students may not receive full credit for both courses MATH UN2010 and MATH UN2015. Students who have taken MATH UN2015 and consider taking higher level Math courses should contact a major advisor to discuss alternative pathways.

Prerequisites: MATH UN1102 and MATH UN1201 or the equivalent. Special differential equations of order one. Linear differential equations with constant and variable coefficients. Systems of such equations. Transform and series solution techniques. Emphasis on applications.

Prerequisites: MATH UN1101 (Calculus I) or equivalent courses. This course introduces students to mathematical modeling through hands-on, project-based learning. Topics include fundamental concepts from linear algebra, multivariable calculus, differential equations, probability and statistics, and introductory machine learning.

Prerequisites: MATH V1102-MATH V1201 or the equivalent and MATH V2010. Mathematical methods for economics. Quadratic forms, Hessian, implicit functions. Convex sets, convex functions. Optimization, constrained optimization, Kuhn-Tucker conditions. Elements of the calculus of variations and optimal control.

Prerequisites: MATH UN1102 and MATH UN1202 and MATH UN2010 or the equivalent. The second term of this course may not be taken without the first. Groups, homomorphisms, normal subgroups, the isomorphism theorems, symmetric groups, group actions, the Sylow theorems, finitely generated abelian groups.

Prerequisites: MATH S1202, MATH S2010, or the equivalent. Students must have a current and solid background in the prerequisites for the course: multivariable calculus and linear algebra. Elements of set theory and general topology. Metric spaces. Euclidian space. Continuous and differentiable functions. Riemann integral. Uniform convergence.

Prerequisites: all 6 MAFN core courses, at least 6 credits of approved electives, and the instructors permission. See the MAFN website for details. This course provides an opportunity for MAFN students to engage in off-campus internships for academic credit that counts towards the degree. Graded by letter grade. Students need to secure an internship and get it approved by the instructor.