Introduction to basic ideas and fundamental laws of probability. Topics include an introduction to combinatorics, probability axioms, conditional probability, discrete random variables, common discrete distributions, expectation, generating functions, limit theorems, and continuous random variables.

1. Probability Spaces and Events: Students learn about probability spaces, which consist of a sample space (set of all possible outcomes) and a probability measure that assigns probabilities to subsets of the sample space (events). They study basic properties of probabilities, such as non-negativity, additivity, and normalization. They also learn about events, which are subsets of the sample space representing outcomes of interest, and operations on events, such as union, intersection, complement, and independence.

2. Random Variables: Random variables are variables that take on numerical values determined by the outcome of a random experiment. Students learn about discrete random variables, which have countable outcomes, and continuous random variables, which have uncountable outcomes. They study probability mass functions (PMFs) and probability density functions (PDFs) of random variables, as well as expected values, variances, and moments.

3. Probability Distributions: Probability distributions describe the possible outcomes and associated probabilities of random variables. Students learn about common probability distributions, including discrete distributions such as the Bernoulli, binomial, geometric, and Poisson distributions, and continuous distributions such as the uniform, normal (Gaussian), exponential, and gamma distributions. They study properties of these distributions, such as mean, variance, skewness, and kurtosis.

4. Joint and Conditional Probability: Students learn about joint probability distributions, which describe the probabilities of combinations of events or random variables. They study conditional probability, which measures the probability of an event given that another event has occurred. They learn about conditional probability distributions, conditional expectations, and conditional independence. They also study Bayes' theorem and its applications in updating probabilities based on new information.

5. Limit Theorems and Law of Large Numbers: Students learn about limit theorems in probability theory, such as the law of large numbers and the central limit theorem. These theorems describe the behavior of sample averages and sums of random variables as the sample size grows large. They study convergence in probability and convergence in distribution, as well as applications of limit theorems in statistics and probabilistic modeling.

Multivariable functions and vector functions are studied as the concepts of differential and integral calculus are generalized to surfaces and higher dimensions. Topics include vectors, parametric equations, cylindrical and spherical coordinates, partial and directional derivatives, multiple integrals, line and surface integrals, and the theorems of Green, Gauss, and Stokes.

1. Vector Functions and Geometry in Space: Students learn about vector-valued functions, which map real numbers to vectors in space. They study vector operations, such as addition, subtraction, scalar multiplication, and differentiation. They also explore parametric equations of curves and surfaces in space, including lines, planes, and curves defined by vector-valued functions.

2. Partial Derivatives: Partial derivatives generalize the concept of derivatives to functions of several variables. Students learn how to compute partial derivatives of functions with respect to each of their variables and interpret them geometrically as slopes of tangent planes. They also study higher-order partial derivatives, mixed partial derivatives, and Clairaut's theorem.

3. Gradient, Divergence, and Curl: Students learn about vector calculus operations, including the gradient, divergence, and curl. They study how these operations characterize properties of vector fields, such as direction, magnitude, and rotational behavior. They also explore applications of these concepts in physics, such as gradient as a directional derivative and divergence and curl in fluid flow and electromagnetism.

4. Multiple Integrals: Multiple integrals generalize the concept of integration to functions of several variables. Students learn about double and triple integrals over rectangular and non-rectangular regions in space. They study techniques for evaluating multiple integrals, such as iterated integrals, change of variables, and polar, cylindrical, and spherical coordinates. They also explore applications of multiple integrals in volume, mass, and center of mass calculations.

5. Vector Calculus Theorems: Students learn about fundamental theorems in vector calculus, such as the divergence theorem and Stokes' theorem. They study how these theorems relate different aspects of vector fields, such as flux, circulation, and surface and line integrals. They also explore applications of these theorems in physics, such as Gauss's law and the circulation of a vector field around a closed curve.

An introduction to using linear algebra techniques and tools to solve and extract data from systems. Topics include: Systems of Linear Equations, Eigenvalues and Eigenvectors and Singular Value Decomposition (SVD), Abstract vector spaces, Matrix invariants, Computer Algebra Systems (CAS).

1. Vector Spaces: Vector spaces are sets of vectors that satisfy certain properties, such as closure under vector addition and scalar multiplication. Students learn about the properties of vector spaces, including vector addition, scalar multiplication, vector space axioms, subspaces, and linear combinations.

2. Matrices and Matrix Operations: Matrices are rectangular arrays of numbers arranged in rows and columns. Students learn about matrix operations, including matrix addition, scalar multiplication, matrix multiplication, and matrix transposition. They study properties of matrices, such as symmetry, skew-symmetry, diagonalizability, and invertibility.

3. Linear Transformations: Linear transformations are functions that map vectors from one vector space to another while preserving certain properties, such as linearity and the origin. Students learn about linear transformations, their properties, and their representations as matrices. They study concepts such as kernel (null space), image (range), rank, and nullity of linear transformations.

4. Eigenvalues and Eigenvectors: Eigenvalues and eigenvectors are key concepts in linear algebra that arise in the study of linear transformations and matrices. Students learn how to find eigenvalues and eigenvectors of matrices, diagonalize matrices using eigenvectors, and apply eigenvalues and eigenvectors to solve systems of linear differential equations and difference equations.

5. Inner Product Spaces and Orthogonality: Inner product spaces are vector spaces equipped with an inner product, which is a generalization of the dot product. Students learn about inner product spaces, orthogonality, orthogonal bases, orthogonal projections, and the Gram-Schmidt process. They study applications of inner product spaces in geometry, least squares approximation, and orthogonalization.

Differential and integral calculus of functions of a single variable. Two semesters in sequence. 

1. **Techniques of Integration**: Students expand their toolkit of integration techniques beyond those introduced in Calculus I. They learn about methods such as integration by parts, trigonometric substitution, partial fractions, and improper integrals. These techniques are used to evaluate a wider range of integrals, including those that cannot be solved using basic methods.

2. **Applications of Integration**: Building on the applications introduced in Calculus I, students explore additional applications of integration. This may include finding the area between curves, volumes of solids with known cross-sections, arc length, surface area of revolution, and physical applications such as work, fluid pressure, and center of mass.

3. **Sequences and Series**: Students learn about sequences, which are ordered lists of numbers, and series, which are sums of the terms of a sequence. They study convergence and divergence of sequences and series, as well as tests for convergence such as the comparison test, integral test, ratio test, and root test. They also learn about power series and Taylor series expansions of functions.

4. **Parametric and Polar Curves**: Students explore parametric equations, which represent curves using separate equations for the \(x\) and \(y\) coordinates as functions of a third variable, typically \(t\). They study how to sketch parametric curves, find tangent lines, and calculate arc length. They also learn about polar coordinates and equations, polar curves, and their conversion to rectangular coordinates.

5. **Vectors and Vector-Valued Functions**: Students learn about vectors and vector operations such as addition, subtraction, scalar multiplication, dot product, and cross product. They study vector-valued functions, which map real numbers to vectors, and explore their properties, calculus of vector-valued functions, arc length, and curvature.

This course will give students a rigorous preparation in algebra. Topics include review of basic algebraic concepts, functions and graphs, polynomial, radical, rational, exponential and logarithmic functions; equations, inequalities, systems of equations and inequalities; applications. 

1. **Linear Equations and Inequalities**: Students learn how to solve linear equations and inequalities, including equations with one variable and systems of linear equations with multiple variables. They study techniques such as substitution, elimination, and graphing to find solutions to equations and inequalities.

2. **Quadratic Equations**: Quadratic equations are equations of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(x\) is the variable. Students learn how to solve quadratic equations using methods such as factoring, completing the square, and the quadratic formula. They also study properties of quadratic functions and their graphs.

3. **Polynomial and Rational Functions**: Students learn about polynomial functions, including their properties, operations (addition, subtraction, multiplication, and division), and graphing techniques. They also study rational functions, which are ratios of polynomial functions, and learn how to find their domains, asymptotes, and intercepts.

4. **Exponential and Logarithmic Functions**: Students learn about exponential functions, which have the form \(f(x) = a^x\), where \(a\) is a constant base and \(x\) is the exponent. They study properties of exponential functions, such as growth and decay, and learn how to graph them. They also learn about logarithmic functions, which are the inverses of exponential functions, and study properties of logarithms, logarithmic equations, and their applications.

5. **Conic Sections**: Students learn about conic sections, which are curves obtained by intersecting a cone with a plane. They study the equations and properties of the four basic types of conic sections: circles, ellipses, parabolas, and hyperbolas. They learn how to graph conic sections and identify their key features, such as centers, vertices, foci, and asymptotes.