- Congruences and Modular Arithmetic a is congruent to b mod n means that n | a -b. Notation: a = b (mod n).
- Properties of Limits There are many rules for computing limits. I'll list the most important ones. There are analogous
- The Comparison and Limit Comparison Tests You can often tell that a series converges or diverges by comparing it to a known series. I'll look first
- Inverse Functions Functions f : X Y and g : Y X are inverses if
- Integration by Parts If u and v are functions of x, the Product Rule says that
- Set Constructions The set constructions I've considered so far --things like A B, C, D E --have involved finite
- The Chinese Remainder Theorem The Chinese Remainder Theorem gives solutions to systems of congruences with relatively prime
- Math 211-04 Review Problems for Test 3
- Review Problems for Test 3 These problems are meant to help you study. The presence of a problem on this sheet does not imply
- Calculus provides information which is useful in graphing curves. The first derivative y
- The Tietze Extension Theorem The Tietze Extension Theorem says that a continuous realvalued function on a closed subset of
- Parametrizing Surfaces 1. What does it mean to parametrize a surface?
- Step Functions The unit step function or Heaviside function is given by
- Tank Problems Here are some additional problems involving flow in and out of a container. These kinds of problems
- Complex Numbers A complex number is a number of the form a + bi, where a and b are real numbers and i =
- Review Problems for Test 2 These problems are provided to help you study. The presence of a problem on this sheet does not imply
- Equations Which Are Quadratic in Form Sometimes an equation is not quadratic as is, but becomes quadratic if you make a substitution. Then
- Factoring Polynomials The opposite of multiplying polynomials is factoring. Why would you want to factor a polynomial?
- A Review of Elementary Solution Methods Here is a list of the kinds of equations I've discussed so far
- Review: Series Solutions Example. Find a series solution about x = 0 for
- Infinite Series An infinite series is a sum
- Fractions and Rational Numbers A rational number is a real number which can be represented as a quotient of two integers.
- Rational Expressions A rational expression is something of the form
- Trig Functions of Special Angles Example. Find the sine, cosine, and tangent of 315 .
- Word Problems Involving Fractions Despite the name, not all of these word problems must be solved using fractions. They can be, but
- Review Problems for Test 1 These problems are meant to help you study. The presence of a problem on this sheet does not imply
- Exponentiation Ciphers and Public Key Cryptography An exponentiation cipher encodes messages using C = Ae
- The Natural Logarithm Let a be a positive number, a = 1, and let x > 0. The logarithm of x to the base a is the number
- Linear Equations in One Variable A linear equation in one variable is an equation involving constants and a single variable which
- Word Problems These word problems involve percents, averages, and peoples' ages. Solving them requires that you set
- You often have to solve a formula --an equation --for a variable. The formula may come from mathematics, but it can also come from another area (such as business, economics, or science). The following
- Absolute Value Equations and Inequalities The absolute value of a number is defined by
- The slope of the line which passes through the points (x1, y1) and (x2, y2) is The slope measures the rate at which a line goes up or down as you move to the right. For example, a
- Polynomials A polynomial (in one variable) is something of the form
- Cartesian Coordinates and Graphing An ordered pair (x, y) of real numbers can be represented by a point in the plane. Construct a pair
- Direct and Inverse Variation y varies directly with x (or: "x and y are directly proportional") if there is a constant k such that
- The Quadratic Formula The quadratic formula is a formual for finding the roots of a quadratic equation. It depends on a
- Review Sheet for Test 1 These problems are provided to help you study. The presence of a problem on this handout does not
- Review Sheet for Test 2 These problems are provided to help you study. The presence of a problem on this handout does not
- Math 101-08 Review Sheet for Test 3
- Review Problems for Test 2 These problems are meant to help you study. The presence of a problem on this sheet does not imply
- Math 161-03/06 1010/112007
- Review Problems for Test 1 These problems are provided to help you study. The presence of a problem on this sheet does not imply
- Math 261-00 Review Sheet for Test 1
- Limits: An Introduction Calculus was used long before it was established on firm mathematical foundations. Limits provide a
- The Limit Definition Having discussed how you can compute limits, I want to examine the definition of a limit in more detail.
- In many cases, you can compute lim f(x) by plugging a in for x
- Differentiation Rules While it's possible to compute derivatives using the definition of the derivative as a limit, it is too much
- Limits and Derivatives of Trig Functions If you graph y = sin x and y = x, you see that the graphs become almost indistinguishable near x = 0
- The Chain Rule The Chain Rule computes the derivative of the composite of two functions. The composite (f g)(x)
- Related Rates Related rates problems deal with situations in which several things are changing at rates which are
- Differentials Calculus is over three hundred years old, but the modern approach via limits only dates to the early
- Concavity and the Second Derivative Test If y = f(x) is a function, the second derivative of y (or of f) is the derivative of the first derivative.
- Definite Integrals The area under a curve can be approximated by adding up the areas of rectangles.
- Finding the Area Between Curves How do you find the area of a region bounded by two curves? I'll consider two cases.
- The Natural Logarithm The Power Rule says
- Inverse Functions Functions f : X Y and g : Y X are inverses if
- Inverse Trig Functions If you restrict f(x) = sinx to the interval -
- Partial Fractions Partial fractions is the opposite of adding fractions over a common denominator. It applies to integrals
- Miscellaneous Substitutions When an integral contains a quadratic expression ax2
- Review of Integration Techniques Example. Compute
- L'H^opital's Rule L'H^opital's Rule is a method for computing a limit of the form
- Finding the Area Between Curves How do you find the area of a region bounded by two curves? I'll consider two cases.
- Volumes of Revolution by Slicing Start with an area --a planar region --which you can imagine as a piece of cardboard. The cardboard
- The work required to raise a weight of P pounds a distance of y feet is P y foot-pounds. (In m-k-s units, one would say that a force of k newtons exerted over a distance of y feet does k y newton-meters, or
- The Ratio Test and the Root Test It is a fact that an increasing sequence of real numbers that is bounded above must converge.
- Review: Convergence Tests for Infinite Series When you are testing a series for convergence or divergence, it's helpful to run through your list of
- Alternating Series If a series has only positive terms, the partial sums get larger and larger. If they get large too rapidly,
- Convergence of Power Series A power series is an infinite series
- Constructing Taylor Series The Taylor series for f(x) at x = c is
- Logical Connectives Mathematics works according to the laws of logic, which specify how to make valid deductions. In order
- Quantifiers Here is a (true) statement about real numbers
- Direct Proofs A direct proof uses the facts of mathematics and the rules of inference to draw a conclusion. Since
- Conditional Proofs A conditional proof uses assumptions which go beyond the basic facts of mathematics. The conclusion
- Existence Proofs An existence proof shows that an object exists. In some cases, this means displaying the object, or
- Proof by Contradiction To prove a statement P by contradiction, you assume the negation P of what you want to prove
- Induction is used to prove a sequence of statements P(1), P(2), P(3), . . . . There may be finitely many statements, but often there are infinitely many.
- Counterexamples A counterexample is an example that disproves a universal ("for all") statement. Obtaining coun-
- Limits at Infinity In this section, I'll discuss proofs for limits of the form lim
- Binary Relations Definition. A binary relation on a set S is a subset of the Cartesian product S S.
- Divisibility Definition. If a and b are integers with a = 0, then a divides b if an = b for some integer n. a | b means "a
- Determinants -Uniqueness and Properties In order to show that there's only one determinant function on M(n, R), I'm going to derive another
- Vector Spaces Vector spaces and linear transformations are the primary objects of study in linear algebra. A
- Linear Independence Definition. Let V be a vector space over a field F, and let S V . S is linearly independent if
- Representing Linear Transformations by Matrices Let f : V W be a linear transformation of finite dimensional vector spaces. Choose ordered bases
- Eigenvalues and Eigenvectors Definition. Let A Mn(F). The characteristic polynomial of A is
- Linear Systems with Constant Coefficients Here is a system of n differential equations in n unknowns
- Inner Product Spaces Definition. Let V be a vector space over F, where F = R or C. An inner product on V is a function
- Fourier Series Imagine a thin piece of wire, which only gains or loses heat through its ends. The temperature u(x, t)
- Matrix Operations with Mathematica In the symbolic math program Mathematica, a matrix is a list whose entries are the matrix's rows. Each
- Mathematical Induction Induction is a method for proving an infinite sequence of statements. In its easiest form, you start by
- Divisibility a | b means that a divides b --that is, b is a multiple of a.
- Greatest Common Divisors The greatest common divisor (m, n) of integer m and n is the largest integer which divides both m
- Linear Diophantine Equations A Diophantine equation is an equation which is to be solved over the integers.
- Solving Linear Congruences A linear congruence ax = b (mod m) has solutions if and only if (a, m) | b.
- Wilson's Theorem and Fermat's Theorem Wilson's theorem says that p is prime if and only if (p -1)! = -1 (mod p).
- Euler's Theorem If n is a positive integer, (n) is the number of integers in the range {1, . . . , n} which are relatively
- Perfect Numbers Definition. A number n > 0 is perfect if (n) = 2n. Equivalently, n is perfect if it is equal to the sum of
- Substitution Ciphers and Block Ciphers A cipher takes a plaintext and encodes it to produce a ciphertext.
- Quadratic Reciprocity Quadratic reciprocity relates solutions to x2
- Continued Fractions A finite continued fraction is an expression of the form
- Infinite Continued Fractions The value of an infinite continued fraction [a0; a1, a2, . . .] is
- Periodic Continued Fractions A quadratic irrational is an irrational number which is a root of a quadratic equation with integer
- Metric Spaces Definition. If X is a set, a metric on X is a function d : X X R such that
- Word Problems: Systems of Equations In this section, I'll look at word problems which give rise to systems of linear equations.
- Row Reduction Row reduction (or Gauss-Jordan elimination) is the process of using row operations to reduce a
- Word Problems Involving Quadratics These word problems involve situations I've discussed in other word problems: The area of a rectangle,
- Properties of Matrix Arithmetic I've given examples which illustrate how you can do arithmetic with matrices. Now I'll give precise
- Logistic Growth The logistic equation is a model of limited population growth. The exponential growth model
- The Fundamental Theorem of Calculus The Fundamental Theorem of Calculus says, roughly, that the following processes undo each other
- Review: Set Theory, Functions, and Relations This is a brief review of material on set theory, functions, and relations. I'll concentrate on things which
- Urysohn's Lemma Urysohn's lemma is often expressed by saying that disjoint closed sets in a normal space can be
- Partial Derivatives The derivative of a function y = f(x) tells us about the rate of change of f . If z = f(x; y) is a function
- Absolute Convergence and Conditional Convergence The convergence tests I've discussed (such as the Ratio Test and Limit Comparsion) apply to positive
- Bernoulli Equations and Riccati Equations An equation of the form
- Homeomorphisms Definition. Let X and Y be topological spaces, and let f : X ! Y . f is a homeomorphism if f is bijective,
- Space Curves You can think of a function ~oe : R! R 3 as a ``machine'' which takes the real line (R), bends and twists
- The Cross Product I'll de ne the cross product algebraically and geometrically.
- Commutative Rings and Fields Different algebraic systems are used in linear algebra. The most important are commutative rings
- Limits at Infinity Here is the graph of f(x) =
- Radians and Degrees An angle consists of two rays emanating from a point
- PathConnectedness and Local Connectedness Definition. Let X be a topological space. A path in X is a continuous function oe : [0; 1] ! X. oe(0) and
- Unitary Matrices and Hermitian Matrices Recall that the conjugate of a complex number a + bi is a -bi. The conjugate of a + bi is denoted
- Quadratic Inequalities In this section, I'll consider quadratic inequalities. I'll solve them using the graph of the quadratic
- Order Relations A partial order on a set is, roughly speaking, a relation that behaves like the relation on R.
- Review Problems on Second Order Equations Example. Solve the equation
- Maxima and Minima for Functions of Two Variables For a function of one variable y = f(x), you look for local maxima and minima at critical points |
- Variation of Parameters Variation of parameters constructs the general solution to a nonhomgeneous equation
- Homogeneous equations A function f(x; y) is homogeneous of degree n in x and y if
- Nonhomogeneous Equations Reduction of Order The solution of a nonhomogeneous secondorder linear equation
- Determinants A matrix is a rectangular array of numbers
- Constant Coefficient Equations: Real and Repeated Roots The full description of these equations is: Linear constant coefficient homogeneous equations. The
- Linear Independence and the Wronskian A set of vectors fv 1 ; : : : ; v n g is independent if
- Trigonometric Functions Take an angle in standard position and draw a circle of radius r centered at the origin. Construct a
- August 15, 2003 11023. Proposed by Wu Wei Chao, Guang Zhou University (New), Guang Dong Province, China. Find all
- Math 161-03/06 919/202007
- Antiderivatives F(x) is an antiderivative of f(x) if
- Series Solutions at Regular Singular Points Consider a secondorder equation
- Connected Subsets of the Real Line Proposition. [0; 1] is connected in the standard topology.
- The Convolution Formula The convolution of f and g is
- Applications of Trig Functions Example. A pepperoni stromboli standing on end subtends an angle of 30 at a distance of 450 feet. How
- Compact Spaces In the real line, compactness is equivalent to being closed and bounded. The HeineBorel Theorem says
- Improper Integrals Roughly speaking, an integral
- Coordinate Transformations A coordinate transformation of the plane is a function f : R2
- Math 261-00 Review Sheet for Test 2
- Trig Substitution Trig substitution reduces certain integrals to integrals of trig functions. The idea is to match the
- Counting and Cardinality The cardinality of a set is roughly the number of elements in a set. This poses few difficulties with
- Hausdorff Spaces If you try to study topological spaces without any conditions, the variety becomes overwhelming ---
- The Countability Axioms Definition. Let X be a topological space, and let x 2 X.
- Double Integrals in Polar. It's often useful to change variables and convert a double integral from rectangular coordinates to polar
- Topological Spaces A function f : R! R is continuous at x = a if and only if
- The Dot Product If #v = #v 1 , v 2 , v 3 # and #
- Solving Right Triangles You solve a right triangle by determining the lengths of the sides and the measures of the angles.
- Divisor Functions Definition. The sum of divisors function is given by
- Determinants -Existence Determinants are functions which take matrices as inputs and produce numbers. They are of enormous
- Linear Inequalities Solving linear inequalities is very similar to solving equations; the only difference is that = is replaced
- Trigonometric Integrals For trig integrals involving powers of sines and cosines, there are two important cases
- Set Algebra and Proofs Involving Sets There are a lot of rules involving sets; you'll probably become familiar with the most important ones
- Lines and Planes Many results from single-variable calculus extend to objects in \higher dimensions". Since lines and
- A Review of Taylor Series A Taylor series about x 0 is an infinite series
- Sums, Products, and Binomial Coefficients Summation notation
- Summation Notation Summation notation is used to denote a sum of terms. Usually, the terms follow a pattern or formula.
- Bases for Vector Spaces Linear independence has to do with the "nonredundancy" of a set of vectors. On the other hand,
- The Fundamental Theorem of Arithmetic The Fundamental Theorem of Arithmetic says that every integer greater than 1 can be factored
- Undetermined Coefficients A nonhomogeneous linear equation with constant coefficients is an equation of the form
- The Extended Euclidean Algorithm The Extended Euclidean Algorithm finds integers a and b such that (m, n) = am + bn.
- Quadratic Residues a is a quadratic residue mod m if x2
- Roughly speaking, a set is a collection of objects. The objects are called the members or the elements of the set.
- The Product Topology If X and Y are topological spaces, the product topology on X \Theta Y is the topology generated by the
- Angular Measure -Applications An angle of radian measure cuts out an arc of length L = r from a circle of radius r
- Quotient Spaces You may have encountered quotient groups or quotient rings in an algebra course. In those situa
- Equivalence Relations and Partitions First, I'll recall the definition of an equivalence relation on a set X.
- The Mean Value Theorem A secant line is a line drawn through two points on a curve.
- Rules of Inference and Logic Proofs A proof is an argument from hypotheses (assumptions) to a conclusion. Each step of the argument
- Absolute Maxima and Minima I'll begin with a couple of examples to illustrate the kinds of problems I want to solve.
- Row Space, Column Space, and Null Space Definition. Let A be an m n matrix.
- Compact Sets and the Real Numbers Theorem. If X is an ordered set satisfying the least upper bound property, then any closed interval [a; b] in
- Interchanging the Order of Integration An iterated integral ZZ
- Radical Equations In this section, I'll discuss how you solve equations involving square roots of variable expressions.
- Substitution You can use substitution to convert a complicated integral into a simpler one. In these problems, I'll
- The Spectral Theorem Theorem. (Schur) If A is an n n matrix, then there is a unitary matrix U such that UAU-1
- Regular Singular Points I'm considering secondorder linear differential equations of the form
- Review Problems on Laplace Transforms Example. Compute L [(t + 1)u 2 (t)].
- Lagrange Multipliers The method of Lagrange multipliers deals with the problem of nding the maxima and minima of a
- Inequality Proofs I'm going to take a functional approach to inequality proofs: Rather than discuss inequalities from
- Tangent Lines and Rates of Change Given a function y = f(x), how do you find the slope of the tangent line to the graph at the point
- Absolute Value and Inequalities | -47| = 47, while |150| = 150 and |0| = 0.
- Fractional Exponents If n is a positive integer, then
- Separation of variables In some cases, you can solve a differential equation
- Closed Sets and Limit Points Definition. If X is a topological space, a subset A ae X is closed if X \Gamma A is open.
- The Separation Axioms Definition. Let X be a space in which singletons are closed.
- Quadratic Functions A quadratic function is a function of the form
- Rational Approximation by Continued Fractions The convergents of a continued fraction expansion of x give the best rational approximations to x.
- The Ring of Integers The integers Z satisfy the axioms for an algebraic structure called an integral domain.
- Equations Involving Fractions By an equation with fractions, I'll mean an equation to solve in which the variable appears in the
- The Exponential Function If a is a positive number and a = 1, the exponential function with base a is
- Definition. A function f from a set X to a set Y is a subset S of the product X Y such that if (x, y1), (x, y2) S, then y1 = y2.
- Error Estimation If the Taylor series for a function f(x) is truncated at the n-th term, what is the difference between
- Trig Functions as Circular Functions You're familiar with functions such as f(x) = x 2 and y = 2x + 5. You've also seen that there are
- Divisibility Tests and Factoring There are simple tests for divisibility by small numbers such as 2, 3, 5, 7, and 9. These tests involve
- Coordinates and Change of Basis Let V be a vector space and let B be a basis for V . Every vector v V can be uniquely expressed as a
- Congruences and Modular Arithmetic a is congruent to b mod n means that n | a -b. Notation: a = b (mod n).
- Linear Transformations Definition. Let V and W be vector spaces over a field F. A linear transformation f : V W is a
- Inverses and Elementary Matrices Matrix inversion gives a method for solving some systems of equations. Suppose
- Integer Exponents I'll discuss some rules for working with integer exponents. Actually, these rules work with arbitrary
- ALMOST CYCLIC GROUPS BRUCE IKENAGA
- Laplace Transforms An integral transform is an operator
- The Divergence Theorem The Divergence Theorem relates flux of a vector field through the boundary of a region to a triple
- First-order linear equations An equation of the form
- Exact Equations and Integrating Factors An equation
- I'll look at vectors from an algebraic point of view and a geometric point of view. Algebraically, a vector is a list of numbers. Here are some 2dimensional vectors
- Limit Point Compactness and Sequential Compactness Definition. Let X be a topological space.
- Inverse Transforms and Initial Value Problems Laplace transforms can be used to solve initial value problems. To do this, I'll need to formula for the
- Local Compactness Definition. Let X be a topological space. X is locally compact if for all x 2 X, there is a compact set C
- Most of linear algebra involves mathematical objects called matrices. A matrix is a finite rectangular array of numbers
- Arithmetic Functions and the Euler Phi Function An arithmetic function takes positive integers as inputs and produces real or complex numbers as
- Euler Equations An equation of the form
- Connected Spaces Definition. A space is connected if it cannot be written as the disjoint union of two nonempty open sets.
- Math 261-00 Review Problems for Test 3
- Linear Homogeneous Differential Equations The full description of these equations is: Linear constant coefficient homogeneous equations. The
- Proof by Cases You can sometimes prove a statement by
- An infinite sequence is a list of numbers. The following examples should make the idea clear. Example. Here is a familiar sequence
- Prime Numbers A prime number is an integer n > 1 whose only positive divisors are 1 and n. An integer greater than
- Systems of Congruences Systems of linear congruences relative to prime moduli can be solved by methods from linear algebra
- Implicit Differentiation Example. The Folium of Descrates is given by the equation x3
- Left and Right-Hand Limits In some cases, you let x approach the number a from the left or the right, rather than "both sides at
- Distance and Velocity If an object moves in a straight line at a constant speed, the distance it travels is
- The existence and uniqueness theorem for firstorder equations Consider a firstorder equation
- Continuous Functions For a function f : R! R, f is continuous at x = a if for every ffl ? 0, there is a ffi ? 0 such that if
- Differential equations Introduction A differential equation is an equation involving a variable and its derivatives with respect to one
- Series Solutions Ordinary Points Consider a secondorder differential equation
- Functions and Graphs A function is a rule which assigns a unique output to each input.
- Increasing and Decreasing Functions A function f increases on a interval if f(a) < f(b) whenever a < b and a and b are points in the
- Counterexamples A counterexample is an example that disproves a universal ("for all") statement. Obtaining coun-
- Miscellaneous Substitutions When an integral contains a quadratic expression ax2
- Eigenvalues and Eigenvectors Definition. Let A Mn(F). The characteristic polynomial of A is
- The Cayley-Hamilton Theorem Terminology. A linear transformation T from a vector space V to itself (i.e. T : V V ) is called a linear
- Integration by Parts If u and v are functions of x, the Product Rule says that
- Equations Which Are Quadratic in Form Sometimes an equation is not quadratic as is, but becomes quadratic if you make a substitution. Then
- Absolute Value Equations The absolute value of a number is defined by
- Solving Absolute Value Inequalities This is a method for solving inequalities like