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5   Introduction to Number Theory

5.1 Divisors

5.2 Representations of Integers and Integer Algorithms

5.3 The Euclidean Algorithm

5.4 The RSA Public-Key Cryptosystem (Skipped)

The part of mathematics devoted to the study of the set of integers and their properties is known as number theory. In this chapter we will develop some of the important concepts of number theory including many of those used in computer science. As we develop number theory, we will use the proof methods developed in [Chapter 2 Proofs] to prove many theorems.

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​(DM Ch. 1, Sets and Logic, Lecture Note)  http://matrix.skku.ac.kr/2018-DM/Ch-1/
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(DM Ch. 1, 동영상강의)
Discrete Math 이산수학 Ch 0 Introduction
https://youtu.be/9ahFnOFTWNQ
Discrete Math 이산수학 Ch 1, 1.1, 1.2 Propositions
https://youtu.be/QgdKqmCFW2Y
Discrete Math 이산수학 Ch 1, 1.3, 1.4 Rules of Inference
https://youtu.be/92siPfThf0M
Discrete Math 이산수학 Ch 1, 1.5, 1.6 Nested Quantifiers
https://youtu.be/7M7w9eX5D0Q

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...

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10 Network Models (if time permits)  http://matrix.skku.ac.kr/2018-DM/Ch-10/
11 Boolean Algebras and Combinatorial Circuits (if time permits)
12 Automata, Grammars, and Languages (if time permits)
13 Computational Geometry (if time permits)
Appendix

DM  Lecture Note & Lab & 동영상강의

******

5   Introduction to Number Theory

5.1 Divisors

5.2 Representations of Integers and Integer Algorithms

5.3 The Euclidean Algorithm

5.4 The RSA Public-Key Cryptosystem (Skipped)

5.1 Divisors

 Definition  1.1 If with .    We say divides      if  ,        ( )  Then we say  is a divisor (or factor) of ,  is a quotient.

divides       is divisor (or factor) of .

does not divide        .

 Example  1.2 Since        divides        .  The quotient is .  We call a divisor of .

Note: , and        .

where

 Theorem 1.3 Let , and are integers.   (a) If and        .   (b) If and        .   (c) If        .

Proof

Proof

(a) By Definition 1.1,

,       .

,        .

.

divides with quotient .

(b) By Definition 1.1,

,       .

,        .

.

divides with quotient .

(c) By Definition 1.1,

,       .

.

divides with quotient .

 Definition  1.4 An integer is prime if is only positive divisors are itself and 1. An integer is called composite  if is not prime.

 Example  1.5 The integer is prime because its only divisors are itself and   The integer is composite  because it is divisible by which is neither nor .

 Example  1.6 Only divisors of are itself and       is prime.  But and divide       is composite.

 Theorem  1.7 A positive integer is composite has a divisor satisfying .

 Algorithm  1.8 Testing Whether an Integer Is Prime This algorithm determines whether the integer is prime. If is prime, the algorithm returns . If is composite, the algorithm returns a divisor satisfying .  To test whether divides , the algorithm checks whether the remainder when is  divided by , is zero.         Input :       Output :

 Example  1.9 To determine whether is prime, Algorithm 1.8 checks whether any of divides .     Since none of these numbers divides , the condition is always false.   Therefore, the algorithm returns to indicate that is prime.      (Similarly is prime.)

Fundamental Theorem of Arithmetic or the unique factorization theorem.

Except for the order of the prime factors, the prime factors are unique.

 Theorem  1.11 Fundamental Theorem of Arithmetic

 Theorem  1.12 There are infinitely many prime numbers.

Proof

 Definition  1.14 gcd (m,n) is the greatest common divisor.

 Example  1.15

 Example  1.16 Find  Using prime factorization Note that is a common divisors of and . And is also a common divisors of and .   is the greatest  common divisors of and .

 COROLLARY If , and are integers, where , such that and , then whenever and are integers.

 Transitivity of Divisibility Prove that for all integers , and , if and then .

Proof

Show that

for some integer .

for some integer .

.

 Theorem  1.17 The prime factorizations of the positive integers and are , .  where each exponent is a nonnegative integer, and where all primes occurring in the    prime factorization of either and are included in both factorizations, with zero      exponents if necessary.  Then is .

 Example  1.18 Find .

Solution

Using prime factorization

and

Find the greatest common divisor

.

 Definition  1.19 Let and be positive integers.  A common multiple of and is an integer that is divisible by both and .     is the least common multiple.     is the smallest positive common multiple of and .

 Example  1.20 is divisible by both and .

 Example  1.21 Find .

Solution

Using prime factorization

Find least common multiple of and

The prime factorization of contain , and as factors

[so that divides ].

The prime factorization of contain , and as factors

[so that divides ].

The smallest number with this property is

.

Therefore, .

 Theorem  1.22

 Example  1.23 Find .

Solution

Using prime factorization

and

Find least common multiple

.

 Example  1.24

 Theorem  1.25

Exercises  11, 24, 29, 31

5.2 Representations of Integers and Integer Algorithms

Integers can be expressed using any integer greater than one as a base, as we will show in this section. Although we commonly use decimal (base 10), representations, binary (base 2), octal (base 8), and hexadecimal (base 16) representations are often used, especially in computer science. Given a base b and an integer n, we will show how to construct the base b representation of this integer.

A bit is a binary digit, that is, a or a .

The binary Number System consists of  symbols(bit).

The octal Number System consists of symbols.

The decimal Number System consists of symbols.

The hexadecimal Number System consists of symbols.

The system is based the base of the number system.

Decimal Number System

Decimal notation is based .

Any integer , and, for each term , each is one of the decimal digits , ,, , , , , , , .

 Place place place place place Decimal Digit Symbol

The decimal number system.

Binary Number System

In the Binary (base ) number system,

Any integer form is

based

where each and each is one of the binary digits , .

 Place place place place place place Decimal Digit Symbol

The binary number system.

 Example  2.1 Computer Representation of Integers , . , express form ,      base expansion of where , , , , are non-negative integers less than , and .

The base expansion of is denoted by .

 Example  2.2 Binary to Decimal

The number consisting of one , no ’s one , one , no ’s, and one .

 Algorithm  2.3 Converting an Integer from Base to Decimal This algorithm returns the decimal value of the base integer .       Input:  , ,     Output:  dec_val     base_b_to_dec (, , )        dec_val        power        for to           dec_val dec_val *power           power power *          return dec_val

Algorithm runs is time .

 Example  2.4 Answer We show how Algorithm converts the binary number to decimal.

Solution

Here , and

,   ,   ,   .

First, is set to , and power is set to .

We then enter the for loop.

Since and ,

* *.

Thus becomes . Executing

*

sets to . We return to the top of the for loop.

Since and ,

* *.

Thus becomes . Executing

*

sets to .

Since and ,

* *.

Thus becomes . Executing

*

sets to .

Since and ,

* *.

Thus becomes . Executing

*

sets to .

The for loop terminates and the algorithm returns , the decimal value of the binary number .

Octal number system

Octal notation is based .

Any integer form is

based

where and each is one of the octal digits , ,, , , , , .

 Place place place place place Decimal Digit Symbol

The octal number system.

Any integer form is

based

where and each is one of the

hexadecimal digits , ,, , , , , , , , 10=,  11=, , , , 15=.

 Place place place place place place Decimal Digit Symbol

 Example  2.6 Decimal to Binary

 Algorithm  2.7 Converting a Decimal Integer into Base This algorithm convert the positive integer into the base integer .    The variable is used as an index in the sequence .  The value of is the remainder when is divided by .  The value of is the quotient when is divided by .       Input:  ,     Output:  ,     dec_to_base_b(, , , )                while ()

Solution

We begin from the right, adding and . This sum is .

We write and carry . At this point the computation is

Next, we add and and . This sum is .

We write and carry . At this point the computation is

Continuing in this way, we obtain

 Algorithm  2.12 Adding Binary Numbers This algorithm add the binary numbers and and stores the sum in .       Input:  , ,     Output:      binary_addition(, , , )        carry        for to

Solution

We begin from the right, adding and . This sum is .

We write and carry . At this point the computation is

Next, we add and and . This sum is .

We write and carry . At this point the computation is

Continuing in this way, we obtain

The straightforward way to compute this power is to repeatedly multiply by

’s

which uses multiplications. We can do better using repeated squaring

Compute .

uses multiplication

The expansion of is power of    the binary expansion is

.

Compute is

which uses additional multiplications for a total of multiplications.

The straightforward technique uses multiplication.

Binary representation of the exponent

For example

 Current Value                     of                   Result Quotient when Divided by Unchanged

Computing using repeated squaring.

 Algorithm  2.16 Exponentiation By Repeated Squaring This algorithm computes using repeated squaring.       Input:  ,     Output:      exp_via_repeated_squaring(, )        result                while ()           if( )           *           *                        return result

The number of times that the while loop executes is determined by .

The variable is repeatedly halved

and when becomes .

 Theorem  2.17 If , and are positive integers, .

Proof   Let , and .

By the quotient-remainder theorem

is divided by       is the remainder

,       .

is divided by       is the remainder

,       .

is divided by       is the remainder

,       .

Now

where .

Therefore,

is divided by       is the remainder

, which translates to

.

To compute , we successively computed

,     ,     ,

To compute ,

,     ,     ,     .

Each multiplication is performed using Theorem .

Compute

Compute

Compute

Compute

 Example  2.18 Compute of .

Solution

.

 Algorithm  2.19 Exponentiation Mod By Repeated Squaring This algorithm computes using repeated squaring.       Input:  , ,     Output:      exp_mod_via_repeated_squaring(, , )        result                while ()           if( )           *           *                        return result

Exercises  31, 38, 56, 59

5.3   The Euclidean Algorithm

Euclidean Algorithm is algorithm for finding the greatest common divisor of two integers.

The Euclidean Algorithm

Let , where , , and are integers. Then .

 Example  3.1 .   .  By inspection .  Therefore, .

 Theorem  3.2 If , , and   (), then .

Proof

By the quotient-remainder theorem,

, , ,     .

The set of common divisors of and is equal to the set of common divisors of and ,

thus proving the theorem.

Let be a common divisor of and .

.

and     .

is a common divisor of and .

is a common divisor of and       and

is a common divisor of and .

.

 Algorithm  3.3 Euclidean Algorithm This algorithm finds the greatest common divisor of the nonnegative integer and , where not both and are zero.       Input :  and (nonnegative integer, not both zero)     Output :  Greatest common divisor of and  1.     2.       // make a largest  3.        4.        ()  5.      while  6.           7.           8.           9.       10.     return  11.

 Example  3.4 Show that .

Solution

Let and    move to line 5

line 6

line 7

line 8

line 6

line 7

line 8

line 6

line 7

line 8

line 6

line 7

line 8

is the greatest common divisor of and .

Analysis of the Euclidean Algorithm

The pair , which requires modulus operations, is input to the Euclidean algorithm.

( modulus operation)

, requires modulus operations        (to make a total of )

and                      (by inductive assumption)

Therefore,

Smallest input pair that requires modulus operations in the Euclidean Algorithm.

 (= number of modulus operations)

 Theorem  3.5 Suppose that the pair , , , requires modulus operations when input to the Euclidean algorithm. Then and , where denotes the Fibonacci sequence.

Proof

Basis Step

The theorem is true if .

Inductive Step

Assume that the theorem is true if .

[Show that the theorem is true for .]

Suppose that the pair , , , requires modulus operations.

At line , we compute .

Thus

(3.2)                         .

The algorithm then repeats using the values and , .

These values require additional modulus operations.

By the inductive assumption,

(3.3)                       and   .

Combining and , we obtain

(3.4)             .

The first inequality in hold because ; cannot equal . because .

Inequalities and give

and    .

 Theorem  3.6 If integers in the range to , not both zero, are input to the Euclidean algorithm,  then at most modulus operations are required.

Proof

Let be the maximum umber of modulus operations required by the Euclidean algorithm for integers in the range to .

Let , be an input pair in the range to that requires modulus operations.

Shows that and that . We may assume that . By Theorem 3.5 .

Thus

.

Since ,

.

Combining these last inequalities, we obtain

.

Taking the logarithm to the base , we obtain

.

Therefore,

.

ÉTIENNE BÉZOUT (1730–1783)  Reading the writings of the great mathematician Leonhard Euler enticed him to become a mathematician.

In 1758 he was appointed to a position at the Académie des Sciences in Paris.

Bézout is also credited with inventing the determinant.

 Theorem 3.7 BÉZOUT’'S THEOREM If and , then there exist integers such that .

 Example  3.8 Express as a linear combination of and .

Solution

We begin with and .

Consider how the Euclidean algorithm compute .

Using the next-to-last division      Express as a linear combination of and .

We find .

The second division is

.

The first division is

.

Taking and , we obtain

.

 Algorithm  3.9 Recursive Euclidean Algorithm This algorithm recursively find the greatest common divisor of the nonnegative integers and , where not both zero and are zero.         Input : and (nonnegative integers, not both zero)       Output : Greatest common divisor of and

First makes largest. If , it correctly returns . Otherwise, Algorithm computes and returns the greatest common divisor of and . Then

.

To compute and , we modify Algorithm .

Modification .

Compute the greatest common divisor

Compute the value of and .

These values are stored in addition parameters named and .

Consider first the case when . Then .

Set .

Since , could be assigned any value; we choose

 Modification of Algorithm 3.9

Next, Algorithm computes and .

Modified Algorithm will computes and . Thus and satisfy

,

where . Compute and in terms of the available values.

be the quotient of divided by ,

Therefore,

where, and .

 Algorithm  3.10 Computing and of Theorem 3.7 This algorithm computes and satisfying , where and are nonnegative integers not both zero, and returns .         Input : and (nonnegative integers, not both zero)       Output : and of Theorem 5.3.7(stored in parameters and ) and the greatest                common divisor of and (which returned)                                                                                           //            //      //            *

Computing  an Inverse Modulo an Integer

For , and such that .

Find , such that .

is the inverse of .

Using Euclidean Algorithm

find numbers and      .

Then, and since , is the remainder.

Thus

.

( may not satisfy )

If fact since, if then , which contradicts .

Since , there exists such that

.

Combining the previous equations, we have

.

Therefore

.

 Example  3.11 Let and .   and     s' n + t' Phi  = -29*100 + 67*231 =1  where s' = -29 and t' = 67.  Thus, .  Here .  The inverse of is .

Show that the number in equation is unique.

Suppose that

,     ,     .

Show that .

Now

.

Therefore, the number in equation is unique.

Exercises  4, 11, 36

5.4 The RSA Public-Key Cryptosystem (Skip)

Q & A

Exercises

(이 강의록은 성균관대 김응기 박사님, 이영수 박사님, 김장수 교수님 등의 기존 강의록과 교재 및 참고자료를 바탕으로 이상구 교수가 새로 만든 2018년 1학기 용 강의록입니다.)

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*** References   (Calculus) ***

< Calculus 미적분학 1, 2 >

http://matrix.skku.ac.kr/Cal-Book/

(Lecture Movie 동영상 강의) SKKU 성균관대

- Calculus with Sage Lectures (Youtube 동영상 강의)

<미적분학 : 동영상 강의> 이상구 교수, 성균관대

- Calculus with Sage Lectures (Youtube 동영상 강의)

1.1 History of Calculus : http://youtu.be/ODfMaHgIhAc

Calculus with Sage Week 1

how to manage our class Review : http://youtu.be/XWEQFlv4jKc

Chapter 1. Functions

http://youtu.be/cl8GqIWIRD0 (동영상 강의)

1.1 Functions and its graph

1.2 Symmetry (문제 풀이)

1.3 Common Functions

1.4 Translation, Stretching and Rotation of Functions

1.5 A Few Basic Concepts

Chapter 2. Limits and Continuity

2.1 Limits of functions : http://youtu.be/VBCeAllP1M0 (동영상 강의)

2.2 Continuity : http://youtu.be/zGxx3PUCTnM (동영상 강의)

Chapter 3. Theory of Differentiation

3.1 Definition of Derivatives, Differentiation : http://youtu.be/A-vDsF9ulTs (동영상 강의)

3.2 Derivatives of Polynomials, Exponential Functions, Trigonometric Functions, The product rule: http://youtu.be/XXMnCESesfQ

3.3 The Chain Rule and Inverse Functions : http://youtu.be/HfScHEsPfKI

3.4 Approximation and Related Rates : http://youtu.be/ViRwEJ0Wfkw

Chapter 4. Applications of Differentiation

4.1 Extreme values of a function : http://youtu.be/mXVU8OqIHJY (동영상 강의)

4.2 The Shape of a Graph : http://youtu.be/cZrAF_77On4

4.3 The Limit of Indeterminate Forms and L’Hospital’s Rule :

4.4 Optimization Problems : http://youtu.be/k0NtkmZFnh8

4.5 Newton’s Method : http://youtu.be/VxCfl2JzMYU

Chapter 5. Integrals

5.1 Areas and Distances : http://youtu.be/mT_oxlD6RSA (동영상 강의)

5.2 The Definite Integral : http://youtu.be/GIm3Oz58Ti8

5.3 The Fundamental Theorem of Calculus : http://youtu.be/Zf1HT2H2fbA

5.4 Indefinite Integrals and the Net Change Theorem : http://youtu.be/E6I3EDzAVuU

5.5 The Substitution Rule : http://youtu.be/h7tmvmNOliU

5.6 The Logarithm Defined as an Integral : http://youtu.be/kD0Z9PqetsA

Midterm Exam : 424일 수요일 4시에 강의실

미적분학 with Sage Midterm Exam, http://youtu.be/QAEI7A2DMMM

Chapter 6. Applications of Integration

미적분학 with Sage Sec-6-1 Areas between Curves, http://youtu.be/o53phm5cqJE

미적분학 with Sage Sec-6-2 Volumes, http://youtu.be/4-ChOAFbJAs

미적분학 with Sage Sec-6-3 Volumes by Cylindrical Shells,

http://youtu.be/qM1izf8qeX8 (동영상 강의)

미적분학 with Sage Sec-6-4 Work : http://youtu.be/u3ZaJWhKy6k

미적분학 with Sage Sec-6-5 Average Value of a Function

Chapter 7. Techniques of Integration

미적분학 with Sage Sec-7-1 Integration by Parts http://youtu.be/WX-6C9tCneE

미적분학 with Sage Sec-7-2 Trigonometric Integrals http://youtu.be/sIR0zNGQbus

미적분학 with Sage Sec-7-3 Trigonometric Substitution http://youtu.be/avTqiEUi8u8

미적분학 with Sage Sec-7-4 Integration of Rational Functions by the Method of Partial Fractions http://youtu.be/KLTHp_7G4cI

미적분학 with Sage Sec-7-5 Guidelines for Integration http://youtu.be/Fgn8U4We60o

미적분학 with Sage Sec-7-6 Integration Using Tables http://youtu.be/tn9jLkgTMp8

미적분학 with Sage Sec-7-7 Approximate Integration http://youtu.be/hg2pw1n1cZI

미적분학 with Sage Sec-7-8 Improper Integrals http://youtu.be/rquxbYrC0Yc

Chapter 8. Further Applications of Integration

미적분학 with Sage Sec-8-1 Arc Length http://youtu.be/7OVqI20z_Bw

미적분학 with Sage Sec-8-2 Area of a Surface of Revolution

미적분학 with Sage Sec-8-3 Applications of Integral Calculus

*미적분학 with Sage Sec-8-4 Differential equations http://youtu.be/uHfOjz8I4-s

Chapter 9. Parametric Equations and Polar Coordinates

9.1 Parametric Equations

9.2 Calculus with Parametric Curves

9.3 Polar Coordinates

9.4 Areas and Lengths in Polar Coordinates

9.5 Conic Section

Chapter 10. Infinite Sequences and Infinite Series

10.1 Sequences and Series

10.2 Tests for convergence of series with positive terms

10.3 Alternating Series and Absolute Convergence

10.4 Power Series

강의에 앞서 : http://youtu.be/YtxYZW3Enko (동영상 강의)

Chapter 11. Vectors and the Geometry of Space

11.1 Three-Dimensional Coordinate Systems

11.2 Vectors

11.3 The Dot Product

11.4 The Vector or Cross Product

11.5 Equations of Lines and Planes

Chapter 12. Vector Valued Functions

12.1 Vector-Valued Functions and Space Curves http://youtu.be/0pvywjBjsQw

12.2 Calculus of Vector Functions

12.3 Arc Length and Curvature

*12.4 Motion Along A Space Curve: Velocity and Acceleration

Chapter 13. Partial Derivatives

13.1 Multivariate Functions

13.2 Limits and Continuity of Multivariate Functions

13.3 Partial Derivatives http://youtu.be/LR89Ct3cEDY (동영상 강의)

13.4 Differentiability and Total Differentials

13.5 The Chain Rule http://youtu.be/r3dGYL1vkEU

13.6 Directional Derivatives and Gradient http://youtu.be/o8L_ShRANjo

13.7 Tangent Plane and Differentiability http://youtu.be/uOf-5YHKGI4

13.8 Extrema of Multivariate Functions http://youtu.be/oDZUkOEszOQ

13.9 Lagrange Multiplier

Chapter 14. Multiple Integrals

14장 앞부분 복습 내용: http://youtu.be/5eCO2GjlJHs (동영상 강의)

Chapter 14. Multiple Integrals

14.1 Double Integrals http://matrix.skku.ac.kr/Cal-Book/part2/CS-Sec-14-1-Sol.html

14.2 Double Integrals in Polar Coordinates

14.3 Surface Area http://matrix.skku.ac.kr/Cal-Book/part2/CS-Sec-14-3-Sol.html

14.4 Cylindrical Coordinates and Spherical Coordinates http://matrix.skku.ac.kr/Cal-Book/part2/CS-Sec-14-4-Sol.html

14.5 Triple Integrals

14.6 Triple Integrals in Cylindrical and Spherical Coordinates

14.7 Change of Variables in Multiple Integrals

14.1 Double Integrals http://youtu.be/jZ2pAmPZYOE (동영상 강의)

14.2 Double Integrals in Polar Coordinates http://youtu.be/olQgihl5aZg

14.3 Surface Area http://youtu.be/p9R0TTLfBzk

14.4 Cylindrical and Spherical Coordinates http://youtu.be/q3FKd2UxV_I

14.5 Triple Integrals http://youtu.be/r1tzH9Ibbqk

14.6 Triple Integrals in Cylindrical & Spherical Co http://youtu.be/xd0U4_C2ePY

14.7 Change of Variables in Multiple Integrals http://youtu.be/INn-bkgXYNg

Chapter 15. Vector Calculus

Chapter 15. Vector Calculus

15.1 Vector Differentiation

15.2 Line Integrals http://matrix.skku.ac.kr/Cal-Book/part2/CS-Sec-15-2-Sol.html

15.3 Independence of the Path

15.4 Green’s Theorem in Plane

15.5 Curl and Divergence

15.6 Surface and Area

15.7 Surface Integrals

15.8 Stokes’ Theorem

15.9 Divergence Theorem

15.1 Vector Differentiation http://youtu.be/q0aVmUCXgTI (동영상 강의)

15.2 Line Integrals http://youtu.be/wHINlpNXYaU

15.3 Independence of the Path http://youtu.be/jGGOL3QDj1Y

15.4 Green’s Theorem in Plane http://youtu.be/WxdTbaSb_ZI

15.5 Curl and Divergence http://youtu.be/IswmJUCTeNA

15.6 Surface and Area http://youtu.be/xX6tNVpegbs

15.7 Surface Integrals http://youtu.be/nrzIrM4doLo

15.8 Stokes’ Theorem http://youtu.be/t4skc_PzJvg

15.9 Divergence Theorem http://youtu.be/3BmcFr81kuQ

(SKKU 선형대수학 Jordan 표준형, SGLee : http://youtu.be/Dn4qBxgUJcA )

**************************************************

<Calclus 미적분학 1, 2>

http://matrix.skku.ac.kr/Cal-Book/

(동영상) <성균관대 학생 문제 풀이 설명 (by Students)>

Chapter 1. Functions

미적분학 with Sage Sec-1-1 Functions and Graph, Problem, 문제풀이 by 황인철 http://youtu.be/rQ2CB8EvkoE

미적분학 with Sage Sec-1-2 Symmetry, Problem, 문제풀이 by 곽주현 http://youtu.be/BNKUzSohiD8

미적분학 with Sage Sec-1-3 Common Functions, Problem, 문제풀이 by 장찬영 http://youtu.be/x0E0ZMxZ3Og

미적분학 with Sage Sec-1-4 Translation, Stretching and Rotation of Functions, Problem, 문제풀이 by 임효정 http://youtu.be/vx7GCWY68Zw

Chapter 2. Limits and Continuity

미적분학 with Sage Sec-2-1 Limits of functions, Problem, 문제풀이 by 장재철-이훈정 http://youtu.be/LZSmRPAAXME

미적분학 with Sage Sec-2-2 Continuity, Problem, 문제풀이 by 이훈정 http://youtu.be/azrkT1RP4-c

미적분학 with Sage Sec-2-2 Continuity, Epsilon-Delta Proof, by 황인철 http://youtu.be/hj8d-j_DGf4

Chapter 3. Theory of Differentiation

미적분학 with Sage Sec-3-1 Definition of Derivatives, Differentiation, Problem, 문제풀이 by 김태현 http://youtu.be/7wTBWuk2CzU

미적분학 with Sage Sec-3-2 Derivatives of Polynomials, Exponential Functions, Trigonometric, Problem, 문제풀이 by 조건우 http://youtu.be/Ei5KGW9vZhE

미적분학 with Sage Sec-3-3 The Chain Rule and Inverse Functions, Problem, 문제풀이 by 유휘의 http://youtu.be/aSKm12922FE

미적분학 with Sage Sec-3-4 Approximation and Related Rates, Problem, 문제풀이 김종민 http://youtu.be/JmBOv6_D6qA

Chapter 4. Applications of Differentiation

미적분학 with Sage Sec-4-1 Extreme values of a function, Problem, 문제풀이 by 김태영 http://youtu.be/_V4MryNEzWY

미적분학 with Sage Sec-4-2 The Shape of a Graph, Problem, 문제풀이 by 김태영 http://youtu.be/SVOWADHlzV8

미적분학 with Sage Sec-4-3 Indeterminate Forms and L'Hospital's Rule, Problem, 문제풀이 by 신종희 http://youtu.be/gR2luDDPsMY

미적분학 with Sage Sec-4-4 Optimization, Problem, 문제풀이 by 이승철 http://youtu.be/AELEV2ElaeQ

미적분학 with Sage Sec-4-5 Newton's Method, Problem, 문제풀이 by 이승철http://youtu.be/fdBHQ46g9RE

Chapter 5. Integrals

미적분학 with Sage Sec-5-1 Area and Distance, Problem, 문제풀이 by 남택현 http://youtu.be/Y_nCn76RPmY

미적분학 with Sage Sec-5-2 Definite Integral, Problem, 문제풀이 by 남택현http://youtu.be/iUsf1h_hTAE

미적분학 with Sage Sec-5-3 and 5-4 Fun Theorem of Calculus Net Change Theorem, Problem, 문제풀이 by 정승찬 & Kim http://youtu.be/Pa4Z38KkDVY

(미적분학 with Sage Sec-5-4 Net Change Theorem, Problem, 문제풀이 by Kim *** )

미적분학 with Sage Sec-5-5 Substitution, Problem, 문제풀이 by 이한울 http://youtu.be/0TMbpCPO4Uc

미적분학 with Sage Sec-5-6 Log and Exponential, Problem, 문제풀이 by 이한울 http://youtu.be/ymDImdIQ90c

Chapter 6. Applications of Integration

미적분학 with Sage Sec-6-2 Volumes, 문제풀이 by 김종민 http://youtu.be/Fd4Mguf2dbU

미적분학 with Sage Sec-6-3 Volumes by Cylindrical Shells, Problem, 문제풀이 by 신영찬 http://youtu.be/gNaKkA0UNHg

미적분학 with Sage Sec-6-4 Work, Problem, 문제풀이 by 김건호 http://youtu.be/SmIo2yaxNsY

미적분학 with Sage Sec-6-5 Average Value of a Function, Problem, 문제풀이 by 신종희 http://youtu.be/BVahd-DJoe8

Chapter 7. Techniques of Integration

미적분학 with Sage Sec-7-1 Integration by Parts, 문제풀이 by 이인행 http://youtu.be/jKCAGJ4HqvQ

미적분학 with Sage Sec-7-2 Trigonometric Integrals, 문제풀이 by 김태현 http://youtu.be/ytETYf1wLbs

미적분학 with Sage Sec-7-3 Trigonometric Substitution, Problem, 문제풀이 by 이훈정 http://youtu.be/utTQHIabTyI

미적분학 with Sage Sec-7-4 Integration of Rational Functions by the Method of Partial Fractions, Problem, 문제풀이 by 장재철 http://youtu.be/SkNW_bax0YI

미적분학 with Sage Sec-7-5 Guidelines for Integration, Problem, 문제풀이 by 김대환 http://youtu.be/-N9Fe_Arp2c

미적분학 with Sage Sec-7-6 Integration Using Tables, Problem, 문제풀이 by 조건우 http://youtu.be/EnEQ9ZS3B_k

미적분학 with Sage Sec-7-8 Improper Integrals, Problem, 문제풀이 by 이인행 http://youtu.be/dfSkjvmSXYo

Chapter 8. Further Applications of Integration

미적분학 with Sage Sec-8-1 Arc Length, 문제풀이 by 남택현 http://youtu.be/A8N-mDD0ja8

미적분학 with Sage Sec-8-2 Area of a Surface of Revolution, 문제풀이 by 정승찬 http://youtu.be/yZFJDJgTJfw

(미적분학 with Sage Sec-8.3 Center of Mass

(미적분학 with Sage Sec-8.4 Differential equations

Chapter 9. Infinite Sequences and Infinite Series

미적분학 with Sage Sec-9-1 Sequences and Series, 문제풀이 by 이원준 http://youtu.be/O6y1v5fJA0k

미적분학 with Sage Sec-9-2 Tests for convergence of series with positive terms, 문제풀이 by 김범윤 http://youtu.be/1flKAnlv9LA

미적분학 with Sage Sec-9.3 Alternating Series and Absolute Convergence ?

(미적분학 with Sage Sec-9.4 Power Series ?

(미적분학 with Sage Sec-9.5 Taylor, Maclaurin, and Binomial Series ?

Chapter 10. Parametric Equations and Polar Coordinates

미적분학 with Sage Sec-10-1 Parametric Equations, 문제풀이 by 임효정 http://youtu.be/Ybs68e0iMZI

미적분학 with Sage Sec-10-2 Calculus with Parametric Curves, 문제풀이 by 장찬영 http://youtu.be/yF5oZOQVnCE

미적분학 with Sage Sec-10-3 Polar Coordinates, Problem, 문제풀이 by 황인철 http://youtu.be/4hoVKvk8dq0

미적분학 with Sage Sec-10-4 Areas and Lengths in Polar Coordinates, Problem, 문제풀이 by 곽주현 http://youtu.be/LRmasW9uqYY

미적분학 with Sage Sec-10-5 Conic Section, Problem, 문제풀이 by 이한울 http://youtu.be/CZ9SHMtqVy4

Chapter 11. Vectors and the Geometry of Space

미적분학 with Sage Sec-11.1 Three-Dimensional Coordinate Systems, 문제풀이 by 김태현 http://youtu.be/_s_2T1VVob8

미적분학 with Sage Sec-11.2 Vectors, 문제풀이 by 오교혁 http://youtu.be/BFgh6irMqsc

(미적분학 with Sage Sec-11.3 The Dot Product ?

(미적분학 with Sage Sec-11.4 The Vector or Cross Product ?

미적분학 with Sage Sec-11.5 Equations of Lines and Planes, 문제풀이 by 구본우 http://youtu.be/lxuGE_Erthg

(미적분학 with Sage Sec-11.6 Cylinders and Quadric Surfaces ?

Chapter 12. Vector Valued Functions

미적분학 with Sage Sec-12.1 Vector-Valued Functions and Space Curves, 문제풀이 by 최양현 http://youtu.be/jvMI6OzdR_I

미적분학 with Sage Sec-12.2 Calculus of Vector Functions , 문제풀이 by 김동윤 http://youtu.be/VS5rPyOjP2I

(미적분학 with Sage Sec-12.3 Arc Length and Curvature ?

(미적분학 with Sage Sec-12.4 Motion Along A Space Curve: Velocity and Acceleration ?

Chapter 13. Partial Derivatives

미적분학 with Sage Sec-13.1 Multivariate Functions, 문제풀이 by 구본우 http://youtu.be/As_0AYApHlM

(미적분학 with Sage Sec-13.2 Limits and Continuity of Multivariate Functions, 문제풀이 by 김건호?

(미적분학 with Sage Sec-13.3 Partial Derivatives, 문제풀이 by 김동윤?

미적분학 with Sage Sec-13.4 Differentiability and Total Differentials, 문제풀이 by 김범윤 http://youtu.be/qDmCWBiXbIA

미적분학 with Sage Sec-13.5 The Chain Rule, 문제풀이 by 김유경 http://youtu.be/vzN5By6qzvM

미적분학 with Sage Sec-13.6 Directional Derivatives and Gradient, 문제풀이 by 김태현 http://youtu.be/2_7TOUuzJoE

미적분학 with Sage Sec-13.7 Tangent Plane and Differentiability, 문제풀이 by 서용태 http://youtu.be/GDkE8OqUvsk

미적분학 with Sage Sec-13.8 Extrema of Multivariate Functions, 문제풀이 by 오교혁 http://youtu.be/FWmk_MasIjE

미적분학 with Sage Sec- 13.9 Lagrange Multiplier, 문제풀이 by 이원준 http://youtu.be/YMGdQWBzyrI

Chapter 14. Multiple Integrals

미적분학 with Sage Sec-14.1 Double Integrals, 문제풀이 by 이인행 http://youtu.be/w8g9fgcEP4A

미적분학 with Sage Sec-14.2 Double Integrals in Polar Coordinates, 문제풀이 by 이지석 http://youtu.be/jpsObxtZ50A

(미적분학 with Sage Sec-14.3 Surface Area, 문제풀이 by 이한울?

(미적분학 with Sage Sec-14.4 Cylindrical Coordinates and Spherical Coordinates, 문제풀이 by 최양현?

미적분학 with Sage Sec-14.5 Triple Integrals, 문제풀이 by 이인행 http://youtu.be/Voq67ooqQJs

(미적분학 with Sage Sec-14.6 Triple Integrals in Cylindrical and Spherical Coordinates, 문제풀이 by 구본우?

(미적분학 with Sage Sec-14.7 Change of Variables in Multiple Integrals, 문제풀이 by 김건호 , 최양현?

Chapter 15. Vector Calculus

(미적분학 with Sage Sec-15.1 Vector Differentiation, 문제풀이 by 김동윤?

미적분학 with Sage Sec-15.2 Line Integrals, 문제풀이 by 김범윤 http://youtu.be/ZdRjCfJeHM8

미적분학 with Sage Sec-15.3 Independence of the Path, 문제풀이 by 김유경 http://youtu.be/TreCe8ESEiU

(미적분학 with Sage Sec-15.4 Green’s Theorem in Plane, 문제풀이 by 김태현?

미적분학 with Sage Sec-15.5 Curl and Divergence, 문제풀이 by 서용태 http://youtu.be/wLTHYaANwtI

미적분학 with Sage Sec-15.6 Surface and Area, 문제풀이 by 오교혁 http://youtu.be/j7F3xVNdHvA

미적분학 with Sage Sec-15.7 Surface Integrals, 문제풀이 by 이원준 http://youtu.be/s_MRgW2By38

(미적분학 with Sage Sec-15.8 Stokes’ Theorem, 문제풀이 by 이인행?

미적분학 with Sage Sec-15.9 Divergence Theorem, 문제풀이 by 최주영 http://youtu.be/vGMLoGWF1Is