C H A P T E R 3

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

by SGLee

Rules for Inequalities

1. If , then .

2. If , then .

3. If , then .

4. If , then .

5. If , then .

6. If , then .

7. If , then .

8. If , then .

9. If , then .

Special Functions

1. Exponential Functions

If and , then a function of the form is called an exponential function.

The number is called the base and is called the exponential.

http://matrix.skku.ac.kr/Mobile-Sage-G/sage-grapher.html

2. Logarithmic Functions

The logarithmic with base and , is defined by .

3. Inverse Trigonometric Functions

4. Hyperbolic Functions

5. Inverse Hyperbolic Functions

http://matrix.skku.ac.kr/cal-lab/cal-1-2-Rational-Fts.html

http://matrix.skku.ac.kr/cal-lab/cal-9-1-example-2.html

Formulas of Trigonometric Functions

1. Addition and Subtraction Formulas

2. Double Angle Formulas

3. Half Angle Formulas

4. Triple Angle Formulas

5. Product-to-Sum

6. Sun-to-Product

7. Linear Combination

where,

where,

Limits

http://matrix.skku.ac.kr/cal-lab/SKKU-Cell-Epsilon-Delta.html

http://matrix.skku.ac.kr/cal-lab/cal-Limit.html

1. Limit Laws

Suppose that the limits and exist and is a constant. Then

if

, where is a positive integer.

, where is a positive integer.

(If is even, we require )

http://matrix.skku.ac.kr/cal-lab/cal-2-1-7.html

http://matrix.skku.ac.kr/cal-lab/cal-2-2-3.html

2. Squeeze Theorem (or Sandwich Theorem)

If when is near and

　 then

http://matrix.skku.ac.kr/cal-lab/cal-2-1-12.html

Derivatives

1. Differentiation Rules

where is a positive integer and is a constant.

2. Chain Rule

If is any real number and is differentiable and , then .

3. Parametric Formula

For the parametric equations: and ,

,

4. Implicit Function

Let be an implicit function.

where .

5. Inverse Function

or

6. Trigonometric Functions

http://matrix.skku.ac.kr/cal-lab/cal-3-3-Exm24.html

7. Inverse Trigonometric Functions

http://matrix.skku.ac.kr/cal-lab/cal-3-2-13.html

8. Logarithmic Functions

9. Exponential Functions

10. Hyperbolic Functions

11. Inverse Hyperbolic Functions

General Rules of Integration

http://matrix.skku.ac.kr/cal-lab/cal-RiemannSum.html

http://matrix.skku.ac.kr/cal-lab/cal-5-1-exs-1.html

http://matrix.skku.ac.kr/cal-lab/cal-6-1-example-6.html

http://matrix.skku.ac.kr/cal-lab/cal-6-3-17.html

http://matrix.skku.ac.kr/cal-lab/cal-12-1-Rotations-B.html

1. Basic Forms

http://matrix.skku.ac.kr/cal-lab/cal-7-7-Exm-8.html

,

http://matrix.skku.ac.kr/cal-lab/cal-5-2-20.html

2. Trigonometric Forms

http://matrix.skku.ac.kr/cal-lab/cal-5-4-exm-7.html

http://matrix.skku.ac.kr/cal-lab/cal-8-3-3.html

3. Hyperbolic Forms

http://matrix.skku.ac.kr/cal-lab/cal-8-1-9.html

4. Forms Involving

5. Inverse Trigonometric Forms

http://myhandbook.info/form_integ.html

Series

1. Taylor Series

http://matrix.skku.ac.kr/cal-lab/cal-10-5-Exm-11.html

2. Maclaurin Series

,

,

,

,

,

3. Binomial Series

If and are any real numbers and is a positive integer, we have

where and

Vectors

1. Dot Product

Let and .

http://matrix.skku.ac.kr/cal-lab/cal-11-3-2.html

2. Projections

Scalar projection of onto : ,

Vector projection of onto :

http://matrix.skku.ac.kr/cal-lab/cal-11-5-20.html

3. Definition and Properties of Cross Product

http://matrix.skku.ac.kr/cal-lab/cal-11-4-Exs-6.html

Let two nonzero vectors and are two sides of a parallelogram,

then the area of the parallelogram is .

http://matrix.skku.ac.kr/cal-lab/cal-11-4-10.html

http://matrix.skku.ac.kr/cal-lab/cal-11-4-16.html

4. Rules of Limits

,

http://matrix.skku.ac.kr/cal-lab/9-5-Example-7.html

5. Rules of Differentiation

http://myhandbook.info/form_diff.html

where is a scalar

http://matrix.skku.ac.kr/cal-lab/cal-4-3-24.html

http://matrix.skku.ac.kr/cal-lab/cal-4-2-9.html

6. Derivative of a Vector Function

If , then

http://matrix.skku.ac.kr/cal-lab/cal-13-2-Exm6.html

http://matrix.skku.ac.kr/cal-lab/cal-12-4-3.html

http://matrix.skku.ac.kr/cal-lab/cal-13-2-20.html

http://matrix.skku.ac.kr/cal-lab/cal-13-2-Exm6.html

7. Integral of a Vector Function

If , then

8. Arc Length

http://matrix.skku.ac.kr/cal-lab/cal-8-1-9.html

http://matrix.skku.ac.kr/cal-lab/cal-8-1-11.html

http://matrix.skku.ac.kr/cal-lab/cal-13-3-2.html

9. Curvature

,

http://matrix.skku.ac.kr/cal-lab/cal-13-3-12.html

10. Equations of Line

: a vector equation

: parametric equations

http://matrix.skku.ac.kr/cal-lab/11-5-Exmaple-7.html

: a symmetric equation

http://matrix.skku.ac.kr/cal-lab/11-5-Exmaple-14.html

11. Equation of Plane

: a standard form of a plane

: a vector version of a plane

: parametric equations of a plane

http://matrix.skku.ac.kr/cal-lab/cal-11-5-20.html

Formulas of Vector Calculus

http://en.wikipedia.org/wiki/Vector_calculus_identities

1. Line Integral

http://matrix.skku.ac.kr/cal-lab/Sec15-2-1.html

http://matrix.skku.ac.kr/cal-lab/Sec15-4-Exs-1.html

2. Path Independent Theorem

Let be a potential function for .

For any piecewise smooth curve from and ,

http://matrix.skku.ac.kr/cal-lab/Sec15-2-Exm-1.html

3. Area of a Plane Region

If has a piecewise smooth boundary with positive orientation,

then the area of is

4. Area of a Surface

http://matrix.skku.ac.kr/cal-lab/cal-8-2-Exm-4.html

http://matrix.skku.ac.kr/cal-lab/cal-8-2-3.html

http://matrix.skku.ac.kr/cal-lab/cal-8-2-4.html

http://matrix.skku.ac.kr/cal-lab/cal-0-a.html

5. Surface Integral

http://matrix.skku.ac.kr/cal-lab/cal-14-5-1.html

http://matrix.skku.ac.kr/cal-lab/Sec15-6-Exm-4.html

http://matrix.skku.ac.kr/cal-lab/Sec15-6-Exm-5.html

http://matrix.skku.ac.kr/cal-lab/Sec15-6-Exm-8.html

http://matrix.skku.ac.kr/cal-lab/Sec15-7-Exm-2.html

http://matrix.skku.ac.kr/cal-lab/cal-15-9-Exam-3.html

6. Gradient

http://matrix.skku.ac.kr/cal-lab/cal-12-2-2.html

7. Divergence

http://matrix.skku.ac.kr/cal-lab/Sec15-5-Exm-3.html

8. Curl

http://matrix.skku.ac.kr/cal-lab/Sec15-8-Exm-2.html

9. Laplace Operator

http://matrix.skku.ac.kr/cal-lab/cal-12-2-6.html

10. Vector Triple Products

http://matrix.skku.ac.kr/cal-lab/cal-11-4-16.html

Theorems on Vector Calculus

1. Green's Theorem

http://matrix.skku.ac.kr/cal-lab/Sec15-4-Exm-1.html

http://matrix.skku.ac.kr/cal-lab/Sec15-4-Exs-10.html

http://matrix.skku.ac.kr/cal-lab/cal-15-2-10.html

2. Stoke's Theorem

http://matrix.skku.ac.kr/cal-lab/Sec15-5-Exm-3.html

http://matrix.skku.ac.kr/cal-lab/cal-14-7-4.html

http://matrix.skku.ac.kr/cal-lab/cal-15-5-5.html

3. Divergence Theorem

div .

http://matrix.skku.ac.kr/cal-lab/Sec15-5-Exs-7.html

http://matrix.skku.ac.kr/cal-lab/cal-14-8-1.html

http://matrix.skku.ac.kr/cal-lab/cal-14-8-5.html

http://matrix.skku.ac.kr/cal-lab/cal-15-8-Exs-8.html

Mobile Sage Grapher : http://matrix.skku.ac.kr/Mobile-Sage-G/sage-grapher.html

http://math1.skku.ac.kr/pub/

http://math1.skku.ac.kr/home/pub/1433/

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