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

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

Define Function

Taylor expansion

 

Equation solving

Approximate solution

Partial fraction

 

Limit

Right hand limit

Left hand limit

Limit at + infinity

Limit at - infinity

 

Derivative

Second derivative

Indefinite integral

Definite integral

 

Series

 

Radius of convergence

 

 

 

Define a vector

Norm of the vector

Distance from yz-plane

Distance from zx-plane

Distance from xz-plane

Distance from x-axis

Distance from y-axis

Distance from z-axis

 

Define a vector valued function

Velocity 

Acceleration 

Speed

Velocity vector at t = 2

Acceleration vector at t = 2

Arc length

Dot product

Cross product

f(x)=exp(-2*x)

f(x).taylor(x, 2, 4)

 

solve(f(x)==0, x)

find_root(f(x), a, b)

f.partial_fraction(x)

 

limit(f(x), x=2)

limit(f(x), x=2, dir='+')

limit(f(x), x=2, dir='-')

limit(f(x), x=+oo)

limit(f(x), x=-oo)

 

diff(f(x), x)

diff(f(x), x, 2)

integral(f(x), x)

integral(f(x), (x, -2, 3))

 

sum((e/10)^x, x, 1, +oo)

 

u(x)=1/(x*2^x)

rho=limit(abs(u(x+1)/u(x)), x=+oo)

1/rho

 

x = vector([3, -4, 5])

x.norm()

sqrt(x[0]^2)

sqrt(x[1]^2)

sqrt(x[2]^2)

sqrt(x[1]^2 + x[2]^2)

sqrt(x[0]^2 + x[2]^2)

sqrt(x[0]^2 + x[1]^2)

 

var('t'); r=vector([2-5*t,4*t,1+3*t])

v=diff(r, t)

a=diff(r, t, 2)

v.norm()

v.subs(t=2)

a.subs(t=2)

integral(v.norm(), (t, 0, 5))

v.dot_product(a) 

v.cross_product(a)



Find gradient

 

 

Double integral

Triple integral

 

Double integral (in Polar coordinate)

 

 

 

Cylindrical 

to Rectangular

 

Spherical 

to Rectangular

 

Graph in plane

Parametric function in plane

Implicit function in plane

Line segment in plane

 

Graph in

Parametric function in

Implicit function in

 

Level curve

 

Vector field

Vector field in

 

Line integral

 

 

 

curl (A)

div (A)

 

Green’s theorem

Stokes’ theorem

Divergence theorem

var('x, y, z')    

f=x*y+y*z^2+x*z^3

f.gradient()

integral(integral(f, (x, 0, y)), (y, 0, 1))

integral(integral(integral(f, (x, 0, y)), (y, 0, 1)), (z, 0, 1))

 

var('r, t')

f=arctan(tan(t))

integral(integral(f*r, (r, 1, 2)), (t, 0, pi/4))

 

T = Cylindrical('height', ['radius', 'azimuth'])

T.transform(radius=1, azimuth= pi, height=2)

 

T = Spherical('radius', ['azimuth', 'inclination'])

T.transform(radius=3, azimuth=pi/6, inclination=pi/6)

 

plot(sin(x), (x, -4, 4))

parametric_plot((sin(x), cos(x)), (x, 0, 2*pi))

var('x, y'); implicit_plot(sin(x)-y==1, (x, -2, 2), (y, -2, 2))

line([(1, 1), (2, 2)], color='red')

 

var('x, y'); plot(x^2+y^2, (x, -2, 2), (y, -2, 2))

parametric_plot3d((sin(x), cos(x), x), (x, 0, 2*pi))

var('x, y, z'); implicit_plot3d(x^2+y^2==5, (x, -3, 3), (y, -3, 3),  (z, -3, 3))

 

var('x, y'); contour_plot(x^2+y^2, (x, -1, 1), (y, -1, 1), cmap='hsv', labels=True)

 

var('x, y'); plot_vector_field((x+y, x), (x, -3, 3), (y, -3, 3))

 

var('x, y, z'); plot_vector_field3d((0, 0, 1), (x, -3, 3), (y, -3, 3), (z, -3, 3))

 

var('t'); r=vector([t-t^3, t^2, 0])

integral(r.dot_product(diff(r, t)), (t, 0, pi))

 

A(x,y,z) = P*i+Q*j+R*k  # conservative if curl(F)=0.

curlA=(diff(R,y)-diff(Q,z))*i+(diff(P,z)-diff(R,x))*j+(diff(Q,x)-diff(P,y))*k 

divA = diff(P,x)+diff(Q,y)+diff(R,z)    # divergence of A

 

integral(integral((diff(N, x)-diff(M, y))*r, (r, 0, 3)), (t, 0, 2*pi))

integral(integral(curl(F).dot_product(-n), (r, 0, 1)), (t, 0, 2*pi))

integral(integral(integral(Div, 0, 3), (y, 0, 2)), (z, 0, 1))


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

       

       

             

             

      

       

 

 

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-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

 

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

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/