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Examples of Fractional Calculus Computer Algebra System 例题

Content

  • Arithmetic 算术
    1. Exact_computation
    2. Complex 复数
    3. Numerical approximations
  • Algebra 代数
    1. factor
    2. expand
    3. Convert
    4. inverse function
    5. inverse equation
    6. polynomial
  • Function 函数
    1. Trigonometry 三角函数
    2. Complex Function 复变函数
    3. special Function
  • Calculus 微积分
    1. Limit
    2. Derivatives
    3. Integrals
    4. Fractional calculus
  • Equation 方程
    1. inverse equation
    2. polynomial equation
    3. Algebra_equation
    4. system of equations
    5. Diophantine equation
    6. congruence equation
    7. Modulus equation
    8. Probability_equation
    9. recurrence_equation
    10. functional_equation
    11. difference_equation
    12. Inequalities
    13. differential equation
    14. fractional differential equation
    15. system of differential equations
    16. partial differental equation
    17. integral equation
    18. fractional integral equation
    19. differential integral equation
    20. test solution
  • Discrete Math 离散数学
    1. Summation ∑
    2. Indefinite sum
    3. definite sum
    4. infinite sum
    5. Series 级数
    6. Product ∏
  • Definition 定义式
  • Numeric math 数值数学
  • Number Theory 数论
  • Probability 概率
  • Statistics 统计
  • Multi elements
    1. list()
    2. vector
    3. and
  • Animation 动画
  • Plot 制图
    1. Interactive plot 互动制图
    2. parametric plot, polar plot
    3. solve equation graphically
    4. area plot with integral
    5. complex plot
    6. Geometry 几何
  • plane graph 平面图
    1. plane graph 平面图 with plot2D
    2. function plot with funplot
    3. differentiate graphically with diff2D
    4. integrate graphically with integrate2D
    5. solve ODE graphically with odeplot
  • 3D graph 立体图
    1. surface in 3D with plot3D
    2. contour in 3D with contour3D
    3. wireframe in 3D with wirefram3D
    4. complex function in 3D with complex3D
    5. a line in 3D with parametric3D
    6. a column in 3D with parametric3D
    7. the 4-dimensional object (x,y,z,t) in 3D with implicit3D
  • programming 编程
  • bugs

      Arithmetic 算术 >>

      Exact computation

    1. Fraction `1E2-1/2`
    2. Big number: add prefix "big" to number
      big1234567890123456789

    3. mod operation
      input mod(3,2) for 3 mod 2

      Complex 复数

      Complex( 1,2) number is special vector, i.e. the 2-dimentional vector, so it can be operated and plotted as vector.
    4. complex numbers in the complex plane
      complex(1,2) = 1+2i
    5. input complex number in polar(r,theta*degree) coordinates
      polar(1,45degree)

    6. input complex number in polar(r,theta) coordinates for degree by polard(r,degree)
      polard(1,45)

    7. input complex number in r*cis(theta*degree) format
      2cis(45degree)
    8. Convert to complex( )

    9. in order to auto plot complex number as vector, input complex(1,-2) for 1-2i, or convert 1-2i to complex(1,-2) by
      convert(1-2i to complex) = tocomplex(1-2i)
    10. input complex number in polar
      tocomplex(polar(1,45degree))

    11. Convert complex a+b*i to polar(r,theta) coordinates
      convert 1-i to polar = topolar(1-i)

    12. Convert complex a+b*i to polar(r,theta*degree) coordinates
      topolard(1-i)


    13. complex 2D plot
      complex2D(x^x)
    14. complex 3D plot
      complex3D(pow(x,x))

      Numerical approximations

    15. Convert back by numeric computation n( )
      n(polar(2,45degree))
      n( sin(pi/4) )
      n( sin(30 degree) )
    16. `sin^((0.5))(1)` is the 0.5 order derivative of sin(x) at x=1
      n( sin(0.5,1) )
    17. `sin(1)^(0.5)` is the 0.5 power of sin(x) at x=1
      n( sin(1)^0.5 )
    18. Algebra 代数 >>

    19. simplify
      taylor( (x^2 - 1)/(x-1) )
    20. expand
      expand( (x-1)^3 )

    21. factorization
      factor( x^4-1 )
    22. factorizing
      factor( x^2+3*x+2 )
    23. tangent

    24. tangent equation at x=1
      tangent( sin(x),x=1 )
    25. tangentplot( ) show dynamic tangent line when your mouse over the curve.
      tangentplot( sin(x) )

      Convert

      convert( sin(x) to exp) is the same as toexp(sin(x))
    26. convert to exp
      toexp( cos(x) )


    27. convert to trig
      convert exp(x) to trig

    28. convert sin(x) to exp(x),
      convert sin(x) to exp = toexp( sin(x) )

    29. Convert to exp(x)
      toexp(Gamma(2,x))
    30. inverse function

    31. input sin(x), click the inverse button
      inverse( sin(x) )
      check its result by clicking the inverse button again.
      In order to show multi-value, use the inverse equation instead function.

      inverse equation

    32. inverse equation to show multivalue if it has.
      inverse( sin(x)=y )
      check its result by clicking the inverse button again.

      polynomial

    33. the unit polynomial
      poly(3,x) gives the unit polynomial x^3+x^2+x+1.

    34. Hermite polynomial
      hermite(3,x) gives the Hermite polynomial while hermite(3) gives Hermite number.

    35. harmonic polynomial
      harmonic(-3,1,x) = harmonic(-3,x)
      harmonic(-3,x)

    36. the zeta polynomial
      zeta(-3,x) is the zeta polynomial.

    37. expand polynomial
      expand(hermite(3,x))

    38. topoly( ) convert polynomial to polys( ) as holder of polynomial coefficients,
      convert `x^2-5*x+6` to poly = topoly( `x^2-5*x+6` )
    39. activate polys( ) to polynomial
      simplify( polys(1,-5,6,x) )
    40. polyroots( ) is holder of polynomial roots, topolyroot( ) convert a polynomial to polyroots.
      convert (x^2-1) to polyroot = topolyroot(x^2-1)
    41. polysolve( ) numerically solve polynomial for multi-roots.
      polysolve(x^2-1)
    42. nsolve( ) numerically solve for a single root.
      nsolve(x^2-1)
    43. solve( ) for sybmbloic and numeric roots.
      solve(x^2-1)
    44. construct polynomial from roots. activate polyroots( ) to polynomial, and reduce roots to polynomial equation = 0 by click on the simplify( ) button.
      simplify( polyroots(2,3) )

      Number

      When the variable x of polynomial is numnber, it becomes polynomial number, please see Number_Theory section.
    45. Function 函数 >>

      Function 函数
    46. Trigonometry 三角函数
      expand Trigonometry by expandtrig( )

    47. inverse function
      inverse( sin(x) )

    48. plot a multivalue function by the inverse equation
      inverse( sin(x)=y )

    49. expand
      expand( sin(x)^2 )
    50. factor
      factor( sin(x)*cos(x) )

    51. Complex Function 复变函数

      complex2D( ) shows the real and imag curves in real domain x, and complex3D( ) shows complex function in complex domain z, for 20 graphes in one plot.

      special Function

      Calculus 微积分 >>

      Limit

    52. click the lim( ) button to Limit at x->0
      `lim_(x->0) sin(x)/x ` = lim sin(x)/x as x->0 = lim(sin(x)/x)
    53. click the nlim( ) button to numeric limit at x->0
    54. click the limoo( ) button to Limit at x->oo
      `lim _(x->oo) log(x)/x` = lim( log(x)/x as x->inf )

    55. one side limit, left or right side:
      lim(exp(-x),x,0,right)

      Derivatives

    56. Differentiate
      `d/dx sin(x)` = d(sin(x))

    57. Second order derivative
      `d^2/dx^2 sin(x)` = d(sin(x),x,2) = d(sin(x) as x order 2)

    58. sin(0.5,x) is inert holder of the 0.5 order derivative `sin^((0.5))(x)`, it can be activated by simplify( ):
      simplify( sin(0.5,x) )
    59. Derivative as x=1
      `d/dx | _(x->1) x^6` = d( x^6 as x->1 )

    60. Second order derivative as x=1
      `d^2/dx^2| _(x->1) x^6` = d(x^6 as x->1 order 2) = d(x^6, x->1, 2)

      Fractional calculus

    61. Fractional calculus
    62. semiderivative
      `d^(0.5)/dx^(0.5) sin(x)` = d(sin(x),x,0.5) = d( sin(x) as x order 0.5) = semid(sin(x))

    63. input sin(0.5,x) as the 0.5 order derivative of sin(x) for
      `sin^((0.5))(x)` = `sin^((0.5))(x)` = sin(0.5,x)
    64. simplify sin(0.5,x) as the 0.5 order derivative of sin(x),
      `sin^((0.5))(x)` = simplify(sin(0.5,x))
    65. 0.5 order derivative again
      `d^(0.5)/dx^(0.5) d^(0.5)/dx^(0.5) sin(x)` = d(d(sin(x),x,0.5),x,0.5)
    66. Minus order derivative
      `d^(-0.5)/dx^(-0.5) sin(x)` = d(sin(x),x,-0.5)

    67. inverse the 0.5 order derivative of sin(x) function
      f(-1)( sin(0.5)(x) ) = inverse(sin(0.5,x))

    68. Derive the product rule
      `d/dx (f(x)*g(x)*h(x))` = d(f(x)*g(x)*h(x))

    69. … as well as the quotient rule
      `d/dx f(x)/g(x)` = d(f(x)/g(x))

    70. for derivatives
      `d/dx ((sin(x)* x^2)/(1 + tan(cot(x))))` = d((sin(x)* x^2)/(1 + tan(cot(x))))

    71. Multiple ways to derive functions
      `d/dy cot(x*y)` = d(cot(x*y) ,y)

    72. Implicit derivatives, too
      `d/dx (y(x)^2 - 5*sin(x))` = d(y(x)^2 - 5*sin(x))

    73. the nth derivative formula
      ` d^n/dx^n (sin(x)*exp(x)) ` = nthd(sin(x)*exp(x))
    74. differentiate graphically

      some functions cannot be differentiated or integrated symbolically, but can be semi-differentiated and integrated graphically in diff2D. e.g.

      Integrals

    75. indefinite integrate `int` sin(x) dx = integrate(sin(x))

    76. enter a function sin(x), then click the ∫ button to integrate
      `int(cos(x)*e^x+sin(x)*e^x)\ dx` = int(cos(x)*e^x+sin(x)*e^x)
      `int tan(x)\ dx` = integrate tan(x) = int(tan(x))

      integrator

      If integrate( ) cannot do, please try integrator(x)
    77. integrator(sin(x))
    78. enter sin(x), then click the ∫ dx button to integrator

    79. Multiple integrate
      `int int (x + y)\ dx dy` = int( int(x+y, x),y)
      `int int exp(-x)\ dx dx` = integrate(exp(-x) as x order 2)

    80. Definite integration
      `int _1^3` (2*x + 1) dx = int(2x+1,x,1,3) = int(2x+1 as x from 1 to 3)

    81. Improper integral
      `int _0^(pi/2)` tan(x) dx =int(tan(x),x,0,pi/2)

    82. Infinite integral
      `int _0^oo 1/(x^2 + 1)` dx = int(1/x^2+1),x,0,oo)

    83. Definite integration
      `int_0^1` sin(x) dx = integrate( sin(x),x,0,1 ) = integrate sin(x) as x from 0 to 1

      fractional integrate

    84. semi integrate, semiint( )
      `int sin(x) \ dx^(1/2)` = int(sin(x),x,1/2) = int sin(x) as x order 1/2 = semiint(sin(x)) = d(sin(x),x,-1/2)

    85. indefinite semiintegrate
      `int sin(x)\ dx^0.5` = `d^(-0.5)/dx^(-0.5) sin(x)` = int(sin(x),x,0.5) = semiint(sin(x))

    86. Definite fractional integration
      `int_0^1` sin(x) `(dx)^0.5` = integrate( sin(x),x,0.5,0,1 ) = semiintegrate sin(x) as x from 0 to 1

    87. Exact answers
      `int (2x+3)^7` dx = int (2x+3)^7

    88. numeric computation by click on the "~=" button
      n( `int _0^1` sin(x) dx ) = nint(sin(x),x,0,1) = nint(sin(x))

      integrate graphically

      some functions cannot be differentiated or integrated symbolically, but can be semi-differentiated and integrated graphically in integrate2D. e.g.
    89. Equation 方程 >>

      inverse an equation

    90. inverse an equation to show multivalue curve.
      inverse( sin(x)=y )
      check its result by clicking the inverse button again.

      polynomial equation

    91. polyroots( ) is holder of polynomial roots, topolyroot( ) convert a polynomial to polyroots.
      convert (x^2-1) to polyroot = topolyroot(x^2-1)

    92. polysolve( ) numerically solve polynomial for multi-roots.
      polysolve(x^2-1)

    93. construct polynomial from roots. activate polyroots( ) to polynomial, and reduce roots to polynomial equation = 0 by click on the simplify( ) button.
      simplify( polyroots(2,3) )

    94. solve( ) for sybmbloic and numeric roots.
      solve(x^2-1)
      solve( x^2-5*x-6 )

    95. solve equation and inequalities, by default, equation = 0 for default unknown x if the unknown omit.
      solve( x^2+3*x+2 )

    96. Symbolic roots
      solve( x^2 + 4*x + a )

    97. Complex roots
      solve( x^2 + 4*x + 181 )

    98. solve equation for x.
      solve( x^2-5*x-6=0,x )

    99. numerically root
      nsolve( x^3 + 4*x + 181 )

    100. nsolve( ) numerically solve for a single root.
      nsolve(x^2-1)

      Algebra Equation

      solve( ) algebra equation, e.g. exp( ) equation,
    101. Solve nonlinear equations:
      solve(exp(x)+exp(-x)=4)

      system of equations

    102. system of 2 equations with 2 unknowns x and y by default if the unknowns omit.
      solve( 2x+3y-1=0,3x+2y-1=0 )

      Diophantine equation

    103. it is that number of equation is less than number of the unknown, e.g. one equation with 2 unknowns x and y.
      solve( 2x-3y-3=0, x,y)

      congruence equation

      By definition of congruence, a x ≡ b (mod m) if a x − b is divisible by m. Hence, a x ≡ b (mod m) if a x − b = m y, for some integer y. Rearranging the equation to the equivalent form of Diophantine equation a x − m y = b.
      x^2-3x-2=2*(mod 2)
      x^2-3x-2=2mod(2)

      Modulus equation

    104. solve( ) Modulus equation for the unknown x inside the mod( ) function, e.g.
      input mod(x,2)=1 for
      x mod 2 = 1
      click the solve button

      Probability_equation

    105. solve( ) Probability equation for the unknown k inside the Probability function P( ),
      solve( P(x>k)=0.2, k)

      recurrence_equation

    106. rsolve( ) recurrence and functional and difference equation for y(x)
      y(x+1)=y(x)+x
      y(x+1)=y(x)+1/x
    107. fsolve( ) recurrence and functional and difference equation for f(x)
      f(x+1)=f(x)+x
      f(x+1)=f(x)+1/x

      functional_equation

    108. rsolve( ) recurrence and functional and difference equation for y(x)
      y(a+b)=y(a)*y(b)
      y(a*b)=y(a)+y(b)
    109. fsolve( ) recurrence and functional and difference equation for f(x)
      f(a+b)=f(a)*f(b)
      f(a*b)=f(a)+f(b)

      difference equation

    110. rsolve( ) recurrence and functional and difference equation for y(x)
      y(x+1)-y(x)=x
      y(x+2)-y(x+1)-y(x)=0
    111. fsolve( ) recurrence and functional and difference equation for f(x)
      f(x+1)-f(x)=x
      f(x+2)- f(x+1)-f(x)=0

      Inequalities

    112. solve( ) Inequalities for x.
      solve( 2*x-1>0 )
      solve( x^2+3*x+2>0 )

      differential equation

      ODE( ) and dsolve( ) and lasove( ) solve ordinary differential equation (ODE) to unknown y.
    113. Solve linear ordinary differential equations:
      y'=x*y+x
      y'= 2y
      y'-y-1=0

    114. Solve nonlinear ordinary differential equations:
      (y')^2-2y^2-4y-2=0

      dsolve( y' = sin(x-y) )

      dsolve( y(1,x)=cos(x-y) )

      dsolve( ds(y)=tan(x-y) )

    115. 2000 examples of Ordinary differential equation (ODE)
    116. more examples in bugs

      solve graphically

      The odeplot( ) can be used to visualize individual functions, First and Second order Ordinary Differential Equation over the indicated domain. Input the right hand side of Ordinary Differential Equations, y"=f(x,y,z), where z for y', then click the checkbox. by default it is first order ODE.
      y''=y'-y for second order ODE

      integral equation

      indefinite integral equation

    117. indefinite integral equation
      input ints(y) -2y = exp(x) for
      `int y` dx - 2y = exp(x)

      definite integral equation

    118. definite integral equation
      input integrates(y(t)/sqrt(x-t),t,0,x) = 2y for
      `int_0^x (y(t))/sqrt(x-t)` dt = 2y

      differential integral equation

      input ds(y)-ints(y) -y-exp(x)=0 for
      `dy/dx-int y dx -y-exp(x)=0`

      fractional differential equation

      dsolve( ) also solves fractional differential equation
    119. Solve linear equations:
      `d^0.5/dx^0.5 y = 2y`
      `d^0.5/dx^0.5 y -y - E_(0.5) (4x^0.5) = 0`
      `d^0.5/dx^0.5 y -y -exp(4x) = 0`
      `(d^0.5y)/dx^0.5=sin(x)`

    120. Solve nonlinear equations:
      `d^0.5/dx^0.5 y = y^2`

      fractional integral equation

    121. `d^-0.5/dx^-0.5 y = 2y`

      fractional differential integral equation

      ds(y,x,0.5)-ints(y,x,0.5) -y-exp(x)=0
      `(d^0.5y)/(dx^0.5)-int y (dx)^0.5 -y-exp(x)=0`

      complex order differential equation

    122. `(d^(1-i) y)/dx^(1-i)-2y-exp(x)=0`

      variable order differential equation

      `(d^sin(x) y)/dx^sin(x)-2y-exp(x)=0`

      system of differential equations

    123. system of 2 equations with 2 unknowns x of the 0.5 order and y of the 0.8 order with a variable t.
      dsolve( x(1,t)=x,y(1,t)=x-y )

      partial differental equation

      PDE( ) and pdsolve( ) solve partial differental equation with two variables t and x, then click the plot2D button to plot solution, pull the t slider to change the t value. click the plot3D button for 3D graph.
    124. Solve a linear equation:
      `dy/dt = dy/dx-2y`

    125. Solve a nonlinear equation:
      `dy/dt = dy/dx*y^2`

      `dy/dt = dy/dx-y^2`

      `(d^2y)/(dt^2) = 2* (d^2y)/(dx^2)-y^2-2x*y-x^2`

      fractional partial differental equation

      PDE( ) and pdsolve( ) solve fractional partial differental equation.
    126. Solve linear equations:
      `(d^0.5y)/dt^0.5 = dy/dx-2y`

    127. Solve nonlinear equations:
      `(d^0.5y)/dt^0.5 = 2* (d^0.5y)/dx^0.5*y^2`

      `(d^0.5y)/dt^0.5 = 2* (d^0.5y)/dx^0.5-y^2`

      `(d^1.5y)/(dt^1.5) + (d^1.5y)/(dx^1.5)-2y^2-4x*y-2x^2 =0`

    128. More examples are in Analytical Solution of Fractional Differential Equations

      test solution

      test solution for algebaic equation to the unknown x by test(solution,eq,x) or click the test( ) button.

    129. test(1,x^2-1=0,x)
      test( -1, x^2-5*x-6 )

      test solution for differential equation to the unknown y by test( ) or click the test( ) button.

    130. test( exp(2x), `dy/dx=2y` )
    131. test( exp(4x), `(d^0.5y)/dx^0.5=2y` )
    132. Discrete Math 离散数学 >>

      The default index variable in discrete math is k.
    133. Input harmonic(2,x), click the defintion( ) button to show its defintion, check its result by clicking the simplify( ) button, then click the limoo( ) button for its limit as x->oo.

      Difference

      Δ(k^2) = difference(k^2)
    134. Check its result by the sum( ) button

      Summation ∑

      Indefinite sum

      ∑ k = sum(k)
    135. Check its result by the difference( ) button
      Δ sum(k) = difference( sum(k) )
    136. In order to auto plot, the index variable should be x,
      `sum_x x` = sum(x,x)

      definite sum

    137. Definite sum = Partial sum x from 1 to x, e.g.
      1+2+ .. +x = `sum _(k=1) ^x k` = sum(k,k,1,x)
    138. Definite sum, sum x from 1 to 5, e.g.
      1+2+ .. +5 = ∑(x,x,0,5) = sum(x,x,0,5)
    139. Infinite sum x from 0 to inf, e.g.
      1/0!+1/1!+1/2!+ .. +1/x! = sum 1/(x!) as x->oo
      sum(x^k,k,0,5)

    140. Definite sum with parameter x as upper limit

      sum(k^2, k,0, x)
    141. Check its result by the difference( ) button, and then the expand( ) button.
    142. convert to sum series definition
      tosum( exp(x) )
    143. expand above sum series by the expand( ) button
      expand( tosum(exp(x)) )

    144. Indefinite sum

      ∑ k
      sum( x^k/k!,k )
    145. partial sum of 1+2+ .. + k = ∑ k = partialsum(k)
    146. Definite sum of 1+2+ .. +5 = ∑ k

      partial sum with parameter upper limit x

      sum(1/k^2,k,1,x)

      infinite sum

    147. if the upper limit go to infinite, it becomes infinite sum
      Infinite sum of 1/1^2+1/2^+1/3^2 .. +1/k^2+... = lim(sum( 1/k^2,k,1,x) as x->oo) = sum( 1/k^2,k,1,oo )
      Infinite sum of 1/0!+1/1!+1/2! .. +1/k!+... = lim(sum( 1/k!,k,0,x) as x->oo) = sum( 1/k!,k,0,oo )
    148. Series 级数

    149. convert to sum series definition
      tosum( exp(x) ) = toseries( exp(x) )
    150. check its result by clicking the simplify( ) button
      simplify( tosum( exp(x) ))

    151. expand above sum series
      expand( tosum(exp(x)) )

    152. compare to Taylor series
      taylor( exp(x), x=0, 8)
    153. compare to series
      series( exp(x) )

    154. Taylor series expansion as x=0,
      taylor( exp(x) as x=0 ) = taylor(exp(x))

      by default x=0.

    155. series expand not only to taylor series,
      series( exp(x) )
      but aslo to other series expansion,
      series( zeta(2,x) )

      Product ∏

    156. prod(x,x)

    157. `prod x`

      Definition 定义式 >>

    158. definition of function
      definition( exp(x) )
    159. check its result by clicking the simplify( ) button
      simplify( def(exp(x)) )
    160. series definition

    161. convert to series definition
      toseries( exp(x) )
    162. check its result by clicking the simplify( ) button
      simplify( tosum(exp(x)) )
    163. integral definition

    164. convert to integral definition
      toint( exp(x) )
    165. check its result by clicking the simplify( ) button
      simplify( toint(exp(x)) )
    166. Numeric math 数值数学 >>

      Number Theory 数论 >>

      When the variable x of polynomial is numnber, it becomes number, e.g.
    167. poly number
      poly(3,2)
    168. Hermite number
      hermite(3,2)
    169. harmonic number
      harmonic(-3,2)
      harmonic(-3,2,4)
      harmonic(1,1,4) = harmonic(1,4) = harmonic(4)
    170. Bell number
      n(bell(5))
    171. double factorial 6!!
    172. Calculate the 4nd prime prime(4)
    173. is prime number? isprime(12321)
    174. next prime greater than 4 nextprime(4)
    175. binomial number `((4),(2))`
    176. combination number `C_2^4`
    177. harmonic number `H_4`
    178. congruence equation
      3x-1=2*(mod 2)
      x^2-3x-2=2mod( 2)
    179. modular equation
      Enter mod(x-1,10)=2 for (x-1) mod 10=2
    180. Diophantine equation
      number of equation is less than number of the unknown, e.g. one equation with 2 unkowns x and y,
      solve( 3x-2y-2=0, x,y )
    181. Probability 概率 >>

    182. P( ) is probability of standard normal distribution
      P(x<0.8)
    183. Phi( ) is standard normal distribution function
      `Phi(x)`
    184. solve Probability equation for k
      solve(P(x>k)=0.2,k)
    185. Statistics 统计

      We can sort list( ), add numbers together with total(list()), max(list()), min(list()), size(list()).

      Multi elements >>

      We can put multi elements together with list(), vector(), and. Most operation in them is the same as in one element, one by one. e.g. +,-,*,/, differentiation, integration, sum, etc. We count its elements with size(), as same as to count elements in function. We only talk about special properties as follows.
    186. list( )
      The list element can be symbol, formula or function. We can sort list( ), e.g.
      sort(list(b,x,sin(x)))
      add numbers to list. e.g.
      2+list(a,b,c)
    187. vector( )
      It has direction. the position of the element is fix so we cannot sort it. vector is number with direction. two vector( ) in the same dimention can be operated by +, -, *, /, the result can be checked by its reverse operation. the system auto plot the 2-dimentional vector.
      vector(2,4)/vector(1,2) =2
      vector(2,4)=vector(1,2)*2
    188. and
      We can plot multi curves with the and. e.g. plot(x and x*x). the position of the element is fix so we cannot sort it.

      Plot 制图 >>

    189. plane curve 2D
    190. surface 2D

      3D graph 立体图 plot 3D >>

    191. space curve 3D
    192. surface 3D
    193. surface 4D

      Animation 动画 >>

    194. Classication by plot function 按制图函数分类
    195. Classication by appliaction 按应用分类

      programming 编程 >>

      There are many coding :
    196. math coding 数学编程
    197. html + javaScript coding 网页编程
    198. online programming 在线编程
    199. plot 函数图
    200. rose 玫瑰花
    201. check code 验证码
    202. calculator 计算器
    203. sci calculator 科学计算器
    204. color 颜色取色器
    205. Chinese calendar 农历日历
    206. calendar 日历
      more examples

      bugs >>

      There are over 500 bugs in another software but they are no problem in MathHand.com
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