j6/lib/calculus.js

Javascript scientific library (like R, NumPy, Matlab)

module.exports = function (j6) {
// 待實現：自動微分 http://chaoya.info/articles/2016-01-03-ad.html
// 既不会产生很高的计算代价，也不会带来额外的误差
// npm 範例: https://github.com/ehaas/js-autodiff
// 測試範例：https://github.com/ehaas/js-autodiff/blob/master/examples.js

/* eslint-disable no-undef */

// =========== Calculus =================
j6.dx = 0.01

// 微分 differential calculus
j6.diff = function (f, x, dx = j6.dx) {
  var dy = f(x.add(dx)).sub(f(x.sub(dx)))
  return dy.div(dx.mul(2))
}

// 積分 integral calculus
j6.i = j6.integral = function (f, a, b, dx = j6.dx) {
  var area = 0.0
  for (var x = a; x < b; x = x + dx) {
    area = area + f(x).mul(dx)
  }
  return area
}

// 偏微分 partial differential calculus
// f=[f1,f2,....] , x=[x1,x2,...] , dx=[dx1,dx2,....]
j6.pdiff = j6.pdifferential = function (f, x, i) {
  f = j6.fa(f)
  var dx = j6.fill(x.length, 0)
  dx[i] = j6.dx
  var df = f(x.add(dx)).sub(f(x.sub(dx)))
  return df.div(dx.norm().mul(2))
}

// multidimensional integral calculus
// f=[f1,f2,....] , a=[a1,a2,...] , b=[b1,b2,....]
j6.pintegral = function (f, a, b) {
}

// 梯度 gradient : grad(f,x)=[pdiff(f,x,0), .., pdiff(f,x,n)]
j6.grad = j6.gradient = function (f, x) {
  var gf = []
  for (var i = 0; i < x.length; i++) {
    gf[i] = j6.pdiff(f, x, i)
  }
  return gf
}

// 散度 divergence : div(f,x) = sum(pdiff(j6[i],x,i))
j6.divergence = function (F, x) {
  var f = []
  var d = []
  for (var i = 0; i < x.length; i++) {
    f[i] = (xt) => F(xt)[i]
    d[i] = F.pdiff(f[i], x, i)
  }
  return d.sum()
}

// 旋度 curl : curl(F) = div(F)xF
j6.curl = j6.curlance = function (F, x) {
}

// 線積分： int F●dr = int F(r(t))●r'(t) dt
j6.vintegral = function (F, r, a, b, dt) {
  dt = dt || F.dx
  var sum = 0
  for (var t = a; t < b; t += dt) {
    sum += F(r(t)).dot(r.diff(t))
  }
  return sum
}

// 定理：int j6●dr  = int(sum(Qxi dxi))

// 向量保守場： j6=grad(f) ==> int j6●dr = f(B)-f(j6)

// 定理： 向量保守場 j6, pdiff(Qxi,xj) == pdiff(Qxj,xi)  for any i, j
// ex: j6=[x^2y, x^3] 中， grad(j6)=[x^2, 3x^2] 兩者不相等，所以不是保守場

// 格林定理：保守場中 《線積分=微分後的區域積分》
// int P dx + Q dy = int int pdiff(Q,x) - pdiff(P, y) dx dy

// 散度定理：通量 = 散度的區域積分
// 2D : int j6●n ds = int int div j6 dx dy
// 3D : int int j6●n dS = int int int div j6 dV

// 史托克定理：  j6 在曲面 S 上的旋度總和 = j6 沿著邊界曲線 C 的線積分
// int int_D curl(j6)●n dS = int_C j6●dr
}