{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Proving a \"calculus approach\" result"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In Table 4.2, Hughes & Hase claim that for the function $Z(A,B) = k\\frac{A^n}{B^m}$, the fractional uncertainty in $Z$ is given by \n",
    "$$\\frac{\\alpha_z}{Z} = \\sqrt{\\left(n\\frac{\\alpha_A}{A}\\right)^2 + \\left(m\\frac{\\alpha_B}{B}\\right)^2}. $$\n",
    "Use the calculus approach to prove this result."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We know that\n",
    "\n",
    "$$\n",
    "\\alpha_Z = \\sqrt{\\left(\\alpha_Z^A\\right)^2 + \\left(\\alpha_Z^B\\right)^2}, \n",
    "$$\n",
    "\n",
    "where $\\alpha_Z^A$ is the uncertainty in $Z$ due to the uncertainty in $A$, and $\\alpha_Z^B$ is the uncertainty in $Z$ due to the uncertainty in $B$. The calculus approach says that\n",
    "\n",
    "$$ \n",
    "\\alpha_Z^A = \\left(\\frac{\\partial Z}{\\partial A}\\right)\\alpha_A\\quad\\mbox{and}\\quad\n",
    " \\alpha_Z^B= \\left(\\frac{\\partial Z}{\\partial B}\\right)\\alpha_B.\n",
    " $$\n",
    "\n",
    "If $Z = k\\frac{A^n}{B^m}$, then \n",
    "\n",
    "$$\n",
    "\\alpha^A_Z = nk\\frac{A^{n-1}}{B^m} \\alpha_A = n\\frac{Z}{A}\\alpha_A,\\quad\\mbox{and}\\quad\n",
    "\\alpha^B_Z = -m\\frac{A^n}{B^{m+1}}\\alpha_B = -m\\frac{Z}{B}\\alpha_B. \n",
    "$$\n",
    "\n",
    "Combining these results\n",
    "\n",
    "\\begin{eqnarray*}\n",
    "\\alpha_Z &=& \\sqrt{\\left(\\alpha_Z^A\\right)^2 + \\left(\\alpha_Z^B\\right)^2} \\\\\n",
    "         &=& \\sqrt{\\left(n\\frac{Z}{A}\\alpha_A\\right)^2 + \\left(-m\\frac{Z}{B}\\alpha_B\\right)^2}\n",
    "\\end{eqnarray*}\n",
    "\n",
    "or\n",
    "\n",
    "$$\n",
    "        \\frac{\\alpha_Z}{Z}= \\sqrt{\\left(n\\frac{\\alpha_A}{A}\\right)^2 + \\left(m\\frac{\\alpha_B}{B}\\right)^2},\n",
    "$$\n",
    "\n",
    "as desired."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}