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anmolj7/calc.ipynb

Last active January 21, 2020 07:31
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calc.ipynb
{
"cells": [
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Collecting sympy\n",
"\u001b[?25l Downloading https://files.pythonhosted.org/packages/ce/5b/acc12e3c0d0be685601fc2b2d20ed18dc0bf461380e763afc9d0a548deb0/sympy-1.5.1-py2.py3-none-any.whl (5.6MB)\n",
"\u001b[K 100% |████████████████████████████████| 5.6MB 311kB/s eta 0:00:01\n",
"\u001b[?25hCollecting mpmath>=0.19 (from sympy)\n",
"\u001b[?25l Downloading https://files.pythonhosted.org/packages/ca/63/3384ebb3b51af9610086b23ea976e6d27d6d97bf140a76a365bd77a3eb32/mpmath-1.1.0.tar.gz (512kB)\n",
"\u001b[K 100% |████████████████████████████████| 522kB 254kB/s ta 0:00:01\n",
"\u001b[?25hBuilding wheels for collected packages: mpmath\n",
" Running setup.py bdist_wheel for mpmath ... \u001b[?25ldone\n",
"\u001b[?25h Stored in directory: /home/emperoraj/.cache/pip/wheels/63/9d/8e/37c3f6506ed3f152733a699e92d8e0c9f5e5f01dea262f80ad\n",
"Successfully built mpmath\n",
"Installing collected packages: mpmath, sympy\n",
"Successfully installed mpmath-1.1.0 sympy-1.5.1\n"
]
}
],
"source": [
"!pip3 install sympy --user"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [],
"source": [
"import sympy\n",
"sympy.init_printing(use_latex=True)"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [],
"source": [
"#Creating symbolic variables\n",
"x, y, u, v = sympy.symbols('x y u v') # At this point I wonder who's the guy\n",
"# who started the convention of using x and y in math xD Also, I am used to \n",
"# u and v because of blackpenredpen"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Let's solve this simple limit :P\n",
"$$\\lim_{x \\to 0^+}\\left(\\frac{\\sin{\\left(x \\right)}}{x}\\right)$$"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle 1$"
],
"text/plain": [
"1"
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sympy.limit(sympy.sin(x)/x, x, 0)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Starting with \"simple\" derrivatives "
]
},
{
"cell_type": "code",
"execution_count": 28,
"metadata": {},
"outputs": [],
"source": [
"function = sympy.sin(180*x)**2*sympy.exp(2*x)"
]
},
{
"cell_type": "code",
"execution_count": 29,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle e^{2 x} \\sin^{2}{\\left(180 x \\right)}$"
],
"text/plain": [
" 2⋅x 2 \n",
"ℯ ⋅sin (180⋅x)"
]
},
"execution_count": 29,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"function"
]
},
{
"cell_type": "code",
"execution_count": 32,
"metadata": {},
"outputs": [],
"source": [
"dv = sympy.diff(function, x, 1)"
]
},
{
"cell_type": "code",
"execution_count": 34,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle 2 e^{2 x} \\sin^{2}{\\left(180 x \\right)} + 360 e^{2 x} \\sin{\\left(180 x \\right)} \\cos{\\left(180 x \\right)}$"
],
"text/plain": [
" 2⋅x 2 2⋅x \n",
"2⋅ℯ ⋅sin (180⋅x) + 360⋅ℯ ⋅sin(180⋅x)⋅cos(180⋅x)"
]
},
"execution_count": 34,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"dv #Yeah, simple actually :P"
]
},
{
"cell_type": "code",
"execution_count": 41,
"metadata": {},
"outputs": [],
"source": [
"function = sympy.exp(x**x)"
]
},
{
"cell_type": "code",
"execution_count": 42,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle e^{x^{x}}$"
],
"text/plain": [
" ⎛ x⎞\n",
" ⎝x ⎠\n",
"ℯ "
]
},
"execution_count": 42,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"function"
]
},
{
"cell_type": "code",
"execution_count": 44,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle x^{x} \\left(\\log{\\left(x \\right)} + 1\\right) e^{x^{x}}$"
],
"text/plain": [
" ⎛ x⎞\n",
" x ⎝x ⎠\n",
"x ⋅(log(x) + 1)⋅ℯ "
]
},
"execution_count": 44,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sympy.diff(function, x, 1) #Math issooooo beautifullllll"
]
},
{
"cell_type": "code",
"execution_count": 45,
"metadata": {},
"outputs": [],
"source": [
"#Lets subsitute the value of x with 9"
]
},
{
"cell_type": "code",
"execution_count": 46,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle 2 e^{2 x} \\sin^{2}{\\left(180 x \\right)} + 360 e^{2 x} \\sin{\\left(180 x \\right)} \\cos{\\left(180 x \\right)}$"
],
"text/plain": [
" 2⋅x 2 2⋅x \n",
"2⋅ℯ ⋅sin (180⋅x) + 360⋅ℯ ⋅sin(180⋅x)⋅cos(180⋅x)"
]
},
"execution_count": 46,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"dv"
]
},
{
"cell_type": "code",
"execution_count": 48,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle 360 e^{18} \\sin{\\left(1620 \\right)} \\cos{\\left(1620 \\right)} + 2 e^{18} \\sin^{2}{\\left(1620 \\right)}$"
],
"text/plain": [
" 18 18 2 \n",
"360⋅ℯ ⋅sin(1620)⋅cos(1620) + 2⋅ℯ ⋅sin (1620)"
]
},
"execution_count": 48,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"dv.subs(x, 9)"
]
},
{
"cell_type": "code",
"execution_count": 50,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle -9958200450.54737$"
],
"text/plain": [
"-9958200450.54737"
]
},
"execution_count": 50,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"#Evaluting .. \n",
"dv.subs(x, 9).evalf() # I don't know if it's the right answer and I won't bother to check :P"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Now let's integrate the derrivative and see if we can get back to the origin function we defined"
]
},
{
"cell_type": "code",
"execution_count": 55,
"metadata": {
"scrolled": true
},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle 2 e^{2 x} \\sin^{2}{\\left(180 x \\right)} + 360 e^{2 x} \\sin{\\left(180 x \\right)} \\cos{\\left(180 x \\right)}$"
],
"text/plain": [
" 2⋅x 2 2⋅x \n",
"2⋅ℯ ⋅sin (180⋅x) + 360⋅ℯ ⋅sin(180⋅x)⋅cos(180⋅x)"
]
},
"execution_count": 55,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"dv"
]
},
{
"cell_type": "code",
"execution_count": 57,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle e^{2 x} \\sin^{2}{\\left(180 x \\right)}$"
],
"text/plain": [
" 2⋅x 2 \n",
"ℯ ⋅sin (180⋅x)"
]
},
"execution_count": 57,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sympy.integrate(dv, x) #Works as expected :P"
]
},
{
"cell_type": "code",
"execution_count": 58,
"metadata": {},
"outputs": [],
"source": [
"# Let's try what it answers when we integrate x^x which everybody knows is not possible :P"
]
},
{
"cell_type": "code",
"execution_count": 59,
"metadata": {},
"outputs": [],
"source": [
"function = x**x"
]
},
{
"cell_type": "code",
"execution_count": 60,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle x^{x}$"
],
"text/plain": [
" x\n",
"x "
]
},
"execution_count": 60,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"function"
]
},
{
"cell_type": "code",
"execution_count": 61,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\int x^{x}\\, dx$"
],
"text/plain": [
"⌠ \n",
"⎮ x \n",
"⎮ x dx\n",
"⌡ "
]
},
"execution_count": 61,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sympy.integrate(function, x)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Uh huh, sympy be like: <img src=\"https://encrypted-tbn0.gstatic.com/images?q=tbn%3AANd9GcSei1-zpMPPO7Dmmr3XBC_afud4wJL2Cb95qGP19NNhkb7iDA5k\">"
]
},
{
"cell_type": "code",
"execution_count": 77,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\frac{d}{d x} y{\\left(x \\right)} = \\operatorname{acos}{\\left(x \\right)}$"
],
"text/plain": [
"d \n",
"──(y(x)) = acos(x)\n",
"dx "
]
},
"execution_count": 77,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# A simple ODE\n",
"x, l = sympy.symbols('x lambda')\n",
"\n",
"y = sympy.Function('y')(x)\n",
"\n",
"dydx = y.diff(x)\n",
"expression = sympy.Eq(dydx, sympy.acos(x))\n",
"expression"
]
},
{
"cell_type": "code",
"execution_count": 78,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle y{\\left(x \\right)} = C_{1} + x \\operatorname{acos}{\\left(x \\right)} - \\sqrt{1 - x^{2}}$"
],
"text/plain": [
" ________\n",
" ╱ 2 \n",
"y(x) = C₁ + x⋅acos(x) - ╲╱ 1 - x "
]
},
"execution_count": 78,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sympy.dsolve(expression)"
]
},
{
"cell_type": "code",
"execution_count": 82,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\frac{\\partial}{\\partial t} u{\\left(x,t \\right)} = \\frac{\\partial}{\\partial x} u{\\left(x,t \\right)}$"
],
"text/plain": [
"∂ ∂ \n",
"──(u(x, t)) = ──(u(x, t))\n",
"∂t ∂x "
]
},
"execution_count": 82,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"x,t = sympy.symbols('x t')\n",
"u = sympy.Function('u')\n",
"pde = sympy.Eq(sympy.diff(u(x,t),t), sympy.diff(u(x,t),x))\n",
"pde"
]
},
{
"cell_type": "code",
"execution_count": 84,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle u{\\left(x,t \\right)} = F{\\left(t + x \\right)}$"
],
"text/plain": [
"u(x, t) = F(t + x)"
]
},
"execution_count": 84,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sympy.pdsolve(pde)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Itegration using scipy module"
]
},
{
"cell_type": "code",
"execution_count": 86,
"metadata": {},
"outputs": [],
"source": [
"from scipy import integrate\n",
"from numpy import exp"
]
},
{
"cell_type": "code",
"execution_count": 87,
"metadata": {},
"outputs": [],
"source": [
"f = lambda x: exp(x)"
]
},
{
"cell_type": "code",
"execution_count": 91,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\left( 145.69487727411754, \\ 1.6175380731966177e-12\\right)$"
],
"text/plain": [
"(145.69487727411754, 1.6175380731966177e-12)"
]
},
"execution_count": 91,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"integrate.quad(f, 1, 5)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### The difference between integration with scipy and with sympy is the symbol computation i.e, sympy just ouputs the evaluation, which is not what we require in some cases. Suppose I made a smart glass with OCR which could solve questions, I'd rather want sympy instead of scipy to cheat in an exam. If I need to compute something, I can just use the evalf function in sympy, it would just make my life easier XD"
]
}
],
"metadata": {
"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": "3.7.5"
}
},
"nbformat": 4,
"nbformat_minor": 2
}
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