{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Higher dimensions"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Finite difference generalizes readily to multiple dimensions, by computing the elements of the gradient individually:\n",
"\n",
"$$\n",
"\\frac{\\partial f}{\\partial x_i} ≈ \\frac{f(x + heᵢ) - f(x - heᵢ)} {2h} $$\n",
"\n",
"Boundary cases are handelled similarly but with the bounded dimension calculated by forward / backward difference.\n"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"\n",
"\n",
"\n",
" \n",
"\n",
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"# prompt: Give me an example of an ugly 2D function, plot it as a surface in plotly and show its gradients in arrows\n",
"\n",
"import numpy as np\n",
"import plotly.graph_objects as go\n",
"import plotly.figure_factory as ff\n",
"\n",
"# Define the function\n",
"def ugly_function(x, y):\n",
" return (np.sin(x) * np.cos(y) + np.cos(x) * np.sin(y)) * np.exp(-(x**2 + y**2) / 10)\n",
"\n",
"# Create a grid of x and y values\n",
"x = np.linspace(-5, 5, 50)\n",
"y = np.linspace(-5, 5, 50)\n",
"X, Y = np.meshgrid(x, y)\n",
"\n",
"# Calculate the function values and gradients\n",
"Z = ugly_function(X, Y)\n",
"dZdx, dZdy = np.gradient(Z)\n",
"\n",
"# Create the surface plot\n",
"fig = go.Figure(data=[go.Surface(z=Z, x=X, y=Y)])\n",
"fig.update_layout(title='Ugly 2D Function', autosize=False,\n",
" width=500, height=500,\n",
" margin=dict(l=65, r=50, b=65, t=90))\n",
"\n",
"# Add gradient arrows to the plot\n",
"arrow_x = X[::5, ::5].flatten()\n",
"arrow_y = Y[::5, ::5].flatten()\n",
"arrow_z = Z[::5, ::5].flatten()\n",
"arrow_u = dZdx[::5, ::5].flatten()\n",
"arrow_v = dZdy[::5, ::5].flatten()\n",
"\n",
"\n",
"fig.add_trace(go.Cone(\n",
" x=arrow_x,\n",
" y=arrow_y,\n",
" z=arrow_z,\n",
" u=arrow_u,\n",
" v=arrow_v,\n",
" w=np.zeros_like(arrow_u),\n",
" sizemode=\"absolute\",\n",
" sizeref=0.1,\n",
" showscale=False,\n",
" colorscale='Blues'\n",
"))\n",
"\n",
"fig.show()\n"
]
}
],
"metadata": {
"language_info": {
"name": "python"
}
},
"nbformat": 4,
"nbformat_minor": 2
}