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quarto/_freeze/differentiable_vector_calculus/scalar_functions/execute-results/html.json
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"markdown": "# Scalar functions\n\n\n\nThis section uses these add-on packages:\n\n``` {.julia .cell-code}\nusing CalculusWithJulia\nusing Plots\nusing ForwardDiff\nusing SymPy\nusing Roots\nusing QuadGK\nusing JSON\n```\n\n\nAlso, these methods from the `Contour` package:\n\n``` {.julia .cell-code}\nimport Contour: contours, levels, level, lines, coordinates\n```\n\n\n\n\n---\n\n\nConsider a function $f: R^n \\rightarrow R$. It has multiple arguments for its input (an $x_1, x_2, \\dots, x_n$) and only one, *scalar*, value for an output. Some simple examples might be:\n\n\n\n$$\n\\begin{align}\nf(x,y) &= x^2 + y^2\\\\\ng(x,y) &= x \\cdot y\\\\\nh(x,y) &= \\sin(x) \\cdot \\sin(y)\n\\end{align}\n$$\n\n\nFor two examples from real life consider the elevation Point Query Service (of the [USGS](https://nationalmap.gov/epqs/)) returns the elevation in international feet or meters for a specific latitude/longitude within the United States. The longitude can be associated to an $x$ coordinate, the latitude to a $y$ coordinate, and the elevation a $z$ coordinate, and as long as the region is small enough, the $x$-$y$ coordinates can be thought to lie on a plane. (A flat earth assumption.)\n\n\nSimilarly, a weather map, say of the United States, may show the maximum predicted temperature for a given day. This describes a function that take a position ($x$, $y$) and returns a predicted temperature ($z$).\n\n\nMathematically, we may describe the values $(x,y)$ in terms of a point, $P=(x,y)$ or a vector $\\vec{v} = \\langle x, y \\rangle$ using the identification of a point with a vector. As convenient, we may write any of $f(x,y)$, $f(P)$, or $f(\\vec{v})$ to describe the evaluation of $f$ at the value $x$ and $y$\n\n\n---\n\nReturning to the task at hand, in `Julia`, defining a scalar function is straightforward, the syntax following mathematical notation:\n\n::: {.cell execution_count=5}\n``` {.julia .cell-code}\nf(x,y) = x^2 + y^2\ng(x,y) = x * y\nh(x,y) = sin(x) * sin(y)\n```\n\n::: {.cell-output .cell-output-display execution_count=6}\n```\nh (generic function with 1 method)\n```\n:::\n:::\n\n\nTo call a scalar function for specific values of $x$ and $y$ is also similar to the mathematical case:\n\n::: {.cell execution_count=6}\n``` {.julia .cell-code}\nf(1,2), g(2, 3), h(3,4)\n```\n\n::: {.cell-output .cell-output-display execution_count=7}\n```\n(5, 6, -0.10679997423758245)\n```\n:::\n:::\n\n\nIt may be advantageous to have the values as a vector or a point, as in `v=[x,y]`. Splatting can be used to turn a vector or tuple into two arguments:\n\n::: {.cell execution_count=7}\n``` {.julia .cell-code}\nv = [1,2]\nf(v...)\n```\n\n::: {.cell-output .cell-output-display execution_count=8}\n```\n5\n```\n:::\n:::\n\n\nAlternatively, the function may be defined using a vector argument:\n\n::: {.cell execution_count=8}\n``` {.julia .cell-code}\nf(v) = v[1]^2 + v[2]^2\n```\n\n::: {.cell-output .cell-output-display execution_count=9}\n```\nf (generic function with 2 methods)\n```\n:::\n:::\n\n\nA style required for other packages within the `Julia` ecosystem, as there are many advantages to passing containers of values: they can have arbitrary length, they can be modified inside a function, the functions can be more generic, etc.\n\n\nMore verbosely, but avoiding index notation, we can use multiline functions:\n\n::: {.cell execution_count=9}\n``` {.julia .cell-code}\nfunction g(v)\n x, y = v\n x * y\nend\n```\n\n::: {.cell-output .cell-output-display execution_count=10}\n```\ng (generic function with 2 methods)\n```\n:::\n:::\n\n\nThen we have\n\n::: {.cell execution_count=10}\n``` {.julia .cell-code}\nf(v), g([2,3])\n```\n\n::: {.cell-output .cell-output-display execution_count=11}\n```\n(5, 6)\n```\n:::\n:::\n\n\n---\n\n\nMore elegantly, perhaps – and the approach we will use in this section – is to mirror the mathematical notation through multiple dispatch. If we define `j` for multiple variables, say with:\n\n::: {.cell execution_count=11}\n``` {.julia .cell-code}\nj(x,y) = x^2 - 2x*y^2\n```\n\n::: {.cell-output .cell-output-display execution_count=12}\n```\nj (generic function with 1 method)\n```\n:::\n:::\n\n\nThe we can define an alternative method with just a single variable and use splatting to turn it into multiple variables:\n\n::: {.cell execution_count=12}\n``` {.julia .cell-code}\nj(v) = j(v...)\n```\n\n::: {.cell-output .cell-output-display execution_count=13}\n```\nj (generic function with 2 methods)\n```\n:::\n:::\n\n\nThe we can call `j` with a vector or point:\n\n::: {.cell execution_count=13}\n``` {.julia .cell-code}\nj([1,2])\n```\n\n::: {.cell-output .cell-output-display execution_count=14}\n```\n-7\n```\n:::\n:::\n\n\nor by passing in the individual components:\n\n::: {.cell execution_count=14}\n``` {.julia .cell-code}\nj(1,2)\n```\n\n::: {.cell-output .cell-output-display execution_count=15}\n```\n-7\n```\n:::\n:::\n\n\n---\n\nFollowing a calculus perspective, we take up the questionLine truncated
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