CalculusWithJulia.jl

CalculusWithJulia

A package to accompany notes on using Julia with calculus.

This package does two things:

• It loads a few other packages making it easier to use (and install) the functionality provided by them and

• It defines a handful of functions for convenience. The exported ones

are e, unzip, rangeclamp tangent, secant, D (and the prime notation), divergence, gradient, curl, and ∇, along with some plotting functions

Packages loaded by CalculusWithJulia

• The SpecialFunctions is loaded giving access to a few special functions used in these notes, e.g., airyai, gamma

• The ForwardDiff package is loaded giving access to its derivative, gradient, jacobian, and hessian functions for finding automatic derivatives of functions. In addition, this package defines ' (for functions) to return a derivative (which commits type piracy), ∇ to find the gradient (∇(f)), the divergence (∇⋅F). and the curl (∇×F), along with divergence and curl.

• The LinearAlgebra package is loaded for access to several of its functions for working with vectors norm, cdot (⋅), cross (×), det.

• The PlotUtils package is loaded so that its adapted_grid function is available.

Packages with extra features added when loaded

The Julia package Requires allows for additional code to be run when another package is loaded. The following packages have additional code to load:

• SymPy: for symbolic math.

• Plots: the Plots package provides a plotting interface.

Several plot recipes are provided to ease the creation of plots in the notes. plotif, trimplot, and signchart are used for plotting univariate functions; plot_polar and plot_parametric are used to plot curves in 2 or 3 dimensions; plot_parametric also makes the plotting og parameterically defined surfaces easier; vectorfieldplot and vectorfieldplot3d can be used to plot vector fields; and arrow is a simplified interface to quiver that also indicates 3D vectors.

The plot_implicit function can plot 2D implicit plots. (It is borrowed from ImplicitPlots.jl, which is avoided, as it has dependencies that hold other packages back.)

Other packages with a recurring role in the accompanying notes:

• Roots is used to find zeros of univariate functions

• SymPy for symbolic math

• QuadGK and HCubature are used for numeric integration

D(f)

Function interface to ForwardDiff.derivative.

Also overrides f' to take take a derivative.

arrow(p, v)

Add vector, v, to plot anchored at point p.

Example

Fn = parametric(t -> [2cos(t), 3sin(t)])
Fnp = t -> ForwardDiff.derivative(Fn, t)
p = plot(Fn, 0, 2pi, legend=false)
for t in 0:pi/4:pi
   arrow!(Fn(t), Fnp(t))
end
p

arrow(p, v)

Add vector, v, to plot anchored at point p.

Example

Fn = parametric(t -> [2cos(t), 3sin(t)])
Fnp = t -> ForwardDiff.derivative(Fn, t)
p = plot(Fn, 0, 2pi, legend=false)
for t in 0:pi/4:pi
   arrow!(Fn(t), Fnp(t))
end
p

arrow(p, v)

Add vector, v, to plot anchored at point p.

Example

Fn = parametric(t -> [2cos(t), 3sin(t)])
Fnp = t -> ForwardDiff.derivative(Fn, t)
p = plot(Fn, 0, 2pi, legend=false)
for t in 0:pi/4:pi
   arrow!(Fn(t), Fnp(t))
end
p

curl(F)

Find curl of a 2 or 3-D vector field.

divergence(F)

Find divergence of a 3-D vector vield.

plotif(f, g, a, b)

Plot f colored depending on g < 0 or not.

plotif(f, g, a, b)

Plot f colored depending on g < 0 or not.

plotif(f, g, a, b)

Plot f colored depending on g < 0 or not.

rangeclamp(f, hi=20, lo=-hi; replacement=NaN)

Modify f so that values of f(x) outside of [lo,hi] are replaced by replacement.

Examples

f(x) = 1/x
plot(rangeclamp(f), -1, 1)
plot(rangeclamp(f, 10), -1, 1) # no `abs(y)` values exceeding 10

riemann: compute Riemann sum approximations to a definite integral. As well, implement trapezoid and Simpson's rule.

Example:

f(x) = exp(x^2)
riemann(f, 0, 1, 1000)   # default right-Riemann sums
riemann(f, 0, 1, 1000, method="left")       # left sums
riemann(f, 0, 1, 1000, method="trapezoid")  # use trapezoid rule
riemann(f, 0, 1, 1000, method="simpsons")   # use Simpson's rule

secant(f::Function, a, b)

Returns a function describing the secant line to the graph of f at x=a and x=b.

Example. Where does the secant line intersect the y axis?

f(x) = sin(x)
a, b = pi/4, pi/3
sl(x) = secant(f, a, b)(x)  # or sl = sl(f, a, b) to use a non-generic function
sl(0)

sign_chart(f, a, b; atol=1e-4)

Create a sign chart for f over (a,b). Returns a tuple with an identified zero or vertical asymptote and the corresponding sign change. The tolerance is used to disambiguate numerically found values.

Example

julia> sign_chart(x -> x/(x-1)^2, -5, 5)
2-element Vector{NamedTuple{(:∞0, :sign_change), Tuple{Float64, String}}}:
 (∞0 = 0.0, sign_change = "- → +")
 (∞0 = 1.0000000000000002, sign_change = "+ → +")

tangent(f::Function, c)

Returns a function describing the tangent line to the graph of f at x=c.

Example. Where does the tangent line intersect the y axis?

f(x) = sin(x)
tl(x) = tangent(f, pi/4)(x)  # or tl = tangent(f, pi/3) to use a non-generic function
tl(0)

Uses the automatic derivative of f to find the slope of the tangent line at x=c.

trimplot(f, a, b, c=20; kwargs...)

Plot f over [a,b] but break graph if it exceeds c in absolute value.

trimplot(f, a, b, c=20; kwargs...)

Plot f over [a,b] but break graph if it exceeds c in absolute value.

trimplot(f, a, b, c=20; kwargs...)

Plot f over [a,b] but break graph if it exceeds c in absolute value.

`unzip(vs)`
`unzip(v1, v2, ...)`
`unzip(r::Function, a, b)`

Take a vector of points described by vectors (as returned by, say r(t)=[sin(t),cos(t)], r.([1,2,3]), and return a tuple of collected x values, y values, and optionally z values.

Wrapper around the invert function of SplitApplyCombine.

If the argument is specified as a comma separated collection of vectors, then these are combined and passed along.

If the argument is a function and two end points, then the function is evaluated at 100 points between a and b.

This is useful for plotting when the data is more conveniently represented in terms of vectors, but the plotting interface requires the x and y values collected.

Examples:

using Plots
r(t) = [sin(t), cos(t)]
rp(t) = [cos(t), -sin(t)]
plot(unzip(r, 0, 2pi)...)  # calls plot(xs, ys)

t0, t1 = pi/6, pi/4

p, v = r(t0), rp(t0)
plot!(unzip(p, p+v)...)  # connect p to p+v with line

p, v = r(t1), rp(t1)
quiver!(unzip([p])..., quiver=unzip([v]))

Based on unzip from the Plots package. Implemented through invert of SplitApplyCombine

Note: for a vector of points, xs, each of length 2, a similar functionality would be (first.(xs), last.(xs)). If each point had length 3, then with second(x)=x[2], a similar functionality would be (first.(xs), second.(xs), last.(xs)).

```

uvec(x)

Helper to find a unit vector.

vectorfieldplot(F; [xlim=(-5,5)], [ylim=(-5,5)], [nx=8], [ny=8])

Create a vector field plot using a grid described by xlim, ylim with nx and ny grid points in each direction.

F(x,y) = [-y, x]
vectorfieldplot(F, xlim=(-4,4), ylim=(-4,4))

vectorfieldplot(F; [xlim=(-5,5)], [ylim=(-5,5)], [nx=8], [ny=8])

Create a vector field plot using a grid described by xlim, ylim with nx and ny grid points in each direction.

F(x,y) = [-y, x]
vectorfieldplot(F, xlim=(-4,4), ylim=(-4,4))

vectorfieldplot(F; [xlim=(-5,5)], [ylim=(-5,5)], [nx=8], [ny=8])

Create a vector field plot using a grid described by xlim, ylim with nx and ny grid points in each direction.

F(x,y) = [-y, x]
vectorfieldplot(F, xlim=(-4,4), ylim=(-4,4))

vectorfieldplot3d(F; [xlim=(-5,5)], [ylim=(-5,5)], [nx=5], [ny=5])

Create a 3 dimensional vector field plot using a grid described by xlim, ylim, zlim with nx, ny, and nz grid points in each direction.

Note: the vectors are represented with line, not arrow due to no implementation of :quiver3d.

F(x,y,z) = [-y, x,z]
vectorfieldplot3d(F, xlims=(-4,4), ylims=(-4,4), zlims=(0,3))

vectorfieldplot3d(F; [xlim=(-5,5)], [ylim=(-5,5)], [nx=5], [ny=5])

Create a 3 dimensional vector field plot using a grid described by xlim, ylim, zlim with nx, ny, and nz grid points in each direction.

Note: the vectors are represented with line, not arrow due to no implementation of :quiver3d.

F(x,y,z) = [-y, x,z]
vectorfieldplot3d(F, xlims=(-4,4), ylims=(-4,4), zlims=(0,3))

vectorfieldplot3d(F; [xlim=(-5,5)], [ylim=(-5,5)], [nx=5], [ny=5])

Create a 3 dimensional vector field plot using a grid described by xlim, ylim, zlim with nx, ny, and nz grid points in each direction.

Note: the vectors are represented with line, not arrow due to no implementation of :quiver3d.

F(x,y,z) = [-y, x,z]
vectorfieldplot3d(F, xlims=(-4,4), ylims=(-4,4), zlims=(0,3))