Parametric, Vector, and Polar Functions

Prerequisite: calculus, physics

What I love most about parametric, vector, and polar functions.. their graphs are sooo pretty!

Parametric

Recall the basics of parametric functions:

x = f(t)

y = g(t)

(dy/dx) = (dy/dt)(dt/dx)

Personally, I like to use ∆ instead of d for visualization. It appeals much more to a physicist. To step up to second order derivative, this is what you do:

y'' = (d/dx)y' = (dy'/dt)(dt/dx)

Basically y'' is the derivative of y', which can be written in the form (dy'/dt). Then simply do the division divination cancellation hocus pocus~

My calculus AB class at school has not yet done arc length, but here is the parametric equation. I find that it helps to think in terms of ∆:

L = ∫ √ (dx/dt)^2 + (dy/dt)^2 dt   [t', t'']

Vector

Having done a load of Newtonian physics the basics are already intuitive, but there are still some new content. The unit vector is a vector of magnitude 1 that points in a direction, given by (v / |v|). Not sure what purpose this serves. Never came across such a thing.

Velocity and acceleration are just derivatives of one another, yet they can be separated into components. That means, so can an integral:

displacement =   <   ∫ vx(t) dt   ,   ∫ vy(t) dt   >   [a, b]

distance =   ∫ |v(t)| dt   =   ∫ √(vx(t))^2 + (vy(t))^2 dt   [a, b]

in which vx(t) is the x component of v(t), and vy(t) the y component.

Polar

Never knew about polar coordinates before.. but I love the curves! Polar functions have fancy names for the origin and x axis, called the pole and initial ray respectively, but they are the same thing. I suppose the naming makes sense since polar functions only need one point to start with and one axis to label radius lengths on. Polar functions are based on:

r = f(θ)

where r is the radius spanning out from the pole, and θ is the angle away from the initial ray.

Rectangular conversion converts polar coordinates into "ordinary" coordinates, which is essentially just taking the x and y position component of the radius.

x = rcosθ

y = rsinθ

and you can call these useful identities:

r^2 = x^2 + y^2

tanθ = y/x

You can also split a polar function into its components to make a parametric function:

x = rcosθ = f(θ)cosθ

y = rsinθ = f(θ)sinθ

This parametrized polar function can then undergo division divination cancellation hocus pocus as well~

(dy/dx) = (dy/dθ)(dθ/dx)

As for integrals, it is not so obvious. Start with the original formula for a sector's area:

A = (1/2)(r^2)θ

Make it differential. It looks funny at first but if you think in terms of ∆, it makes more sense.

dA = (1/2)(r^2)dθ

And of course if you take the integral of this, you get the sector area.

A = ∫ (1/2)(r^2) dθ   [a, b]

Really, go look at some pictures of polar graphs. They are sooo pretty > <