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 > <

L'Hôpital's Rule and Improper Integrals

Prerequisite: calculus

Improper integrals are definite integrals of infinite partitions. Meaning, the interval of integration stretches on to infinity but still has a definite area. Whuuut? Yep. Provided that the infinite end approaches a limit. You will see~

Due to the infinite nature of improper integrals, you need L'Hôpital's rule to further deal with limits. It is important when you run into results of (0 / 0), (∞ / ∞), ∞•0, ∞-∞, and the such, which are indeterminate forms. The rule to finding their limits is this:

[lim x-->a] f(x) / g(x)   =   [lim x-->a] f'(x) / g'(x)

provided that f(x) / g(x) is an indeterminate form.

And sometimes, you may have to derive many times over until you get to an actual number, or a dead end indicating that the limit does not exist.

As a side note, it is useful to get rid of logarithms that lead to 1^∞, 0^0, ∞^0, and the like. For example:

[lim x-->a] ln(f(x)) = L   becomes   [lim x-->a] f(x) = e^L

Next, you should know about the comparison test. If a function converges, then it approaches a limit and has a definite integral. On the other hand, a function that diverges does not approach a limit and does not have a finite integral.

The comparison test is similar to the sandwich theorem. Consider functions f(x) and g(x) that are continuous on [a, +∞) and 0 ≤ f(x) ≤ g(x):

1)  ∫ f(x) dx [a,+∞) converges if ∫ g(x) dx [a,+∞) converges.

2)  ∫ g(x) dx [a,+∞) diverges if ∫ f(x) dx [a,+∞) diverges.

It is fairly intuitive. Nothing crazy.

Now for solving improper integrals. Say, you want to take the integral of f(x) over interval [a,+∞). This is what you do:

Set up.

∫ f(x) dx   [a,+∞)

Antidifferentiate.

f(x) --> F(x)   [a, +∞)

Deploy integration evaluation theorem.

∫ f(x) dx   [a , +∞)    =   F(+∞) - F(a)

Find limit of F(+∞).

= [ lim(x --> +∞)   F(x) ] - F(a)

If the limit diverges, then the integral is not finite. If the limit does exists, evaluate the total to get the integral. There~

Say, you want to take the integral of f(x) over the interval (-∞, +∞). Just split the integral at 0.

∫ f(x) dx  (-∞, +∞)

= ∫ f(x) dx  (-∞, 0]   +    ∫ f(x) dx  [0, +∞)

Then do the same~

Confuci-us

Prerequisite: none

As a Chinese, the translation of "孔夫子" (kongˇ-foo-zhi˙) to "Confucius" (kon-fyoo-shus) has always been somewhat confusing.. very kon-fyoo-sing indeed! The last syllable has puzzled many a Chinese. Why would "zhi˙" become "shus"?

I thought this quirky translation was unique to Confucius until I found out yesterday that 孟子 (mengˋ-zhi˙) is also known as "Mencius" (men-shus). It finally clicked when I searched "Suncius" (孫子, author of The Art of War) and it showed up in "Vicipaedia".

Latin.

The "-us" at the end is a masculine suffix. Julius. Marcus. Brutus. Confuci-us. As for pronounciation in Ecclesiastical Latin:

1) C before O is a hard C (as in k)

2) U is pronounced "oo"

3) C before I is a soft C (as in ch)

The last U is probably a short "oo". So instead of "kon-fyoo-shus", it should be:

"kon-foo-chi-us"

And leads me to wonder.. what is my Latinized Archaic name?

First of all, Archaic Chinese. Females more commonly used "氏" (shiˋ), which they attached to their father's or husband's surname. Problem is, my last name is from my mother's side (special case) and I am not married yet. "子" (zhi˙) was used for respected or scholarly people, although the fact that it was more explicitly used for males was due to the lack of female scholars.

There are more choices but they get very specific about status and age, and are not convenient for public use. For my case I better use "子".

My surname is 孫, so 孫子.

I wrote The Art of War, whoopee~

On the Latin part, pick any female Latin suffix of choice. Let me see..

Suncia

Suncilla

Suncilia

Suncilea

Suncina

Suncilina

Suncianna

Suncissa

Suncietta

Sunciella

I would not go for more than three syllables because that just sounds too princess-like.

Suncia

Suncilla

Suncina

Suncissa

Maybe something not so "flowery".

Suncia

Suncissa

Ehh.. I would rather not name myself after some furniture. "Cissa" is actually a genus of magpies, but preferable.

Suncissa