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

Physical Optics

Prerequisite: algebra2, physics

This is the diffraction equation from regular physics:

dsinθ = mλ (m = 0, 1, 2, 3..)

in which d is the distance between two slits

and m is the order at which there is a maximum.

Here is a twin equation:

Dsinθ = mλ (m = 1, 2, 3, 4..)

in which D is the distance of a slit's width

and m is the order at which there is a minimum.

This twin equation can be shifted by half a wavelength to describe maximums:

Dsinθ = (m - 1/2)λ (m = 1, 2, 3, 4..)

in which D is the distance of a slit's width

and m is the order at which there is a maximum.

And the regular can be shifted to describe minimums. So there. The basis of the equation is this:

dsinθ = mλ

path difference = phase difference

This equation can be adapted for soap bubbles and the like, applicable only for perpendicular light rays (note that the diagram slants the rays for demonstration purposes):

path difference = phase difference

2t = (λ / n)

Where t is the medium's thickness

and n is the index of refraction through a medium.

And really, you can use any combination you like~

dsinθ = mλ = 2t = (λ / n)

Here are some laws on polarization. Malus' law tells the output intensity of a light ray going through polaroids at a certain angle, which is essentially the component aligned with the transmission axis. Brewster's law tells the incidence angle at which light is polarized. Interestingly, the sum of the angle of polarization and angle or refraction is 90˚ (θp + θr = 90˚).

If you find polarization fascinating, you ought to know about Quantum Cryptography~ In fact, everyone ought to know about it!

Geometric Optics

Prerequisite: geometry, physics

The only new material in this chapter is combined lenses in which light travels through more than one lens. It is surprisingly easy, which makes me wonder why it was not included in regular physics.

The key is this: the image through the first lens becomes the object for the second lens. This holds true even if the first image happens to cross the second lens.

And if you remember using microscopes in biology class, the total magnification is the product of the lenses' individual magnifications.

That is all~

女の子

First find out about introspective.

onna no ko
engulfed in flames
as a daisy in a desert
under the merciless sun
parched of its living water
singed of its white purity
vapourized with scattered winds
denied a resting place on earth
fire that nurtures
fire that kills
an uncontrolled wildfire
extends its deathly grasp
north and south
east and west
all four corners of the world
will burn
with children victim to their vulnerabilities
and fire setters to their mistaken justices:
a life for life
a death for death
from there young shoots are set alight
baked and shrivelled to mere ashes
restoring the earth with dead love
which from its cradle no love grows
even a daughter of water
may sizzle in the blaze
so only God can save
kız çocuğu

The Imagery

a scorching desert
fire, fire, fire
a burnt pit where plants no longer grow

The Content

As you can tell I am a fan of FazıI Say. I wrote this a while back, in September probably. Thinking not only of war, but of the deadliness of human fire.

Natural fires, whether, seasonal or volcanic, restore the Earth and bring new life after the destruction. But war fire leaves a singe that impoverishes and does not heal quickly enough. It is no natural fire but a synthetic fire, such as the one that bombed Hiroshima. And unlike natural fires in which the dead are returned to the earth, the atomic bomb simply vapourized the dead. There was not a trace of dignity left for the deceased. War fire is like radioactive radiation in the sense that nothing grows in the aftermath.

I thought the line "and fire setters to their mistaken justices" was particularly intriguing. It seems that I side with Mencius (the Eastern counterpart of John Locke) on the matter of human nature, preferring to describe violent impulses as bad mistakes that came from good intentions.

Spring Nicht

First find out about introspective.

spring nicht

for the breeze is gentle

and the scent is soft

the spring has only just come

a bud on the heights

the night has only just passed

a sunrise on the horizon

but to miss hope around the corner

give in to gravity over the edge

and fall vacant into the abyss

where the spring is withheld

where the night is eternal

where surrender is the last surrender

so vulnerably

with just a moment of chill

before spring night

the icicles hurt, they did

needles of denial

drips of doubt

the splinters crushed, they did

shards of affection

slivers of endearment

the precious dream marred

from end to end

when it has hardly begun

so why shatter it further

so why cripple it further

to make it true

hold on for dear life

and God forbid

do not jump

The Imagery

breezy twilight on the heights
warm glow in the distance
dark depths
fractured cold ice

The Content

This entry was written in August. It is based on a wordplay between "spring nicht" and "spring night", in which the former actually means "do not jump". Looking back, this piece is more dramatic than I realize.

Maybe you all be worried at this dark theme, but it is my form of optimism. My favourite sections are "but to miss hope around the corner.. where surrender is the last surrender" and "the precious dream marred" onwards. The former tells the consequences, and the latter gives incentive to hold on. "And God forbid, do not jump"!

As you can tell by now, the cold really bothers me. I love getting cozy on a cold night, but without a source of warmth it is just plain depressing. At the time I was writing this I had a mild trauma psychologically conditioned with cold nights; just thinking about cold nights got me anxious. It no longer bothers me anymore but that is a story I would rather leave alone.

Matthew 6:2

"Thus, when you give to the needy, sound no trumpet before you, as the hypocrites do in the synagogues and in the streets, that they may be praised by others. Truly, I say to you, they have received their reward".

Despite what this verse says, I will tell a story I have never told:

It was the first week of school: Thursday, 6th of August, 2015. Mum needed to pick up brother so she dropped me off at the overpass to walk home. It was about 4:00pm; cars rushed beneath the overpass while it was deserted up top, except for one beggar.

My family does not do much giving. Dad is all talk and no action. He is the greatest preacher of hunger, war, and global warming, rambling on and on at the dinner table, yet the most gluttonous, prejudiced, and wasteful of us all. Mum is more silent on the matter of poverty, since she does not have a career of her own. I have never seen my parents give. When we passed by beggars, dad tugs me along with words of pity while holding onto his "you win or you lose" philosophy.

Before the day I walked that overpass, I used to ponder. If my parents are not around to restrain me, what will I do for these unfortunate people? Will I give heartily as I always wanted to? Or will I walk on by like my parents did? More still, what if the money is of my own labour and pain? It is easy to give something that is not mine, but will I be able to give away money of my own?

So it was time to find out.

I stopped three meters from him and considered how much I should give. Then I snapped to my senses and asked myself: why am I withholding? Somebody is in need here! I rummaged my backpack for every note and coin there is. Surely I must have something, and after having searched my bag inside out and upside down, I confirmed that I only had one hundred baht.

Call me stupid, or gullible, but if anyone is willing to sit all day on a dirty overpass for a meager donation, I say that person is desperate enough to receive some help. His right hand was a stump. He held up to his forehead and I put the hundred into his cup. Then I left the overpass.

That ought to have settled my unease, having proven myself capable. But on my way home I just got upset. All I could think was "I only had one hundred baht". If only I had more to give. Why did I not have more to give? What did I spend it on? Nothing, really. I spend forty baht a day for lunch at school, and that is all.

The very next day I stopped having lunch. At least that is what I can do as a student. And as far as health goes, it has not killed me.

At home we work by a "when you need more" basis. Mum always gives me more than I need, two hundred baht a week and one thousand baht for trips. I tell her honestly that I still have money, but she stuffs it in my hand anyways, and I usually end up returning them all at the end of the year. But this time I add all the money to an envelope written "Yi Ting physics" (that envelope is another story to itself. A couple months ago I donated a total of 1,500 to help the Nepalese earthquake recovery 2015 from my talent show and science fair prizes).

By November I had accumulated a little over five thousand baht. When Habitat for Humanity came around, I did not withhold. I gave the whole envelope away. In December when the donation for Northern Thai hill tribes came around, I did not withhold and gave my one thousand.

It was the first week of semester two: Wednesday, 6th of January, 2016. Mum needed to pick up brother so she dropped me off at the overpass to walk home. It was about 4:00pm; cars rushed beneath the overpass while it was deserted up top, except for one beggar.

It had been exactly five months since, and he is still there. I did not withhold and gave my two thousand baht.

So I thought of this story because of the recent "Asia Cold Snap" (East Asia Cold Wave January 2016). It is a real thing. Having lived in the tropics all our lives, the cold was utterly unbearable. Temperatures can go as low as fifteen degrees Celsius on that open-air overpass. I hope the two thousand helped him through.

And never mind that I lost my "reward" telling this story, because it is more important that you know. It is possible to give, even when you think you have nothing to give. I have a serious lack of "practical smarts" but over the course of one year, I gave away a total of 10,000 baht. That is a lot, especially for a jobless, not-so-street-savvy seventeen year old.

You have the power to give. You do.

Fluid Mechanics

Prerequisite: physics 1, algebra 2

All things "flowable" are fluids, which may be liquids, gases, plasmas, ion soups.. but for now, liquids and gases. Specific gravity is the ratio of two substances' densities, which comes in handy with buoyancy.

One main property that fluid mechanics considers is pressure. Here are two laws of fluid statics, regarding pressure of stationary fluid on an object:

The first one is quite understandable, since pressures need to be uniform in order for fluids to be stationary. The second one is somewhat confusing, but the logic is that since nothing is moving, the net force must be zero. Backwards reasoning. Sorry for the misleading picture, here is a better one:

There are many explanations out there from torque to average force, but my understanding is this: force is acceleration times mass, so if a fluid exerts a diagonal force as shown above, the fluid mass is accelerating with a y component and therefore moving. In order for the fluid to not be moving, it can only accelerate horizontally against the rigid bottle.

A more detailed formula for pressure can be derived as follows:

P = F / A
    = (mg) / A
    = (DV)g / A
    = D(l*w*h)g / (l*w)
    = Dhg
    = Dgh

This formula is for incompressible fluids, in other words, fluids that have the same density at any depth. For compressible fluids, ∆P = Dg∆h.

 Another common formula is this:

Atmospheric pressure is obviously pressure due to the atmosphere. Absolute pressure is the pressure inside a container regardless of the outside pressure. Gauge pressure is in a sense the "net pressure".

But I rather like to think of it as P(atm) - P(abs) = P(g), in which the pressure balance between the outside and inside determines the "apparent pressure" P(g). Dunno. Makes more intuitive sense to me.

Pascal's Principle is applied in the hydraulic lift in which since the pressure is the same at both ends, the out force increases with a maximized area so that the pressures match. The increased force can then be used to lift objects.

In terms of work, the in force is smaller but needs to cover a greater distance, in order for the greater out force to cover a short distance. No glitches, hacks, nor dirty tricks. Just physics.

Another property that fluid mechanics studies is buoyancy, the ability of an object to float. By Archimedes' Principle, an object's buoyant force is the weight of displaced fluid. If you divide constant g from both sides, you have D(fluid)V(displacement) = DV of object.

In fluid dynamics, flow rate can be expressed in mass over time, and if density does not vary, volume over time. The volume rate is easier to visualize while the mass rate uses density to convert volume units into mass units.

And here, Bernoulli's Principle: pressure is inversely proportional to pressure. I like to think of it as driving, or traffic. The cars are mashed up together in the slow lane but widely spaced out in the fast lane. Planes can fly because the bumps in their wings cause air to flow at a greater velocity over the wing than under, so the pressure is greater under than over, then lift off~ And there, the equation is derived by setting work and ∆PE equal to ∆KE.

So there, the basics of fluid mechanics~

Proverbs 17:3

"The crucible is for silver, and the furnace is for gold,
and the Lord tests hearts"

Many things are going to happen this year. For the first half, there is the extra Calculus BC and Physics 2 AP exams that I need to self study. As of now I already dashed through fluid mechanics, sequences, L'Hôpital's rule, and improper integrals within two weeks. Burn, burn, burn!

Questioning my choices too. Life can be so easy if I pursued humanities as my profession. I guess easy topics have no appeal to me. Sciences are more gruelling, but the elegance of the universe's ways is far more aesthetic than all the humanities. Humanities are earthly, while sciences are transcendental. So this is why.

The burning is not so much in making time for studies, but making time for God. The test is not about whether I can retain my sanity, but whether I can keep God a part of my life. To be honest, I almost canceled this post to do more studying. But hey, priorities~

It is very tempting to take credit for myself, which is why there is the push and pull of success and failure. Hopefully by the end I will remember how I got there. As I have learned last month, no temptation is too great to overcome. I can combat the weakness of my flesh and humble myself. This is another test unto itself. The way is not so simple, but I will not be alone~

For the second half of this year and onwards, I will not be in Thailand. For sure. I have not applied for any Thai universities so it will be a whirl leaving my home of (by then) eighteen years. Living with my parents is a sweet little haven, a nice and protected bubble. Whereas being alone out there, no one will be around to rein me in. What I do is up to me. The Lord tests hearts.

But until then, these APs. Burn away, honey.