I didn't Remember the Formulas for either of them all I remembered was n!.

go to wikipedia there you will find why the natural log is called natural and many properties which make it fundamental to logs, reciprical functions and integration.

We have 3x + 5 = 7

we want to "untangle" this by subtraction, division, etc. to get "x" by itself.

ax^2 + bx + c ... now there's a big tangle! We have a special technique to help. Idea... to represent that area as a square, then we "untangle" by taking just one of the sides as x. Boom.

Calculus behaves according to very simple, beautiful rules:

Why are these "allowed" to work? We have limits and infinitesimals to justify these rules. Almost like we have physics to "justify" why a bumblee can fly. Don't get too caught up in it!

All changesets are kept, even those on deleted branches, so you can cherry pick from a deleted branch.

i need to know the USES of the pythagorean theorem.

If speed was represented by the magnitude of the vector, it would be lost after rotation.

For example in your Example 1: Growing Crystals, how shall we modify the "Growth Formula" if those baby crystals start to grow after a delay, let's say n hours, or any crystal cannot produce new crystals for n hours?

I never really understood how i worked until now. My algebra teacher told us that it was the square root of -1, but it makes so much sense to think of it as a two-step process to get to -1, where there was previously only a one-step way to get there, and the fact that it sort of adds a second dimension seems to fit with that perfectly. Also, i is cyclic! Like sine! There are always so many more connections within mathematics than I expect.

If i had read these articles in my school days it could have saved me so much of agony and i would have loved math instead of being distressed by it. I would suggest that wiki would be great to share new ideas. Thank YOU.

Extra: Pre-chain rule

Now, f and his buddy g could actually be part of a bigger machine (call it h). Heck, h could get fancy and link the output of f to the input of g:

input: x => f: x^2 => g: x^3 => output: y

Whoa. We have a "squaring machine" feeding a "cubing machine". What do we get?

input:2 => f(2): 4 => g(4): 64 => output:64

We have a "6th power" machine, written g(f(x)) or (x^2)^3. A bigger system is made from smaller parts.

The "behavior" of the system depends on who you ask: f, g, or h? (f says "I'm squaring the input", g says "I'm cubing the input", and h says "I'm raising the input to the 6th power").

Understanding derivatives means seeing how these "points of view" fit together.

---

The jumble of rules for taking derivatives never truly clicked for me, despite years of using them in classes. Addition rule, product rule, quotient rule -- how do they fit together? What are we even trying to do?

Here's my take:

• We have a system to analyze, our function f
• The derivative f' (df/dx) is the moment-by-moment behavior. This is easy to find for simple polynomials
• It turns out f is part of a bigger system (h = f + g, h = f * g, etc.)
• Using the behavior (derivative) of the parts, can we get the behavior of the whole?

We can. Every part has a "point of view" about the entire system changes. Combine every point of view to get the overall behavior.

Ah. And why not analyze the entire system at once? Well, why not eat a hamburger in one bite? Small parts are easier to wrap your head around.

Instead of memorizing rules, today's goal is building fluency for how they fit together:

Let's see how to "combine the perspectives" of each part. This article covers weeks of calculus class, so take it in pieces!

Functions: Anything, Anything But Graphs

It's easy to write "f(x) = x^2", shove a graph in your face, and start lecturing. But does this really help our intuition?

Not for me, not in this case. Graphs, the go-to visualization, squash the input and output along a single curve, hiding the machinery that turns one into the other.

But the rules for derivatives are understood by seeing how the machinery fits together! Let's be explicit. I imagine a process of "input(x) => f => output(y)":

Think of f as a machine with an input lever "x" and an output lever "y". As we adjust x, f sets the height for y. (Another analogy: x is an input signal, f receives it, does some magic, and spits out signal y. Use whichever analogy clicks best)

Wiggle Wiggle Wiggle

The derivative is the "moment-by-moment" behavior, but what does that mean? (And don't mindlessly tell me "derivative is the slope"! See any graphs around these parts, stranger?)

The derivative is a wiggle. The lever is at x, we "wiggle" it (just a little bit), and we see how much y changes. "Oh, we moved the input lever 1mm, and the output moved by 5mm".

The result can be seen as "output wiggle per input wiggle" or "dy/dx". This is usually a formula (not a static value) and depends on your current input setting.

For example, if f(x) = x^2, the derivative is 2x. Yep, you've memorized that. What does it mean?

If our input lever is at x=10 and we wiggle it slightly (dx = 0.1), the output should change by dy. How much, exactly?

• We know f'(x) = dy/dx = 2 * x
• At x = 10 the "output wiggle per input wiggle" is = 2 * 10 = 20. The output moves 20 units for every unit of input movement.
• If dx = 0.1, then dy = 20 * dx = 20 * .1 = 2

And indeed, the difference between 10^2 and (10.1)^2 is about 2. The derivative estimated how far the output lever would move (a perfect, infinitely small wiggle would move 2 units; we moved 2.01).

The essence of each derivative rule is setting up the system, wiggling it, and seeing how the output moves.

Addition and Subtraction

Time for our first system:

What happens when the input changes?

In my head, I think "Function h takes a single input. It feeds the same input to f and g and adds the output levers. f and g don't even know about each other!"

Functions f and g never interact, and don't know their inputs are being combined. We, the prowling overseers that we are, know their moment-by-moment behavior is added:

Now let's see this from each "point of view"

• The overall system has behavior dh
• From f's perspective, it contributes df to the whole [it doesn't know about g]
• From g's perspective, it contributes dg to the whole [it doesn't know about f]

In general: every change to the system is due to one of the variables changing (f or g in this case). If we add the contributions from each possible variable, we've described the entire system.

When do we use df vs df/dx?

Sometimes we use df, sometimes df/dx -- what gives? (This confused me for a while)

• df is a general notion of "however much f changed"
• df/dx is a specific notion of "however much f changed, relative to how much x changed"

The generic "df" is helpful to see overall behavior, and we can scale it down to "df/dx" when we need an exact formula / greater detail.

Imagine you're driving across the country. You might measure the distance traveled, and check your tank to see how much fuel you used. Finally, you can compute your "miles per gallon". You got the output (distance), the input (gasoline), and can do the division. You figured out each contribution separately -- you didn't jump into the gas tank to get the rate on the go!

In calculus, we don't always want to think about the final "x" (input) we'll be comparing against: we want to visualize actual changes, not ratios. Working at the "df" level gives us breathing room to think about how the function is changing overall, and eventually scale it down in terms of a specific input.

The addition rule above can be written:

Multiplication (Product Rule)

Next puzzle: suppose the parts "f" and "g" are multiplied. How does the system behave?

Hrm, tricky -- the parts are interacting more closely. However, the strategy is the same: see how each part contributes from its own point of view:

• total change in h = f's contribution (from f's point of view) + g's contribution (from g's point of view)

Check out this diagram:

What's going on?

• We have our system: f and g are multiplied, giving h (the area of the rectangle)
• Input "x" changes by dx off in the distance. f changes by some amount df (let's think about the absolute change, not the rate!) and g changes by dg (some absolute amount, and probably different from dg). Because f and g changed, the area of the rectangle will change too.
• What's the area change from f's point of view? Well, f knows he changed by df, but has no idea what happened to g. From f's perspective, he's the only one who moved and will add a slice of area = df * g
• Similarly, g doesn't know how f changed, but knows he'll add as lice of area "dg * f"

The total change in the system (dh) is the two slices of area:

Now, like our miles per gallon example, we "divide by dx" to show this in terms of how much x changed:

(Aside: Divide by dx? Engineers will nod, mathematicians will frown. Technically, df/dx is not a fraction, it's the entire operation of taking the derivative (with the limit and all that). But intuition-wise, infinitesimal-wise, we are "scaling by dx". I'm a smiler.)

The key to understanding the product rule is recognizing you're adding two "slivers of area", one from each point of view.

Gotcha: Hey, isn't there some effect from both f and g changing simultaneously (df * dg)?

Yep. However, this area is an infinitesimal * infinitesimal (a "2nd-order infinitesimal") and invisible at the current level. It's a tricky concept, but (df * dg) / dx vanishes compared to normal derivatives like df/dx.

The Chain Rule: You've Known It All Along

The chain rule is a fancy name for a concept you've used most of your life.

Imagine you have a shop that earns $70 each week. What's your daily income?

• $70 / week = $10 / day

Easy. From the day's point of view you're e of view of a day, you're earning "10", but from the poit

From the "point of view" of a day, you're earning "10". From the point of view of the week, you're earning "70". Each "point of n of viewTo change points of view, multiply by 7

From the "point of view" of a day, you're earning $10. From the "point of view" of the week, you're earning $70. To change points of view, multiply by 7. Each "point of view" has a different scale.

Now, we're used to scales that work with simple multiplication or division (days to weeks, grams to ounces, etc.). But changing your point of view can be non-linear!

Let's say I'm painting a square wall: I can take a "surface area" or "wall height" point of view. Switching between the two means I square (or square root) to get the new number.

If I change my input by "1", what happens? Well, if we're in the "area" point of view, the area increases by 1. Simple.

But if we're in the "height" point of view, the area changes by 2x:

• Height: 10 feet => 11 feet (+1)
• Area: 100 feet => 121 feet (+20)
• Change in area ~ 2x * change in height = 2x [because change in height = 1]

Each change in perspective involves a conversion factor. However, the conversion factor may involve a function, not a straight multiplication.

The chain rule can be written:

In my head: "Wiggle x, which wiggles v, which wiggles u, which wiggles h". Each "wiggle" involves a potential conversion (dv/dx).

A few notes:

• The variable "x" isn't always the end of the line! x could depend on w, which depends on z, and so on. When we take the derivative we say "I want the derivative with respect to x (from x's point of view). Yes, I know x could depend on something else later on -- I don't care." It's like saying "Give me the miles per hour, I don't care about miles per minute or miles per second. Just stop once you have miles per hour."

• Remember, the derivatives dv/du and du/dx can be functions! That is, they aren't static conversions ("multiply by 7"), but more complex operations (like square roots) to change your perspective.

When to add, when to multiply

How come we multiply derivatives with the chain rule, but add them for the others?

The regular rules are about combining points of view to get an overall picture. What change does f see? What change does g see? Add them up for the total.

The chain rule is about going deeper into a single part (like f) and seeing if it's controlled by another variable. It's like looking inside a clock and saying "Hey, the minute hand is controlled by the second hand!". We're staying inside the same part.

Sure, eventually this "per-second" perspective of f could be added to some perspective from g. Great. But the chain rule is about diving deeper into "f's" root causes.

Division (Quotient Rule)

Armed with the chain rule, we can tackle the quotient rule, i.e. the one nobody can remember. You might have learned it with a song like "Low dee high, high dee low...", but come on -- that's not understanding.

Let's understand the division rule (who says "quotient" in real life?). First, let's recognize that division is the inverse of multiplication:

[diagram]

We have a rectangle again, but the sides are "f" and "1/g". Remember, x is still changing off on the side, f and g are changing, but we're taking 1/g instead of straight-up g.

It's a bit funky though. g might change by dg: how does 1 / g behave? I'm already getting confused, isn't that another division?

The chain rule to the rescue. Let's pretend 1/g is its own system, m. So we have a machine, m, and inside it has a division -- let's not worry about that. We want to know:

• f changes by df, and is multiplied by m [df * m = df * (1 / g)]
• m changes by dm, and is multiplied by f [f * dm = ?]

Ok. How do we write dm (how much our 1/g fraction changed) in terms of dg (how much the denominator changed?).

We're trying to find the difference between neighboring values of g: 1/g and 1/(g +dg). Let's break it down:

• What's the difference between 1/4 and 1/3? (1/12)
• How about 1/5 and 1/4? (1/20)
• How about 1/6 and 1/5? (1/30)

How does this work? Get the common denominator:

• 1/3 = 4/12
• 1/4 = 3/12
• Difference: 1/12

The difference between neighbors is 1 / [x * (x + 1)] -- see if you can work out why. If we make our model perfect by estimating what happens when there's no change (removing the +1) we get:

(Aside: this is semi-useful. The difference from 1/100 to 1/101 is about 1/(100 * 100) = one ten thousandth).

Ok, back to reality:

• g changes by dg
• 1 / g becomes 1 / (g + dg)
• The difference is "size of change * derivative" = dg * (-1 / g^2)

A few gut-check questions:

• Why is the derivative negative? As dg increases, we have a bigger denominator and a smaller value overall (just like 1/4 is less than 1/3)

• Why is it -1/g^2 * dg not just -1/g^2? (This confused me at first). Remember, the amount -1/g^2 is the conversion factor to go from the "g" scale to the "m" scale. Fine. But we still need to convert something! We need to take how far we went on the g scale (dg) and convert that to some distance on the m scale (dm).

Phew. To convert a "dg" wiggle to a "dm" wiggle do:

And we get:

Yay! Now, your overeager textbook may simplify this to:

and we hates it! We hates it!

This "simplification" hides the notion that the division rule is just a variation of the product rule. The division rule involves 2 slivers of area, just as before:

• The "f" (numerator) sliver grows as expected
• The "g" (denominator) sliver is negative (a diminish fraction) and needs -1/g^2 conversion rate the the "overall" point of view.

With this intuition, you know it's the denominator that's adds the negative term.

Power Rule

What's the derivative of x^4? 4x^3? Great. You brought down the exponent and subtracted one. Now explain why!

Hrm. There's a few approaches, but here's my favorite: x^4 is really x * x * x * x. It's the multiplication of 4 "independent" components. Each x doesn't know about the others, it might as well be x * u * v * w.

Now think about the first x's point of view:

• It changes from x to x + dx
• The overall change is (x + dx)[u * v * w] - (x)[u * v * w] = dx[u * v * w]
• The change on a "per dx" basis is [u * v * w]

Similarly,

• From u's point of view, it changes by du. It contributes (du/dx) * [x * v * w]
• v contributes (dv/dx) * [x * u * w]
• w contributes (dw/dx) * [x * u * v]

And the curtain is unveiled: x, u, v and w are all the same! The "point of view conversion factor" is 1 for du/dx, dv/dx, and dw/dx, and the total contribution is

In a sentence: the derivative of x^4 is 4x^3 because x^4 has four identical "points of view" which are being combined. Booyeah!

Exponents

e is probably my favorite number. It has the property

which in English means "e constantly changes by 100% of its current amount" (read more here).

Now, this pretty formula assumes we're looking at changes from x's point of view, the exponent. But what if the exponent is made up of smaller pieces?

Interesting. v wiggles, so x wiggles, so e^x wiggles. Again, because of the chain rule, we need a conversion factor when changing from x's point of view down to v's point of view:

Originally, I thought the derivative would be "v * e^v * dv/dx". No -- the derivative of e^foo is e^foo. Nothing else.

But if foo is controlled by something else (we aren't really wiggling foo, we're wiggling x which shows up in foo!) then we need to multiply by that conversion factor (d(foo)/dx).

Natural Logarithm

The derivative of ln(x) is 1/x. Just like that (more here).

My intuition is that ln(x) is the time it takes to grow to x: ln(10) is the time it takes to grow to 10, assuming you're growing 100%, exponentially. But how long does it take to grow to x + 1, or 11?

Well, when you're at 10 you are growing 10 units per second, so it takes roughly 1/10 of a second (1/x) to get to the next value. And when you're at 11, it takes 1/11 of a second to get to 12. The additional time to the next value is 1/x.

Bonus: Partial Derivatives

We'll save the meat for another day, but let's say we have a function which depends on two variables

z = f(x,y)

The derivative of z can be seen from x's point of view (x assumes it's the only "real" variable who is changing) or y's point of view (y assumes it's the only "real" variable who is changing). They're mathematical Solopists.

We can combine these 1-dimensional "points of view" to get an understanding of the entire multi-dimensional system. Whoa.

Examples

Time for the rubber to meet the road. You can plug in derivatives into Wolfram Alpha (like so) to check your work.

Let's explore:

d/dx x^2 + x^3

This is a simple addition rule: x^2 contributes 2x, x^3 contributes 3x^2, for a total of

2x + 3x^2

d/dx x^x

This is a bad mamma-jamma. There's two approaches:

1) Rewrite everything in terms of e.

Ok. So now we have the derivative of e^foo, where "foo" is ln(x) * x.

Since we want the derivative in terms of "x" (not foo), we need to convert to x's point of view: multiply by d(foo)/dx:

for a result of

Yay! Ok, now for the other approach:

2) Assume x^x is made from two different functions that interact (u^v): we'll add their points of view. We're sneaky, u and v are the same, but don't let them know!

From u's point of view, v is just another number (like u^3) so we have:

(Yeah, yeah, bring down the exponent and subtract 1).

From v's point of view, u is just some base (like 5^v) which we can rewrite into base e, and we have

We add both points of view for the total behavior:

And now the big reveal: u = x and v = x! There is no conversion factor to change into this point of view (du/dx = dv/dx = 1), so we have

Which is the same as before! I was pretty excited to be able to approach x^x from a few different sides.

Take a Break

This is a beefy topic: let it percolate into your brain. The essence of the derivative:

See the "point of view" of each variable, and use the rules to merge the points of view into the overall behavior.

Happy math.

One of the best explanation on the globe ever ! Thanks for your kind work ...

Hi Khalid, I am crazy about all articles you wrote, but surprisingly I found this article very confusing. It is a very important and intriguing topic so please clarify, especially with the example you used (the imaginary number i).

please add examples

Understand the intuition for why it works. In my head: an engine inside an engine :). Multiply the rates of change.

Wiggle x... which wiggles y ... which wiggles the output function.

If you wiggle x twice as much, that impact effects how much y wiggles, and pushes to the very end.

x => drives f(x)

f(x) => drives g(x)

Can look for "causes". Find the cause. Find that cause's cause. If you double the initial cause... it should percolate through. And maybe there is a non-linear effect:

I'll plot out a square according to something's height.

Ruler doubles -- area quadruples (ultimate function quadruples).

What drives the ruler? Something else. Can look for causes, and causes... and the effects multiply based on the current derivative. (Derivative is a linear approximation, so combine linear approximations).

Other items to cover: http://en.wikipedia.org/wiki/List_of_calculus_topics

More here: http://aha.betterexplained.com/posts/71/think-of-the-chain-rule-as-wiggling-a-variable-fgx-means-you-w

I have to research the rule of 72 for an assignment, and this website was great! thank you!

It really helps to read this article when you read Calculus Madness of Square Numbers. To me, the imperfect meauserment v/s perfect measurement model really rings a bell.

BTW, is it weird if "you can't measure a quantity w/o some meauserement error" reminds me of Quantum Mechanics, or more precisely, reminds me of "the more precisely you measure the position of a particle, he less sure you are of the speed and vice versa"?

It really helps to read this article when you reach the Calculus Madness sections. To me, the imperfect meauserment v/s perfect measurement model really rings a bell.

BTW, is it weird if "you can't measure a quantity w/o some meauserement error" reminds me of Quantum Mechanics, "the more precisely you measure the position of a particle, he less sure you are of the speed and vice versa"?

The table comparing negative numbers and complex numbers was awesome, and the idea of rotation was something I was taught recently, yet it eluded me till now. And that real life example of the boat and its heading was fantastic. We were never even taught any application for complex numbers in school at all. Now that I think about it, that is quite dissapointing, some of my friends still struggle with it, and its not that they aren't smart, its just not very good teaching...

I get the impression from many school text books that the writers' of these books are not really very familiar with probability and often fail the test of translating the subject into understandable English. This article was able to do that and gave an ordered set of examples which weren't mixed up, as often happens, that compared and contrasted the two concepts of permutation and combination. It would be nice to see an explanation of n to the power r.

My answer is this: I don't believe that anybody needs to be rich to get to a place where they can be happy.......Although I DO think I could get there a lot faster in a new Ferrari!!!

Wealth was put into perspective for me back in 1999, when the richest man in the world was billionaire Bill Gates. My math teacher at the time told me..."If Bill Gates wanted to spend 1 billion dollars in 1 year, he would have to spend 2,739,726 dollars A DAY!!!".

I think Kanye West put it best...."Having money's not everything...not having it is!"

BTW--Bill Gates had a net worth of $92 billion that year!!!

This is an awesome article! Really well explained and reasoned through - wish all tutorials were this helpful!

Real numbers have to satisfy

but think about (this is impossible because any number multiplied by zero is zero, and if we allow division by zero this equation proves 0 = 1).

I haven't completed Algebra 2 yet, so I'm probably missing something, but the measurement of the circumference of a circle is a finite quantity, isn't it? How can it be irrational?

I knew circles and triangles were related, because my Geometry teacher told us they were, but..... That makes sense!

Greetings! I read this and many other articles here before some time. I have to say they are fabulous, and have helped me love mathematics even more.

When I read the part of the article that describes sine as a function that at every point accelerates in the opposite direction and in direct proportion to its height, I couldn't help but thing of (say) Earth's gravitational field.

If one drills a hole through the diameter of the Earth and then falls through it, then one's position as a function of time would be described by a perfect sine wave, because at every point of the diameter of the Earth, one would accelerate in proportion to one's position and in the direction of one's velocity.

It makes even more sense to think about this in the case when one takes the gravitational force to be repulsive instead of attractive. If one were to fire an object from a point at infinity toward the (repulsive) Earth, the path of the object would be described by the hyperbolic sine. Since the second derivative of the hyperbolic sine is the hyperbolic sine itself, one concludes that the object, at each point in space, would accelerate in proportion to its position, but in the direction of its velocity.

The speed of this hypothetical object would be described by the hyperbolic cosine. This connection between the regular sine and the hyperbolic one reflects the function of the constant "i" in various mathematical formulas, as was shown by your analysis of Euler's identity. Since the regular and the hyperbolic sine differ only by the factor of "i" in the denominator, it is possible to say (rather informally) that what the "i" is doing is just the periodization of the function.

In some sense, the "i" brings the system to a bounded and stable state, whereas when we lack an "i" (as in the case of the hyperbolic sine), one has a system that is unstable.

I really look forward to hearing your insights on this matter.

There's always plenty of bugs to go around. We can log them here.

This thread can store ideas for new types of topics (Example: for good problems, Followup: for good topics to write about afterwards, etc.).

This analogy separates the difference between the "measurement" (integrals) and the art (anti-derivatives, finding the original function).

The Fundamental Theorem of Calculus tells us they are the same (the measurement can be predicted if you can find the anti-derivative).

This seems obvious now but, if you missed some key maths lessons, some symbols can be intimidating. The %% \sum %%, for example:

wasn't intuitive for me for a long time - something had to click.

As a programmer I had been using for-loops, for (i=0; i<=n; i++), for many years before I finally realised that the %% \sum %% was a mathematician's version of the same thing:

#include <stdio.h>
int main() {
    int X[] = {10, 20, 50};
    int s, i;
    s = 0;

    for (i=0; i<=2; i++) {
        s = s + X[i];
    }

    printf("sum=%i\n", s);
}

or:

X = [10, 20, 50]
s = 0
for i in range(3):
    s = s + X[i]
print "sum=%i" % s

And %% \sum %%%% \sum %%, which I used to find off-putting, is nothing more than a nested for-loop:

Take a function, F(x).

Integration is "taking a function, applying it along the x-axis".

You can recreate function F taking its derivative F' = f(x), then integrating. You'll recreate the function, up to a constant.

Think of the integral as "building up" a function.

If we integrate x, what do we build up? 1/2 x^2. That might have been the original, up to a +C [initial starting conditions... the original may have been 1/2 x^2 + 37].

Basically, integration gives us the method to UNDO the derivative. We have the set of changes, where's the shortcut for the function itself (the anti-derivative)?

What is the site about?

I will share what actually worked in language I'd actually use.

Not "Let f(x) be a..."

Not pretending to intuitively understand topics I don't ("Oh, i is the square root of -1... simple as pie.").

Not mindlessly restating technical definitions from Wikipedia, showing some procedures, and calling that learning.

Treat math as a language. It exists to convey ideas. How do you test fluency?

• Are we understanding what it's saying?
• Are we focused on vocab and grammar drills?
• Are spelling bee champions the most fluent? The most eloquent? How many award-winning novels, or poetry, or literature, or... are written by spelling bee champions? Any? (I'd be surprised). Calculation is not understanding.

My goal is to increase our fluency.

Yes, yes, proper spelling is important, avoiding mistakes is important -- but expressing ourselves, clearly understanding the core concepts is more important! We have spelling correctors (calculators), we shouldn't rely on them of course. But see their use!

Math is a "computable" language. You can write "The sky is green" in English (a grammatically valid statement) and have no idea if it's true.

In math you can write "1 + 1 = 3" [a grammatically valid statement] and compute if it's true. Amazing!

Some statements in English are undefined / have no meaning ("Justice is red"). Same with math: 1 / 0 [undefined]. But... if you start defining what you mean "Justice is metaphorically red, red with blood of tyrants" you can make some sense (1 / 0 = grows without bound, taking the limit of 1 / x as x shrinks to zero...).

It's really about being clear with what you mean!

Understanding math means becoming fluent, not memorizing advanced vocabulary words (formulas) that you don't really understand and could never use!

Lots and lots of parallels here in teaching. How effective are we at math education? How effective is the typical high-school Spanish class? Were you taught correctly? Is the method appropriate?

I'd rather people fluent with kindergarten-level Spanish terms than sputtering out "The shellfish antagonized me". (Fluent with algebra vs. hating calculus).

Same principle: Someone who loves spanish and is fluent at a 2nd-grade level will be fluent at a 10th-grade level in a few years. Someone who sputters along will hate the subject and never return.

Link: http://ideas.time.com/2011/11/23/why-guessing-is-undervalued/ (Guessing vs. exactness).

Don't 'recite' integrals, read them! (Goal is mathematical fluency, learn by immersion, make some grammar mistakes, that's ok. You'll clean them up!). How do you intuitively read an integral?

or

"I'm traveling along the x-axis (how do you know? It's dx). I want to "multiply in" f(x) at each instant. I want to "multiply in" x^2 at each instant. Hear the language that we're trying to communicate here.

Simple case:

We're moving along the x-axis (notice our dx!) but are multiplying-in the implicit "1". If we move x units, we've multiplied-in "1" x-number of times, and get x.

If you are traveling along "dx" but have a totally unrelated variable (w), which doesn't change based on our x-value, then it doesn't matter. Multiply it in piece-by-piece (inside the integral), or all at the end. %%w^2%% can be extracted. Same for a constant factor (%%a%%).

Another way to put it:

• I'm multiplying the x-axis by a changing f(x)

A test is supposed to reveal areas of improvement so you can get better! Doesn't make sense to mindlessly test and not fix the problem.

They are smoke detectors / engine lights that show problems to fix. Turning them into a competition / punishment metric means we are missing that diagnostic information.

It's like a blood test -- please, TELL ME if I have a disease, so I can get it fixed.

My aha moment came as soon as I scrolled down to the visuals. Everything was self-explanatory after that.

The visualizations make the topic easier to understand.

One sentence: An essay on harnessing your emotions / analogies for math.

1) Math done by humans involves emotions (joy, honesty/truth, beauty). Like learning music theory without having it move you.

2) Rider and the elephant -- emotions dominate. Get the elephant to understand.

3) Examples of emotions / empathy

• Emperor's clothes: acknowledging what was difficult
• Creating the connections in your head: Speaking with analogies (you have the neural net you can apply to this new situation)
• Visceral understanding vs. intellectual understanding

Notes

Why do analogies work so well? One theory:

• We have a conscious mind (rider) and subconscious mind (elephant).
• If the elephant is docile, the rider can nudge it and generally control the thought process
• If the elephant resists, there's not much the rider can do (you are overwhelmed by emotions, distracted, etc.). I can feel this resistance.

Analogies are lessons the elephant, not just the thinker, has accepted. When using an analogy, you are comparing something new (scary!) to something warm and familiar.

This familiarity helps the elephant accept the concept. Here's an example:

How can a number be less than nothing? (New! Scary!)

"Well, a hole in the ground is beneath the floor, so it's negative" (Oh... familiar)

"Well, debt is like having negative money" (Oh... sadly familiar)

And so on. The concept "Less than nothing" could be axiomatically acceptable to a robot. But, to a human, it's new and unfamiliar. We want an existing example of this new concept before we can really "grok" the idea and accept that it's even possible.

And you know what's cool? This new abstraction (negative numbers) becomes a concept on its own. In the future, we don't need to go to the "dig a hole" level, we can compare new concepts (imaginary numbers) to negative numbers, which the elephant now accepts.

If we don't get concepts at this deep, visceral level, we never fully accept them and aren't fluent. It's like a kid in intro Spanish, translating phrases word-by-word in his head, vs. thinking in Spanish. "Donde... esta... el... bano?" Yes, that works. But not "Adonde va el coche rojo?" (Where is the red car going?) Because the word order and articles are different. The word-by-word translation fails, you need to "think in the language".

We need to "think in the analogy".

And yes, analogies are flawed. That's why they're analogies, not the theory.

"All models are flawed, but some are useful." -- George Box.

Start with a known, flawed understanding and improve it.

Addendum: Testing

Tests can help verify that your elephant understands the idea. Do you feel nervousness? Resistance?

Powering through and somehow answering the question doesn't help. The goal of a test is to FIND weak areas so you can shore them up. Tricking the tester that you understand something you don't isn't helpful! (Like fooling the doctor who is doing a malaria test... who is missing out?). The problem is tests are used for punishment, not diagnostics, so the reward(Passing test) > reward(Finding I have a cognitive error and correcting it).

Also: Our brains are relational machines. We relate things to other things, make connections! When you speak in analogies, you are giving examples of the types of connections to make it similar to. This hits you more directly, vs having to say "Here are some facts, now try to decipher them without making errors, and put those connections in your head".

No, I can give you the connections directly!

More emperor's clothes:

• It's normal to learn something and forget it the next year?
• It's normal to focus on passing the test? (Not the knowledge learned?)
• It's normal to have explanations that aren't really satisfying? (I.e., hit those connections)

Let's be honest, really honest about what works for us. Did you get it? No? Ok, let's see why. It's a smoke detector, something we want to know about!

If I have an explanation that makes no sense... it's because 1) I was unclear or 2) I don't understand it that well. Both are signs that we can improve it.

Example: imaginary numbers. Site after site, lesson after lesson -- how many show an example beyond "factoring a polynomial?". Why? Is it (gasp) because the author doesn't have an intuitive understanding? They need to parrot out stilted examples?

We're not doing anyone any favors by pretending we get it.

Even worse: we take a "ho-hum" attitude about these giant discoveries. Yes, negatives can have square roots. La de dah. Matter and energy are the same. The universe is a figment of your imagination, all matter is energy on a slow vibration. Do problems 2-20, evens.

What?! Can we even acknowledge the struggle we had to realize these points?

• Eigenvectors
• Transposes

Posted here:

http://betterexplained.com/articles/calculus-building-intuition-for-the-derivative/

The derivative is the heart of calculus, buried inside this definition:

So, what does this mean -- in English?

Imagine I listed out the daily stock market changes for the next few years (up 1%, down 3%, up 5%...). You could apply the changes one-by-one, plot out all future prices, and buy low / sell high. This might entice the monkeys throwing darts at newspapers (they do well, better than most managers) and you'd become a robber barron.

The derivative isn't simply the "slope of a function". Like the stock list, it gives a total understanding of a system. With this knowledge, you can plot out the past/present/future, make predictions, discover miniums/maximums, and yes, staff your simian workforce.

So what's with the gnarly equation above? Let's step back.

Calculus developed over thousands of years: that definition is the "DNA" description of a cat. Let's start at the "cats are furry and cute" level and work our way up.

In a sentence: derivatives create a perfect model of change by refining an imperfect one.

We all live in a shiny continuum

Infinity is a constant source of paradoxes (aka "headaches"):

• A line is made up of points? Sure.
• So there's an infinite number of points on a line? Yep.
• How do you cross a room when there's an infinite number of points to visit? (Gee, thanks Zeno).

Hrm. My intuition is to fight infinity with infinity. Sure, there's infinity points between 0 and 1. But I move two infinities of points per second (somehow!) and I cross the gap in half a second.

Distance has infinite points, motion is possible, therefore motion is in terms of "infinities of points per second".

Instead of using absolute counts (How many points did you move through?) we can compare rates ("How fast are you moving through this continuum?").

Analogy: See division as motion through the continuum

It's weird, but you can think of

as "I need to travel 10 "infinities" in 5 segments of time. To do this, I travel 2 "infinities" for each unit of time".

What's after zero?

Another brain-buster: what's the next number after zero? .01? .0001?

Hrm. Anything you can name, I can name smaller (take your number and halve it... nyah!).

We can't construct or point to the number after zero, but it must be there. Right? (I hope so).

Let's name the mystical gap to the next number "dx". I don't know what it is, I can't say how big it is, but it's there!

Analogy: dx is the "gap" to the next number.

Measuring change

The derivative is about predicting change. Consider the following:

Officer: Do you know how fast you were going? Driver: I have no idea. Officer: 95 miles per hour. Driver: But I haven't been driving for an hour!

95 mph is an instantaneous rate of change: you don't need to go a "full hour" to get your speed!

And how do we measure speed anyway? We take a before and after (over 1 second, let's say). If you go 140 feet in that one second you're at 95 mph. Simple enough.

Not exactly. Imagine a video camera aimed at Clark Kent. It records 24 pictures per second (40ms between photos), and shows him perfectly still. In one second he doesn't move, and his speed is 0mph... right?

Wrong! Between each 40ms picture, he changes into Superman, solves crimes, and returns to his chair. He moves too fast for your camera!

(Imagine a cop seeing your car in your driveway at 9am and 5pm, and assuming it didn't move).

Analogy: Like a camera watching Superman, the measured speed depends on the instrument!

Running the Treadmill

Now we're near the chewy, slightly tangy philosophical center. The change we measure can't be trusted -- it depends on how we ran the experiment!

Imagine a shirtless Santa on a treadmill. We're going to measure his heart rate in a stress test: we wire up dozens of heavy, cold electrodes (circa 1890) and let him start jogging.

He huffs, he puffs, and his heart rate is 190 beats per minute. Tada!

Sorry. See, the very presence of heavy, cold electrodes probably increased his heart rate! We measured 190, but who knows what it'd be if the electrodes weren't there?

It's more accurate to say:

• measurement = actual amount + measurement effect

Ah. If we do lots of studies, we might discover "Oh, we add an extra 10bpm for every kg of electrode". With this, we can subtract the impact and get a "perfect" measurement.

Analogy: Remove the effect of your electrodes when making a measurement!

Understanding the derivative

Phew. Armed with these analogies, we can understand how the derivative predicts changes:

• Start with some system to study: f(x)
• Change it the smallest about possible (dx) to get a before and after: f(x + dx) - f(x)
• I don't know how small "dx" is, but we can get the rate of moving through the continuum: [f(x + dx) - f(x)] / dx
• dx, however small, adds measurement error. Predict what happens if it's not there (take the limit as dx goes to zero).
• Get your final rate of change at every point, dy/dx

This last step is the key. We have two approaches:

• Limits: what happens when dx shrinks to nothingness, beyond our error margin?
• Infinitesimals: What if dx is a tiny number, undetectable to us?

Both are ways to say "What happens when dx disappears (to us)?". Early critics of calculus argued "Calculus is illogical: is dx zero or not? How can it be non-zero, you divide by it, and suddenly you throw it away?"

Example: f(x) = x^2

Nothing like an example to shake loose the cobwebs. How does f(x) = x^2 change?

Well, we can plug it into the derivative:

We're describing how x^2 changes over time. Note the difference in the last 2 equations: one has the error built in (see the dx!), the last is the "real" change where our measurements have no effect on the outcome.

Time for some numbers: here's the values for x^2, with intervals of dx = 1:

• 1, 4, 9, 16, 25, 36, 49, 64...

The absolute change between each result is:

• 1, 3, 5, 7, 9, 11, 13, 15...

(In this case, the absolute change is the "speed" since the interval was 1.)

Look at the difference from x=2 to x=3 (3^2 - 2^2 = 5). What is 5 made of?

• Measured rate = Actual Rate + Error
• 5 = 2x + dx
• 5 = 2(2) + 1

Sure, we measured a rate of "5 units moved per second" because we went from 4 to 9. But our instrument tricked us: 4 units of speed was the real change, and 1 unit was due to shoddy instruments (1.0 is a large interval, right?).

If we limit ourselves to the integers, then 5 is the perfect speed measurement from 4 to 9. There's no "imperfection" assuming dx = 1 because yes, that is the smallest interval between two points.

But in the real world, we could move a smaller amount! What if our dx was .1? Would we get a better measurement? Considering the change from x=2 to x=2.1:

• 2.1^2 - 2^2 = 0.41

But remember, 0.41 is the rate we traversed our interval of 0.1. Our speed is really 0.41 / .1 = 4.1. And again we have:

• Measured rate = Actual Rate + Error
• 4.1 = 2x + dx

This time, the measure vs actual rate is close (4.1 to 4) compared to 5 to 4 when using the giant steps of dx = 1.

Using this pattern, we see that throwing out the electrodes (letting dx = 0) lets us unearth the true rate of 2x.

In plain English: We analyzed f(x) = x^2, discovered the "imperfect" model of 2x + dx, and the perfect model of 2x.

Gotcha: The Many meanings of "Derivative"

The term "derivative" has a few meanings:

• "The derivative of x^2 is 2x" means "At every point, you are changing by 2x (twice your current position)"

• "The derivative is 44" means "At the point we're considering, we're changing at a rate of 44." For example, f(x) = x^2 at x=22 has an absolute rate of change of 44.

• "The derivative is dx" means "The tiny, hypothetical change we're considering is dx". Technically, dx is called the "differential" but I can tell you: the terms get mixed up, and people will say "derivative of x" and mean dx.

Gotcha: Integration doesn't exist

This blew my mind: the derivative has a specific definition. The integral, the opposite of the derivative, does not have a definition.

Here's an analogy:

• I can break a plate (the function) and see a pile of shards (the derivative)
• If you show me a pile of shards, I can guess what plate it may have come from (by secretly breaking a similar plate in the back room, and seeing if the shard pile matches)

See, there's no algorithm to find the integral, or anti-derivative (anti-derivative is a better term, I think). We essentially have a lookup table and say "Well, 2x is the derivative of x^2. Oh, your derivative is 10x? Well... scribble scribble... it looks like that came from 5x^2".

Finding derivatives is mechanical; finding integrals is art. Sometimes we throw up our hands: here's the list of changes (the stock market prices), apply them piece by piece, and give me an estimate of the original pattern (here's the pieces, use some computer modeling to recreate the plate).

Onward

Math is a language, and I'm still learning to "read" calculus -- to see the message, not the grammar, behind the definitions.

My biggest aha! was seeing the transient nature of dx: it takes a measurement, and is removed to make a perfect model.

Limits and infinitesimals are formal ways to "make dx go away". But don't get caught up in them -- they are safety mechanisms to protect us against the criticisms of Bishops. Newton seemed to do ok without them.

Why do we need limits? To make the derivative more accurate. Mindlessly introducing limits before the derivative is like giving spelling tests in a language you don't speak.

When you get the key idea, questions suddenly become interesting:

• What are the rules for making "dx go away?" (How do infinitesimals and limits work?)
• We can describe numbers without being able to write them down! ("The next number after 0"). Whoa! (Beginnings of analysis)

When the analogies are in your head, the questions become interesting. Happy math.

Extra Notes

Now, this isn't perfect: sometimes the effect of "putting dx in and taking it out" gives a different answer from dx never being there in the first place.

Functions where this happens are called "discontinuous" -- they're often jumpy and don't behave nicely! Thankfully, most "real world" (i.e., what you see in science) equations are nice and smooth, and our models of what happens when dx disappears is truly accurate.

Pretend you have a video camera aimed at, recording 24 pictures per second

• We need to make a perfect model of how something changes
• We make a guess with flawed instruments
• We modify the guess to see what'd happen if our instruments were perfect

However, it's taken humanity thousands of years to create the math:

The derivative lets us predict behavior.

roll our eyes, and begin learning to use it.

The "The essence of calculus can be understood with a few metaphors". In a phrase:

1) Calculus is based on a funky definition: here it is [derivative]

• Show analogies for each part of the definition

2) We can decipher each part with a few analogies

• Measure changes with the continuum with a rate
• Need a before-and-after
• Want the change to be small
• Use the model to predict a perfect-model

3) Tada: Put the analogies together: "Use a flawed model to predict the behavior of a perfect one".

• Generalization!
• Taking surveys (there is a "survey effect")
• Diets, people's behavior, etc. Your measurements effect the outcome!
• Figure out how large the "survey effect" is and correct for it.
• Publication bias, etc. (we get filtered reality).

4) Notes / caveats

• No analytic way to find the integral [really just trial and error]
• Difference from limit / infinitesimal -- justifying the step

==============

I've struggled to be fluent with Calculus. Sure, anyone can memorize stock phrases (Donde esta el bano?) and translate in your head -- how do we become fluent?

Immersion, analogy, and relating to what we already know. Calculus is the culmination of thousands of years of math and philosophy. Let's not pretend it's this simple 1-semester class we can memorize and move on from.

No, it's a way of thinking. Let's build it with analogies.

The essence:

We want to measure a changing thing. But the only way to measure it is to change it.

Did our measurement effect the result? Probably. So, let's make an estimate of what the measurement would be... if we had never measured (ooh, meta).

The continuum is infinite. Yes. But our rate of traveling through the continuum is infinite as well. These things can cancel.

Division gives us a rate of traveling through the continuum. We just don't call it that.

A number line has an awful lot of points

You've probably heard of Zeno's paradoxes:

• How can you walk across a room? First, you have to get halfway there (call it B). But to get to B, you have to get halfway there, to C. But to get to C, you have to get to D... and since you can't move through an infinite number of points, you can't arrive anywhere at all.

Most curious. And yet we move.

My resolution: Fight infinity with infinity.

Motion is also through a continuum. If going 10 feet means traveling through "ten infinities of points", well, that's ok: I'm moving "1 infinity of points per second" and will get there in 10 seconds.

What's after zero?

Take a ruler and shave off the bottom number (0). Drop it on the ground.

What position hits first? .1? .01? .0000000000000001? What's the smallest number?

I don't know. I can't constructively say how we move from 0 to the next smallest number. But "motion" somehow figures this out, and we can move smoothly from 0 to 10 in ten seconds.

Math phrase: Let's call the distance to the next point "dx". No, we don't know exactly how small it is. But it's there, and moving through the continuum makes it "ok".

Measure change with change

Imagine two arrows. I shoot one from a bow, and drop the other towards the ground.

And... time freeze! Like the Matrix (or Saved By The Bell), I halt all time, the arrows are stuck in place, and I walk around.

What's the difference between the arrows? How can I tell, when time is frozen, which is going to fly straight ahead and which will drop? Is there some property in the atoms that says "Hey, when this bozo remembers to un-time-freeze, keep going forward at 200 mph".

I don't think so. The "solution" to figuring out which arrow is which:

• Unfreeze time by a tiny, teensy, amount and see which direction each arrow nudges forward. Use that to figure out their speed (if the arrow nudges forward 1 millimeter in one trillionth of a second, I can figure out its speed).

Math phrase: Let's take the difference between two points in the continuum: x, and the next point, x + dx. f(x + dx) - f(x)

What's up ahead? Division vs. Subtraction

If I move 10 feet in 2 seconds, it means I moved

"10 infinities of feet in 2 infinities of seconds"

Sort of weird, right? But it gives us a rate:

10 feet / 2 seconds = 5 "infinities per second"

Even for generic "units"

10 units / 2 units = 5 "infinities of units per unit"

I.e., for every unit I go, I'm moving through 5 infinities of units.

Arithmetic takes on different meanings depending on what we're counting with. With integers, sure, multiplication is repeated addition. But it can be scaling (reals) or rotation (complex numbers) as we get more advanced.

With integers, division is "splitting into groups". Fine. But with real numbers and decimals, division is more "What is your speed moving through the continuum?"

10 / 5 = 2 means "If you divide 10 into 5 equal units, through each unit your speed is 2".

Now, we could give the distance from each unit to the next:

10 / 5 = 2 means "Start at the origin and move 2 ahead to get to each unit". In fact, subtraction is the raw distance, and "rate" is the distance / time.

The problem, as we saw before, is we have trouble with raw distances when dealing with continuums (what's the raw distance from 0 to the smallest positive number?). But we can deal with rates (the number line advances from 0 to 1 at a rate of "1". I don't know how, but I do know the rate).

It's a bit like saying "Oh, your next exit is about 5 minutes up the road." You didn't give a mile distance, you assumed a rate and gave a time to follow that rate.

It's definitely tricky. These are deep philosophical paradoxes. I'm trying to wrap them into the language of calculus.

Measurements are there and not there

Suppose you're measuring someone's heart rate in a stress test: hop on the treadmill and wire up the electrodes.

Now, imagine the electrodes weigh 25 lbs and randomly electrocute the patient.

What happens to the measurement? You might get a heart rate of 190 when their real heart rate is 140. The presence of the electrodes is tiring and scaring the patient, raising their heart rate! What you measure is not what is real.

The formula might be like this:

Measured heart rate = Real heart rate + weight of electrodes * 2

so

190 = 140 + 25*2

Clearly, you want the electrodes to be small and unintrusive -- but there's always some effect you need to account for.

Ah. There's a paradox: you need the electrodes to make the measurement but afterwards adjust the measurement as if the electrodes weren't there.

This is the "is dx zero or non-zero?" argument made against Calculus (the ghosts of departed quantities). My answer:

We use dx to create a model. Using the model, we predict what would happen if dx wasn't there! This is our estimate for the "true" speed (derivative).

Pretty wild, eh? By the way, this is a general concept:

• When doing studies, people are on their best behavior. Do you account for the fact that you're doing a study and adjust the results afterwards? Everyone sticks to their diet more when they're being watched!

In math: We try to measure the speed, over interval "dx":

We measured a difference of: f(x + dx) - f(x) / dx

and then we try to get the "real" distance: what would the speed be if dx wasn't there?

In math terms, we can say 1) take the limit as dx -> 0 or 2) let dx be an infinitely small number (smaller than we could ever measure).

Here's the rub: sometimes this limit exists, sometimes it doesn't. When the limit does exist, and our prediction works out, we call the function "differentiable". But some functions are finnicky (thankfully not many in the real world) and the "predicted speed when dx=0" is not "the actual speed at that point".

We have to make a measurement to see some change (remember the arrow paradox). And most of the time, we can account for the size of our change and remove it. But some finicky functions aren't well behaved, and we can't predict their speed when our dx is 0 (by a limit or by infinitesimal).

These are the subtleties of learning calculus!

Now, we can learn to read the definition of the derivative, of the integral, and how they're related:

• Derivative: Let's predict the speed of moving through the continuum at every point (distance / time = speed)
• Integral: Let's move through the continuum at that speed (multiply speed * time = distance)

They are inverses, which is the fundamental theorem of calculus. But... the derivative gives "change in distance" not absolute, so if you had a starting value it would be lost (changing from 10 to 20 looks the same as 20 to 30, they are both +10). So you account for that with the initial conditions, C.

Gotchas

There are lots of subtleties when someone says "derivative"

• They can mean "dx", the modeled change to the next point
• They can mean dy/dx, the speed you are moving through the continuum (a general formula for your speed at any point, like dy/dx (x^2) = 2x)
• They can mean a specific value (dy/dx(x^2) at x = 5 is 10)

The idea of the derivative: If you follow the path x^2 [at every time x, you are at position x^2], what is your speed? (2x)

The idea of the integral: If you keep your speed at 2x [at every instant, make your speed to 2x], what path will you take? (x^2)

Derivative: Given distances, get speed. Integral: Given speed, get distance

Notice how they break down for simple cases: Derivative of 2x is 2 -- at every instant, your speed is 2. (The pattern 0, 2, 4, 6, 8... has a derivative of 2... each neighbor is 2 away).

Integral of 2 is 2x. If your speed is 2, then at time x you've gone 2x.

Key insight:

• dx, dy, dz are miniature scaffolding we use to make a model, get an estimate, and then remove (with limits / infinitesimals)
• Continuous functions are ones where this scaffolding trick "works" (the estimate for when dx = 0 is actually what happens when dx = 0)

The goal is to get an intuition for what the derivative and integral are doing.

The derivative is breaking plates

The derivative "breaks" a function into its speed:

2x becomes 2.

The integral tries to recreate the function given its speed. And it's not easy! The operation is implied, i.e. "what function, when broken, looks like this?"

If I ask "What is the integral of 2" you say "What function, when breaking down its speed, gives 2? Oh, it's 2x. So the integral is 2x".

See how you worked backwards? It's like saying "What plate, when broken, gives this pattern? (And you show some shards on the ground)."

You start breaking plates and hoping one of them gives a pattern like the one on the ground. "Oh, it turns out that this green plate here, when broken, gives the pattern you're seeing."

It's an art, especially when the shards get complicated! It's trial and error to some extent, with heuristics (and computers) to help along the way.

Sometimes it's not possible to get an "analytic" function which behaves that way, so you have to just follow the speed and create your own function (following the speed you're giving us seems to have this path... so we'll call that the original).

Summary

Build a model to see how fast we traverse the continuum [whoa, pretty out there].

Notes

On Zeno's paradoxes:

"...and surely an infinite number of tasks cannot be completed in a finite period of time?"

No. We are moving through the continuum, always. We are always doing an infinite number of tasks. 4 infinities vs. 2 infinities.

Great vid explaining the essence of a tensor -- matching up components with basis vectors.

Rank 0 (scalar) : component, but no basis vectors Rank 1 (vector) : component & 1 basis vector (1 x 3) Rank 2 (array) : component matched with 2 basis vectors. (3x3)

(1 basis vector could be direction of surface, the other is the force applied in the x-direction).

Key useful property (which I need to understand): the combination of components & basis vectors can be agreed-upon by all reference frames (moving, rotating, etc.). What does the "combination" refer to? ("Coordinate-free" -- express relationship between vectors). More on coordinate-free geometry: http://www.cs.brown.edu/~ls/teaching08/LN04_Coordfreegeom.pdf

No absolute coordinates -- you just have a "point" and a vector (with direction and magnitude). But it's not on some grid with x and y components. Coordinates are needed to implement, but not reason about, the coordinate system.