Learning to 3D Model : 6 Days in December

6 Days in December is the steamy paranormal romance zombie... {read more}

...apocalypse rabies-sex movie recently announced by newly united writing trio Christine Feehan, Cecily von Ziegesar and Michael Bay.


A series of mysterious, slow-motion plane crashes in the Rocky Mountains has left thousands of Texas housewives in turn-of-the-century gowns with no oily-chested rogues to screw in the stables, and an army of sex-hungry vampire brides descends on the crash sites where the victims have become baby robot zombies through a chemical interaction with the plane's smoking fuselage.  Meanwhile our hero and his sweetie are mere miles away in an isolated cabin supplied with only the necessary birth control, cakes, and military-grade explosives.


    von Ziegesar: "Basically we've taken Jane Smiley's Ten Days in the Hills, and turned it into something people want to see."
  So you kept the sex.
  "Right.  Plus some vampires, haughty girls, machine guns.  Some kind of plague?  Dunno.  My job was the bitches."


The most intriguing part of 6 Days is the introduction of a new theater technology championed by Michael Bay.  He calls it AntethInc.  We caught up with Mr. Bay at an Adidam colony in Bermuda and he graciously agreed to an interview. {Continued in MOVIES: HILARY HAHN IN THE HILLS} 

_____________________________________________

MODELING THE HUMAN FIGURE

December 6th is my birthday.  This last winter, I gave myself a present.  For the first week of December, I modeled the human body.  Every waking hour.  I rose early, and stayed up into the tiny hours.  Remember to eat.  I slept little.
"Blenederella" is a DVD tutorial by Angela Gannette.   40 hours of screencast are condensed to 8 hours of video with timelapse and voice over.  She models directly from reference images.

I'm not an artist.  I  am learning to see in the reverse order:  surface normal, tension, line, cheek.  I practice seeing, and I will learn to draw.  Angela Guenette's approach was perfect for me: she is a subdivision vert-pusher.


It is not designed to teach anatomy or subdivision modeling.  It is for artists.  I used it to learn anatomy and organic subdivision modeling anyway.  Everything was new.   Time-lapse meant "...and so on," for the knowledgeable artist.  I have no such knowledge; I copied the time-lapse, vert-by-vert.  The video for the hand is 20 minutes long.  I spent over a day with it.  Often, I would advance through a few seconds of footage in an hour.


The DVD includes the finished model.  I examined it three times, for details not clear in the videos.  I did not use it as to place vertices, I borrowed no geometry.   I only corrected  what I could see.

 Thursday, December 1
Friday, December 2

Prepare the reference images: front, side, back, 3/4 profiles.  Align them in Photoshop: match height and body positions.  

Begin with the eyes.  Raccoon mask face, the bridge of the nose, trailing off at the cheeks.  Eyes.  Eyelids.  The socket.  Eyeball.

It took me three days to complete the face.  It was nothing like hard surfaces.   Thursday, on just the eye and socket.  Each new form caused me trouble.  The transition from the eye socket to the forehead is my mortal enemy.  We fight in space.

End of Day 2

Saturday, December 3

The chronology is off.  I must have started on the last day of November.  I spent three days on the head.  And I spent Saturday on the ear.

I was thoroughly vexed three times: the ear, the elbow and the hand.  All Saturday, the ear.  Perhaps sixteen hours. The ear is compact, folded, and thin.  I screwed it up.  Some subdivision mistakes can escalate. I made them.  Fixed it.  Screwed it up again. When I moved verts enough change the shape, my mesh became unruly.  I had made a mess of  the base geometry.  

Irreparable.

No. 

I almost gave up, which made me angry and I reworked the entire ear.  I talked to myself.

Why is it doing that?

Because you're telling it to.  Ask a better question.

Damnit! Okay.  Think.  Describe what is going wrong.  Describe how you want it to change.  Look at your geometry.  What do you need to fix?

Here is a consistent strain.  The most difficult obstacles, which caused me the most doubt and fear have been those of which, when overcome, I am most proud.  Not because I did it.    I look back, and it is my best work.

Sunday, December 4

Now I was worried I wouldn't finish by my birthday.  I  planned it out, set limits for the amount of time I could fiddle with a problem. 

The torso.  Here I learned the most about human form.  After building a rough cage, she added guides along the body.

Guide lines of the body; shaping the shoulders

Orthographic and perspective views

I spent most of my time with two viewports: orthographic on the left, with reference image.  Perspective on the right, shaping the body in space, and adding detail.  The back is shaped with few vertices: Guenette described each adjustment, the bones and muscles she was following.  I want more than ever to learn anatomy; to draw.

This is when I started to understand what can be done with subdivision.   When I removed the guides, I was stunned.  Even my mistaken placement, causing an extra, odd bulge, shows clearly as if the musculature and bone were there.

Monday, December 5

The elbow caused me no end of trouble: I never got it right. To move on, I promised myself a return visit, with pencil, paper, and anatomy books.

Okay, look.  I knew the hand would be hard.  I had plenty of warning.  Artists tell stories about hands.  And I work with my hands.  About the motion of fingers, I have a very good idea of the difficulties.  I still had no idea how hard it would be to model.

First go at the hand

The first hand.  I didn't like what I was building.  My lines were wrong and I wanted more detail. My tendons looked like metal rods.  So I started again.  Her fingers, I left the same as here.  But I deleted the palm, turned my hand over, and worked it out, using the muscles and creases of skin as guides. 

Tuesday, December 6

I stitched the wrist to the rest of the body on my birthday.

Of all my 3D work, I am most proud of this hand.  

Test render of the final mesh

Follow-Up: Hair, Clothes

Rigging and Animation

I've done the hair and the basic clothes.  I will finish the clothes from the tutorial, which includes boots, to cover her missing feet.  Still have the entire process of rigging a bone system, and beginning to make basic movement.  After that, materials to replace the placeholders shown here.

After that, there is still a lot to do before I have a Snow Queen.  Clothing is altogether a new matter.  What I intend, I have never seen.

A Promise

205 left to go.

 ■

__________________________________

{PARANORMAL ROMANCE cont'd - read more}

R:    How's Adidam treating you?
M.Bay: {grimaces} Too much Bulgar wheat.  Where are the goddamn spaceships?
R:     ...sssspace?
MB:   Astral messiah my ass. The little fat guy over there keeps painting himself like an Asian and trying to pork my wife.
R:    So we've heard rumors about "Six Days in December."
MB:   Well, first of all it's "6" Days in December.  And second, it's the first ever AntethInc movie.
R:    And what exactly is AnteThink?
MB:   It's post-thought.  The evolution of story to its highest form.  In a post-thought world your brain can stop completely.  No message.  Just 102 minutes of Hilary Hahn's bare ass, brainthirsty-
R:    Wait... Hilary Hahn?
MB:   -zombies, and giant robots.   And, yes.
R:    The violinist?
MB:   The what?
R:    Violinist
MB:   Ungaramist!
R:    .... curly hair?
MB:   Vagosaurus!
R:    ...plays Bach like the sky is
MB:   Paleidobak!  BOOM!
R:    -trying to spea-
MB:   AERODROME!
R:    CHACONNE!
MB:   ROBOTS.  Giant fucking robots. Giant robots, fucking.  Bigger-Than-A-House-bots.  Zomboloid Bots.  Talking robots that talk. Talkybots.  They say, 'BOOM', and 'POW', and they run on energy,  but mostly blow things up.  They're made out of babies I mean zombies. They're made out energy fucking zombie babies.  It's cold and they kill shit and the zombies spread rabies by infecting the robots with spores.  It is the greatest thing I have-
R:    Um.   You were saying about AntethInc-?
MB:  ever done.  Yes.  Right.  For one hundred two straight minutes you will not think at all. We're passionate about this goal.  Thinking will cause you pain.  Before, I would have said, No, that's not possible.  But after meeting Ms. von Ziegesar...... well..... she opened my eyes to a whole new world.  102 minutes of ass.
R:    ....Hilary Hahn?
MB:   Yeah.
R:    ...the violinist.
MB:   Flabbelotist?  Magzamanist?  Speak English.  Stop wasting my time.  I order you to stop thinking.
____________
Ed:  ...The Violinist.
Try on the Bach Chaconne.

Snell's Law, Part IV: Arbitrary N

             {<−− Part 3}

The angle measure between vectors remains unchanged if the systems of equations which define them are translated, scaled or rotated.  Might as well check.

In the image below, N is no longer aligned with the z-axis.  The surface has been rotated by an arbitrary, positive angle, φ  (phi).

Phi, fie, fo fum. 

Ummmmmmmmm.... 

Oh.  Right.  The math.

We have, again in two parts which may be done in any order:

I. Calculate T for any η₁, η₂,  N and I

The dot products:


And so, as before:

The sign of the square root is taken positive, as defined in the last section.  I makes a positive angle with N less than 90° (π/2 radians).

II.  Give T as the sum of two vectors

               αI         +       βN         =        T,               α, β constant:


Using Cramer's Rule:


Solve for alpha:


Solve for beta:


Radness.  And at last:

THE RESULT

...being the same equation as Parts I and II.  So don't do it this way, because it's harder.
(It's over now.)

{CodeCogs you are the warp and weft of my Latex heart}

{<−− Part III}                                                                        {Part V upcoming}

Spatialization 01a

The following simple question has become a test.  I ask it, when I meet Audio People.

_________________________________

There are two objects:

1) street

2) streetlight

As the player walks around the light, the sound changes, as it does in real life.

_________________________________

I am speaking to a Unity 3D rep.  I have asked what tools they have, to spatialize objects.  We have gone through the usual misunderstandings.  No, I do not consider line pass-through a "high level" feature.  No I don't mean canned reverb.  I want to use the game environment.

Why don't you tell me what it is that you want to do.  He has twenty years experience as a studio recording engineer.   He is sure he can handle any questions I have.  He sticks his thumbs in his belt, relaxes into it.  He stresses the word, engineer.  I have asked the right person.

It takes him awhile to understand what I am asking.  Not Doppler.  Not panning.  We could walk outside right now and record it.  Eliminate the other variables: Walk in a circle.  Point the microphone directly at it, all the way around.  Record in mono.  The sound of the light still changes as you walk around it.

This is when he scowls.  He has no idea what I am talking about, and he has just put his thumbs in his pants.    His responses are surly.   I have not been rude; I'm excited.  I have used no jargon.  I have reduced the question to its simplest form.  He didn't know it existed.

So I will begin at the beginning:

Stationary Object, Moving Observer

This question is the simplest case of motion in 3-Dimensional space.*  Assume we have a physical space, and a source of some vibration.  Waves propagate in space, from the source.  Assume the geometry is static, and the source immobile.  Integrating and differentiating the field at any one point is no more difficult than another.  The most important reflections for a particular observer position can be precomputed.  Or the the entire field can be baked.

So what are the transforms necessary to render this scene?

______________________

{

*Moving Object, Stationary is {I believe} more complex.  The source of the field moves.  I will visit fields in depth later, when I am not new to them myself.

}

Who Crapped in My Gauss?

{Grumpiness omitted.  New title}

Knickers: A Case Study of panted Bunnies

   (begin music)

  (begin bunnies)

Modern abstract mathematics.  I should have started here:

Gauss' Disquisitiones Arithmeticae

From the introduction:

"If, for a number of difficult questions, I have used synthetic demonstrations and left out the analysis which led me to them, I have decided to do so out of the desire to reduce the length, to which I had to conform as much as possible."

Synthetic proofs obscure the path.  They are like a magic trick: the apparent simplicity is a lie.  Pick it up, turn it over, and it will close its mouth as you hunt for a method for solution.  Clever little lens.

At least if Gauss does it to me, it will be on purpose.  That is a challenge I can accept.  It will be a nice change. 

(end bunnies)

....music.....

Learning to 3D Model

Ten months ago, I made my first 3D model.   I have no training or skill at art.  Bust out the crayons and I hold my own with the nine-year-olds.  I don't know perspective.  I can't shade.  I can't make an outline of a head.  But I'm stubborn.  How to explain?

"Giving up makes you angry."

Right!  That's just it.  Thanks, Peter.
Giving up makes me angry.  So I refuse to do it.  I have learned  to 3D model.

Blender

I don't steal software.   It shouldn't have to be said, but it does.

3D Studio is a $3000 application.

Blender is free.  It weighs in at 80MB of hard drive space.  It is a fully functional, workstation-class 3D suite.  It includes a Z-Brush style sculpting interface with full, pressure-sensitive artist tablet support.  IK/FK bone systems, animation.  Blender is only infrequently a compromise.  Compared with 3DStudio Max, there are some things I wish I had.  The biggest oversight, to me, is the lack of N-Gons.  The community is large, and the last year has been spent in a push to consolidate the feature base, integrate new features, including a cleaner, friendlier workspace.  The improvements are impressive.

Did I mention  Blender is a professional, workstation-class 3D suite?  Just download it. http://www.blender.org
Don't worry about hardware.  I do all of my work on a 2-year old Mac Mini.

If you have tried Blender in the past, it is not the same application you used.  I began with version 2.49b, less than a year ago.  The current version is simply a different program.

The next thing to do is watch tutorial videos.  Do not attempt to use the online manual, or you will lose your mind.  You have been warned.

In addition, I found this thread at Polycount (about Subdivision modeling) to be a self-contained course in 3D modeling in general.


Andrew Price Tutorials
{Blender Guru website}
Andrew Price taught me to 3D model.  He is funny.  He works quickly and efficiently.  His work is professional.  The final renders for most of his tutorials look like photographs.  He opens each video with a screenshot of the scene to be built, talks about it, and then does something wonderful.  He closes the final project, opens a brand new, empty scene, and builds it again.
And at the end of a one-hour video, although it may take me six hours to complete, I have built the same scene.  He covers all aspects of the 3D process: modeling, lighting, textures and materials, rendering, compositing; rigging, animation, simulations.
I do not know how to thank you enough, Mr. Andrew Price.

I. Subway Tunnel

Final Render, Composited

I learned a lot about modeling and modifiers from the Subway scene.  He works very quickly, making efficient use of arrays and paths.  Vague outlines of objects become the correct shape like magic, with a SubD modifier.   There will be a train in the heart of the Asteroid, but I still needed to adapt the scene a bit...

I have only made a few changes.  My goal was to prepare the assets for a game engine.  Utility tunnel systems will ramify throughout the Asteroid Museum, and I sought an aesthetic which would translate well to a game engine.  in this case, a design where unnatural, hard shadows are an integral and descriptive part of the world.  The reflections still have to go, but  mostly I need the sheen, and a sliver of light, which can be faked in the floor material.  The important issue here is that these lights can be dynamic, without compromising the lighting quality.  The geometry is extremely simple, and enclosed, allowing more complex and dynamic lights.

II.  Asteroid

This was a fascinating tutorial.  If you know how to 3D model, you can guess just how easy this was to make.  Modern 3D programs are incredible.  The smaller rocks are a particle system; they are animated in time.  In Blender, the scene looks like this:

The black box in the upper left corner is the camera.
As time passes, the rocks fly past the camera.  I have not rendered an animation, but it is as simple as pushing a button.  The dispersion camera effect is done after the scene is rendered, in the Compositor.

III.  Bullets! 

Final Render - Composited

Another astonishingly easy, eye-opening tutorial.  This is all texturing.  And the textures are not from bullets.  A grungy wall texture, some dirty concrete, and an image of deep scratches in some uknown material.  The actual modeling of the bullet takes 2 minutes.  The camera effects are all done in the compositing stage.

IV. Architecture - Modern Korean Kitchen:

Here, I have done a lot of my own work.  I made my own toaster, knife, utensils, bottles, oven, which are not covered in the tutorial.  The red pot, and the cute little Sorapot below are both from scratch, from reference images.  For a game environment, I modeled the inside of cabinets and drawers so they can be opened.  I also added surrounding hallways and rooms, a window. This room will live in a full house, in the game world.  

The tutorial moves on to a physically-based renderer.  All the materials are simple, and rely upon accurate lighting and reflection.  None of this will do for a game engine, so I stuck with Blender's internal renderer, and changed the setting.  In the following images, I am weaning myself off reflections, and toward soft, white moonlight and specular highlights.

    What to do about reflections?                   ...Soft moonlight to the rescue!

              

I have not completed the transition, but I think the approach is clear.

IV. Realistic Earth

Sometimes you just have to be done.  I am used to freely manipulating shader nodes in UDK, and I kept trying things Blender's aging  (and now replaced) renderer was not equipped to do.

So I'm done.  I will work out an approach to 3D starscapes when I finish modeling Saturn's Ice Queen moon, Helene.  The Cassini Orbiter's public image library is now sufficient to construct a full, 3 dimensional model.  Incredible.

V. Fluid Simulation

Meet Blender's new render engine, Cycles:

Now, these are meshes.  The water is a fluid simulation/animation, stopped at a particular frame, and rendered.  It is a series of solid objects.  Andrew has a clever approach to modeling ice cubes.

VI.  Rendering with Cycles

The tutorial is an introduction to lighting and materials with Blender's new render engine, Cycles.  He spends no time on modeling; this is the only time I have used the reference meshes, which weigh in at a total of just over 1600 vertices.  Tiny.

To me, this scene is an advertisement for Cycles.  I don't have the hardware to explore Cycles in depth, and  time spent this way is of little practical value for game engines.

I have three uses for such work.  First, the more I know about real lighting, the better  I can manipulate shader code to fake it.  Physically accurate render engines are that good.  I can ask it questions about light, and it will show me answers.
Second, I will be ferrying light from place to place in the Asteroid using natural and artificial prisms and vault lights, liquids, scattering over distances and through different media.  I can build concept art in Cycles to help guide me, and test the physical behaviors to construct the materials and scenes.

Third, Cycles allows direct manipulation of node-based shader materials, in a physically accurate environment.  This is an excellent way to learn about the capabilities of graphics hardware and languages.

More to come... 

Some or All Roots Imaginary

Chapter 4: Linear Differential Equations of Order Greater Than One {pg 196}
About this point in Tenenbaum and Pollard's Ordinary Differential Equations, my copy starts to fall apart.  The Linear Differential Equation of Order Greater than One is the fundamental groundwork for harmonic motion.  Oscillators.  Elementary electric circuits.  I collect here the relationships I review most.

     Definition 18.1.      A linear differential equation of order n  is an equation which can be written in the form


where f₀(x), f₁(x), . . . , f_n(x), and Q(x)  are each continuous functions of x defined on a common interval I, and y⁽ⁿ⁾ denotes the nth derivative of y.    y and its derivatives each has (ordinary) exponent one.  An ordinary differential equation of order n cannot have terms such as y³ or  [y^(k)]².

If f_n(x) is everywhere zero, the equation is of order n-1.
If  Q(x) is not everwhere zero, the equation is nonhomogeneous.  
If Q(x) is everywhere zero, we have the

Homogeneous linear differential equation of order n:


This is the case of interest to me.  At least until I reintroduce offset and other transfer functions which music tools take care to avoid or remove, but which result from many common forms of motion.  Q(x), you shall have your revenge.

SOLUTIONS

If we further assume that each function f_k(x) is a constant, then we obtain the following expression {pg 212}:


We can cleverly suppose y = e^mx is a solution, (or let someone else do it, and read their book), and find this is no sooner said than done.  The question then remains, for what values of m?  Some straightforward differentiation, and at last we have an answer to the problem.  Each value of m which satisfies the algebraic expression


will make y = e^mx a solution of (20.1).  (20.14) is the characteristic equation of (20.1).

 ________________

... skipping Roots of Characteristic Equation Real.....

Alas.

 _______________

LESSON 20D.  Some or All Roots of the Characteristic Equation Imaginary.

My solutions are not real. They are imaginary.  All of them.  Oh good!  What could possibly go wrong?  At this point in the film, he ignores the music, opens the closet, and is killed by the monster.

My plan is to eat the monster.  Act casual.
Hey.   
So... you eat people in the closet.   
I like what you've done with the place.  No, really, it's very 1950 East Berlin.
Is that what happened to my cords?
I see you left the lamé.   No.  I totally disagree.  I think the gold would have been a nice accent.
Where do you--  UPSTAIRS?  You mean that wasn't the CATS?  No, I don't know any teenage gir- look, I'm not having this--  Who's holding the Cthulu talking mask?  Huh? That's right, I am. {It's a DQ crown.  Don't tell} You can talk when it's your turn, or be eaten by space AIDS. 
 That's what I thought.  Look, I'm not trying to be a bitch here, but I had an imaginary monster in my closet when I was 5.
I was over at Steve's the other day  and the All Roots Real in his closet called you a woos. He said you butcher like a baby.

Signal processing books assume knowledge of the equivalence between an exponential solution and one written in sines and cosines.  For the second order equation, there are four forms.

Second Order Equation:

If the coefficients in the characteristic equation are real, the imaginary roots must occur in conjugate pairs.  For the second order equation, call the two roots  α + iβ  and  α - iβ.  Then the general solution is
          

Using Euler's formula,

Substituting these values into the first form and simplifying:

Let new constants

 Then we have
          

The imaginary part has been subsumed into c₂.  The substitution is trivial as far as it goes, but we are heading somewhere with this substitution.  In what follows, I must be clear about the operation of complex numbers in the triangle inequality (Pythagorean theorem).  I find this part continues to cause me some discomfort, which is part of the reason for this review.
Next we multiply by 1:
               

....right?  Hehehe.   Okay, but seriously.  Let's build the right triangle with legs of length c₁ and c₂.  It looks like this:

Fig. 1 - a truck of c's

Which gives the identities:

Time to take stock.  There is not yet any significance to the triangle drawn.  The constants have been rearraned into two new quantities: an angle and a magnitude.  Also,  c₂ lies on the imaginary axis.  I'm grumpy.  I demand that these two new quantities, and their arrangement in space, have physical meaning.  I expect

1) the magnitude be a real-world magnitude: a nonnegative scalar quantity which is significant to the solution to a problem, which is equally the sum of component (forces).

2) An angle measured along an imaginary circle perpendicular to the real plane of motion, the sine of which remains measurable in physical space.

O, physic!
I have played into your hand again!

The equations now read:
          

And the same procedure can be used to construct
          

where c is amplitude and δ is phase angle (offset).   The value of δ will be different  in form 4:  in Fig.1, we now measure from the upper right (corresponding) angle.  The two functions 3 and 4 are everywhere identical.

Continuity: Regions

IN THE BLUE CORNER, Advanced Calculus:

"

Multiple Integrals  

§ 1. Introduction 

In this chapter, we shall discuss double and triple integrals.  We shall follow as closely as possible the analogy with the theory of simple integrals developed in the previous chapter. 

1.1 REGIONS
     We have already discussed in Chapter 1 regions of the plane.  Let us collect here the notations which will be needed in the present chapter.
     A domain D is an open connected set of points.  That is, every point of D is the center of some δ-neighborhood, all of whose points are points of D; and any two points of D can be joined by a broken line having a finite number of segments, all of whose points are points of D.  A domain is bounded if all its points lie inside some square.
     A region R is a closed point set consisting of a bounded domain plus its boundary points.  We shall assume further that the boundary of R consists of a finite number of closed curves that do not cross themselves nor each other.  Note that the regions here defined and designated by the letter R are special cases of the more general ones of §3.1, Chapter 1.  In practical problems, R will usually be given in terms of its boundary curves.  For example, R might be the set of points between two concentric circumferences plus the points on the circumferences.  More frequently, we shall meet regions that can be most simply described by use of functions. Accordingly, we shall have a special notation for these.
     Let φ(x) and ψ(x) ∈ C in a ≤ x₁ ≤ b and φ(x) <ψ(x) in a < x < b.  Then the region R_x or R[a, b, φ(x), ψ(x)], is the region bounded by the curves   

x = a,     x = b,     y = φ(x),     y = ψ(x) 

If (x₁, y₁) is a point of R_x, then a ≤ x₁ ≤ b and φ(x₁)≤ y₁ ≤ ψ(x₁).  A line x = x₁, a  < x₁ < b cuts the boundary of R_x in just two points.  For example, the region R[-1, 1, − √(1 − x²), √(1 − x²)] is the circle x² + y² ≤ 1.  We could define in an obvious way a region R_y.  The region R described above as lying between two concentric circles is neither an R_x nor an R_y. It could be divided into four regions R_x, for example, by two vertical lines tangent to the inner circle.  These vertical lines would be counted twice, as the boundary of adjoining regions.
     A region R is simply connected if its boundary consists of a single closed curve.  The concept of the area of a region R will be assumed known.  Of course, the area of R_x is known from elementary calculus, and the area of R could be defined by use of a limiting process.  The diameter of a region R is the length of the longest line segment that joins two points of R.  In the case of a circle this coincides with the elementary notion of the diameter.  Observe that, if a region R varies so that its diameter approaches zero, then its area also approaches zero.  The converse is not true. 

1.2 DEFINITIONS
     We begin  by dividing a given region of R of area A into subregions.
As in the case of simple integrals, we introduce certain simplifying notations.
"
-Daivd Widder, Advanced Calculus

IN THE RED CORNER,  Euclid of Alexandria.

"The extremities of a surface are lines."
-Euclid, The Elements, Definition 6

_______________________________________ 

Notes

There are important differences between the two quotes.  We need the symbols necessary to operate algebraically, and to perform the Calculus.  To define arbitrary surfaces.  Our pencil is the function.  Still.  The surfaces we will be operating on in Chapter 6 of Widder's Advanced Calculus are precisely those which are defined in Euclid.  The definition is rigorous.  And so,

I have some questions for you, Modern Abstract Algebra.  And I don't mean Gauss.

1. What is the difference between a Dedekind section and Euclid's Definitions?
2. What is the relationship between infinite division and continuous measure?
   (Hint: No matter what anyone says, I can prove false by counterexample any mathematical argument which concludes by proof that I, Ryan, can measure.  All I need to do is disagree.  Likewise, a proof that I cannot measure.  All I need to do is say, "one apple."
   And likewise, I can prove any assertion that integral, unbroken measure does not exist, by waiting for a new instant in time, unbroken from the previous.  Or taking a step, some distance in some direction.  Left foot, right foot.  I defeat you, fake divisor-paradox!  Time for a nap.)
3. What is the logical status of a postulate?
4. I say, mathematics is not a logical circle.  Why?

I think, Mr. Modern, you do not answer these questions accurately.  Prove me wrong.  I'm asking for it.  See?

{Thus goad I a concept.
 Poke. 
 Thus}

Vectors: Grad, Div and Curl

Another chapter down!  This is my favorite part.  It's like someone is reading my mind.

MATH: Were you looking for this?
RYAN: Um.  I dunn-
                  Omigod.  ..... Ummmmm.  This is totally awesome.  Yes. Yes I-
MATH: There's more, if you'd --
RYAN:  ...was.   wait.  What?
MATH ... stop talking.
RYAN:   Oh.
MATH:  Check this out:

A scalar function φ is said to be harmonic if it is continuous,

 has continuous second partial derivatives, and satisfies Laplace's Equation:

      

RYAN:  How did you know I-
MATH: I'm just getting warmed up.  How about this.

RYAN: But how -?
MATH: ...do you define the gradient of a vector field?  Turn the page.
                    {rustle}

     Like that.  Now try this:

     And you have a formula for computing Del squared of a field:

     You should probably check the partials by hand.
RYAN:  Yeah, I-
MATH: Oh, and you're welcome.
RYAN:  Hey! I was go-
MATH: Sure you were.
RYAN:  This shit is fucking amazing.
MATH: {Yawn}Yeah.  {Stretch}  I know.  So, I see the Indian girl is still at the next table.  You know it's not because she wasn't going to drink it, right?
RYAN:  Wait. What?
MATH:  The beer.  You should talk to her. She's taking a long time to put her hair up for you.  You should probably ask her out.
RYAN:  Look, buddy. I'm not a mind reader.  If she wants to talk to me, I'm right here.
MATH:  Spare me.    And good luck with that.
RYAN:  With what?
MATH:  ....
            ....
 RYAN:  What?
MATH:  You are a complete idiot.

____________________

I am a complete idiot.  Anyway, Vectors continue to contain more Radness per component of space than any other container.  Expressed in terms of  BMX,

Where grad(BMX) is the direction of maximum increase in BMX, in mega-wheelie-hours.

I tend to double, triple, sextuple-check identities.  That has to stop.  The vector algebra identities involving Del take an Awful Lot of Symbols to verify by hand.    I did them all. A few results will live here so I can find them.  Like

curl (curl f) =

Notes:

1) The proof is just a verification of consistency:

If the operator distributes partials according to

       1) the usual product rules for vectors,

        2) swapping dot product order  to allow Del to operate on something

Then a result is obtained which is algebraically consistent.  In other words, it might as well be a definition.

I think it is self-evident that the intermediate, undefined form grad(f)  preserves measure if immediately operated on by dot (scalar) product.    It is not a proof that grad(f) has any general meaning.  

I assume the justification for adoption of the algebra itself lies in physical measurement of space. 

2)  I have almost found my difficulty with partial derivatives, implicit functions, and multiple integration.  The place where all of the differentials come together remains unclear.  There is more than one way to integrate multiple variables. It depends on the relationship between all quantities, there are many such quantities, and I cannot convince myself that substitution of a variable is trivial.  I think very soon I will have to go back and build diagrams with every little dx, dy, dz,  and work out each connection to a partial or substituted variable.

Grrrr.

In the meantime, moving forward is helping, so I will keep doing so.  On to Vector Integration.

Your Concern That It Is Awful

 to be me is misplaced.

Nine days ago, in the evening, and after throwing off the bear shoulders, I grew a celesta. My lungs were glass bells. Washboard, guitar string hips.  Accordion bellow arms.  I waltzed the orange light as she slipped upward through  my hands, opening to gray-blue. Alone with the cool-waisted dusk, I ran wood wrists against my grooved, silver legs.  I sang.

But you were not worried about the dreams.  It is the machine-work you do not like. It is plain on your face.  "Strange."  Say it again.

"I would hate to think { } caused you to think this way."
It has not.  My mind is what I have made.  There are two persons in your frown, one broken and one fixed, and I am neither.   I know it seems to you I am unwell. And if you were wrong?  How would you check?  I do not say to you, "my wood wrists."  I said nothing about the Snow Queen.  I said, that I reason. That it is not difficult, it is not intelligence.  The times I am misunderstood are oftenest when I use words in an ordinary way.  I do not expect people to think "on the same level."  I expect them to mean what they say, and when they choose to declare truth, when the words are "therefore" and "because", I expect those words to hold.  I am not the one who has chosen them.  This is strange to you when I say it, that I choose to reason.  This conversation is no longer about me.

You write it down,
"... the idea that what I understand to be thought; what it means to me to be alive and possess a mind, is somehow fundamentally wrong or broken..."  You like this sentence.

"Are you sure you've always thought this way?"

"Yes.  In fact, I think this way so completely, that I did not even realize--"

"I mean, even when you were younger?  When you were 10?"

Oh.  I see. The window has shut.
"Yes."

This question is irrelevant; I am not broken.  And I am not a child.  I have had a lifetime to consider; to choose what I will think and how.  Those things about my mind, body and heart which are beautiful, of which I am the most proud, did not exist when I was 10.  I built them.  It is the one thing  I have done which is the most beautiful.

Vector surfaces

I haven't forgotten about the schoolgirl problem or Snell's Law.  I (finally) have a  correct general solution for Kirikman's schoolgirls: a puzzle consisting of simple blocks with which any of the particular solutions can be built.

I intend to finish Applied Vector Analysis in a month.  8 Chapters remain; most of the math I need is here.  

• DifferentiationSpace curves
  Surfaces
  Tangent, curvature, torsion

• Divergence, Gradient, Curl

• Integration
  Line, surface and volume integrals
  Gauss' (Divergence) Theorem
  Green's Theorem
  Volume-to-surface; surface-to-line transformations
  Irrotational Fields
• Orthogonal Curvilinear Coordinates
  Cripple Special coordinate systems

• Fluid Mechanics:
  Equations of continuity, motion, and energy
  Fluid States
  Steady flow and streamlines
  Vortex flow and circulation

• Electromagnetic Theory:
  Electromagnetic field
  Maxwell's equations
  Potential
  Energy and the Poynting vector
  Static fields

Rad.
I'll take notes here.

PARAMETRIC SURFACES:  NORMAL VECTOR

 The normal N to any surface represented by a vector function r(x,y) is given by

     
where
     


I. Explicit function of x, y
Let a surface S be given by the equations
     
where x, y, z are the coordinate axes.  Then the position vector of any point on the surface is given by the vector function
     
Compute the partial derivatives,
     

 Compute the cross product:
     

And the magnitude, for normalization:
     

The normal vector to any point on the surface is then
     


II. Implicit function of x, y.
I'm still not comfortable with partial derivatives of implicit functions.  I lose the intuition quickly, and return to the definitions often.
Fix it, Ryan.
Okay.  The following result makes me happy.

"

Problem 4-11  Find a unit normal vector n to a surface S that is represented by                                                

"

Go one step at a time.  This is a surface.  Any surface can be represented by a (series of) function(s) of two variables.  So, the given function has two independent variables.  Phi implicitly defines a function of two variables.  By convention, I assume the independent variables are listed first, and are therefore x and y.

Phi defines z implicitly as a function of x and y.


Differentiate with respect to x:
     

Solve for the partial of z with respect to x:
     

Differentiate with respect to y, and solve for the partial of z with respect to y:
     

Now, we have implicitly defined z as a function having the same form as the explicit equation in section I. Substitute the conspicuously Jacobian values (1) and (2) into the equations from I, and we now have the desired partial derivatives:
     

Again, the cross product:
     
 The relationship is clearer, and normalization simplified, if this is rewritten as:
     

Normalize!
     

Goodbye, extra z-partial.

     

Notes:
1. It is not clear to me why the implicit function is differentiated without reference to the unit vectors i, j and k. "Because it's implicit".  Yeah.  I get it.  But rendering z = z(x,y) as a vector function is independent of the function's status as implicit or explicit.  The answer is about measure, and I need to find it.

2.  In the implicit function, it doesn't matter what x, y and z are. One of the three partial derivatives of φ is nonzero, or it does not meet the general definition of a surface.  This nonzero partial corresponds to the dependent variable.  I assume the partial derivative of φ with respect to z is nonzero.  If it turns out to be x or y, the result can be computed in the same fashion.
The convention is for simplicity: list the dependent variables last.  It saves the following step: suppose I intend to solve problem 4-11 and I require the dependent variable in the third position.  Then I will solve the system

instead, where w is the dependent variable.  Map the variables x, y, z in problem 4-11 to u, v, w accordingly as the surface function is known.  I do not have to change a letter of the solution.