Popper's Dreamland

Thursday, March 8, 2012

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

$N= \frac {\boldsymbol{r}\!_x \times \boldsymbol{r}\!_y } {\left| \boldsymbol{r}\!_x \times \boldsymbol{r}\!_y \right|}$
where
$\boldsymbol{r}\!_x = \frac {\partial \boldsymbol{r} } {\partial x}, \;\;\;\;\; \boldsymbol{r}\!_y = \frac {\partial \boldsymbol{r} } {\partial y}$

I. Explicit function of x, y
Let a surface S be given by the equations
$x = x, \;\;\; y=y,\;\;\; z = z(x, y)$
where x, y, z are the coordinate axes. Then the position vector of any point on the surface is given by the vector function
$r(x,y)= x \boldsymbol{\hat\imath} + y\boldsymbol{\hat \jmath} + z(x,y)\boldsymbol{\hat k}$
Compute the partial derivatives,
$\begin{align*}\boldsymbol{r}\!_x = \boldsymbol{\hat\imath} \;+ \frac{\partial z}{\partial x}\boldsymbol{\hat k} \\ \boldsymbol{r}\!_y = \boldsymbol{\hat \jmath} \;+ \frac{\partial z}{\partial y}\boldsymbol{\hat k} \end{align*}$

Compute the cross product:
$\boldsymbol{r}\!_x \times \boldsymbol{r}\!_y \;\;=\;\; \begin{vmatrix} \boldsymbol{\hat \imath} &\boldsymbol{\hat \jmath} &\boldsymbol{\hat k} \\ 1 &0 &\dfrac{\partial z}{\partial x} \\ 0 &1 &\dfrac{\partial z}{\partial y} \end{vmatrix} \;\; = \;\; -\frac{\partial z}{\partial x} \boldsymbol{\hat\imath}\; - \frac{\partial z}{\partial y} \boldsymbol{\hat\jmath}\; + \boldsymbol{\hat k}$

And the magnitude, for normalization:
$\begin{align*} |\boldsymbol{r}\!_x \times \boldsymbol{r}\!_y| &\;\;=\;\; \sqrt{ (\boldsymbol {r}\!_x \times \boldsymbol {r}\!_y)\cdot (\boldsymbol {r}\!_x \times \boldsymbol {r}\!_y)} &\;\;=\;\; \sqrt{ \left( \frac{\partial z}{\partial x}\right)^2 + \left( \frac{\partial z}{\partial y}\right)^2 +1 } \end{align*}$

The normal vector to any point on the surface is then
$N \;=\; \frac{ -\dfrac{\partial z}{\partial x} \boldsymbol{\hat\imath}\; - \dfrac{\partial z}{\partial y} \boldsymbol{\hat\jmath}\; + \boldsymbol{\hat k}} {\sqrt{ \left (\dfrac{\partial z}{\partial x}\right)^2 + \left( \dfrac{\partial z}{\partial y} \right)^2 +1}}$

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 $\phi (x, y ,z) = 0$

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:
$\frac {\partial \phi}{\partial x} + \frac{\partial \phi}{\partial z} \frac{\partial z}{\partial x} \;\;=\;\; \phi_x +\; \phi_z \frac{\partial z}{\partial x} \;\;=0$

Solve for the partial of z with respect to x:
$(1)\;\;\;\;\;\frac{\partial z}{\partial x} \;=\;-\frac{\phi_x}{\phi_z}$

Differentiate with respect to y, and solve for the partial of z with respect to y:
$\begin{align*} &\frac {\partial \phi}{\partial y}\; + \frac{\partial \phi}{\partial z} \frac{\partial z}{\partial y} \;\;=\;\; \phi_y \, +\, \phi_z \frac{\partial z}{\partial y} \;\;=\;0 \\ &(2)\;\;\;\;\; \frac{\partial z}{\partial y} \;\;=- \frac{\phi_y}{\phi_z} \end{align*}$

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:
$\begin{align*} \boldsymbol{r}\!_x = \boldsymbol{\hat\imath} \; - \frac{\phi_x}{\phi_z}\boldsymbol{\hat k} \\ \boldsymbol{r}\!_y = \boldsymbol{\hat \jmath} \; - \frac{\phi_y}{\phi_z}\boldsymbol{\hat k} \end{align*}$

Again, the cross product:
$\boldsymbol{r}\!_x \times \boldsymbol{r}\!_y \;\;=\;\; \begin{vmatrix} \boldsymbol{\hat \imath} &\boldsymbol{\hat \jmath} &\boldsymbol{\hat k} \\ 1 &0 &-\dfrac{\phi_x}{\phi_z} \\ 0 &1 &-\dfrac{\phi_y}{\phi_z} \end{vmatrix} \;\; = \;\; \frac{\phi_x}{\phi_z} \boldsymbol{\hat\imath}\; + \frac{\phi_y}{\phi_z} \boldsymbol{\hat\jmath}\; + \boldsymbol{\hat k}$
The relationship is clearer, and normalization simplified, if this is rewritten as:
$\frac{1}{\phi_z}\left(\phi_x \boldsymbol{\hat\imath}\; + \phi_y \boldsymbol{\hat\jmath}\; + \phi_z\boldsymbol{\hat k}\right)$

Normalize!
$N = \frac{\boldsymbol {r}_x\times \boldsymbol{r}_y} {|\boldsymbol{r}\!_x \times \boldsymbol{r}\!_y|} &\;\;=\;\; \frac {\phi_x \boldsymbol{\hat\imath}\; + \phi_y \boldsymbol{\hat\jmath}\; + \phi_z\boldsymbol{\hat k}} {\sqrt{(\phi_x)^2 +(\phi_y)^2 + (\phi_z)^2}}$

Goodbye, extra z-partial.

$= \frac{ \dfrac{\partial \phi}{\partial x}\boldsymbol{\hat\imath}\; + \dfrac{\partial \phi}{\partial y}\boldsymbol{\hat\jmath}\; + \dfrac{\partial \phi}{\partial z}\boldsymbol{\hat k}} { \sqrt{ \left(\dfrac{\partial \phi}{\partial x}\right)^2 + \left(\dfrac{\partial \phi}{\partial y}\right)^2 + \left(\dfrac{\partial \phi}{\partial z}\right)^2 }}$

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

$\phi (u, v ,w) = 0$

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.

Tuesday, March 6, 2012

Double integration equivalence

Time to clarify my assumptions about integration.

The double integral,

is shorthand for:

We begin at the innermost integral. Integrate with respect to x, holding y constant. Then integrate the result with respect to y (holding x constant). Each step is an ordinary, single-variable integral. We may also change the order of integration:

Just remember to keep x-bounds with x, and y-bounds with y.

These are convenient results, and after hammering through partial differentiation, integrating in any order makes intuitive sense. This is the next chapter of Advanced Calculus, but I think the result is self-evident, and I want to clarify that assumption so I can find any mistakes.

I have heard equality (2) referred to as Fubini's Theorem, but the statement before me is, the two integrals are equivalent. The only Theorem I have any business with must be the Fundamental Theorem of the Calculus. If I break up (2) into infinite sums, the sum on the right side of the equal sign must be the same as that on the left. Right?

Variable bounds, paramaterization, partial summation with discrete and continuous quantities are a few pages away. I need a working, base example: a continuous function with constant bounds. Let me build that now with the simplest case: a dobule integral of the form (1), with constant bounds.

A FUNCTION OF A SINGLE VARIABLE

Let F(x) be a function whose derivative, F'(x) = f(x) is continuous over the closed interval [a, b], i.e.

Then the Fundamental Theorem of Calculus states
$(3) ∫{a}^{b} f(x)dx = F(b) - F(a)$

PROOF: This approach is abstract, but concise. By continuity, we can write the integral as the limit of a sum
$(4)∫{a to b} f(x)dx = lim(δ −> 0) Σ{k=1 to n} f(ξ_k)(x_k − x_{k-1})$

Where δ is the largest subinterval and ξ_k lies in the subinterval [x_k - x_k-1].
Now, we choose for our ξ_k the points in each subinterval where the line tangent to F(x) has the same slope as the segment joining the endpoints, F(x_k ) and F(x_k-1):
$(5) F'(ξ)(x_k − x_{k-1}) = F(x_k) − F(x_{k-1})$
(Rise/Run) (Run) = (Rise)

Then the right hand side of (4) becomes,
$(6) Σ_{k=1}^{n} f(ξ_k)(x_k − x_{k-1}) = f(ξ_1)(x_1 − x_0) + f(ξ_2 )(x_2 − x_1) + . . . +f(ξ_n)(x_n-x_{n-1}) = [F(x_1)−F(x_0)] + [F(x_2)−F(x_1)] + . . . + [F(n) − F(n−1)]$

Rearranging the terms, we have
$(7) F(x_n) − F(x_{n-1}) +F(x_{n-1}) − F(x_{n-2}) + . . .+ F(x_2) − F(x_1) + F(x_1) − F(x_0)$
_____________________________________________________

And the theorem is proved. Let me try to extend this method myself.

A FUNCTION OF TWO INDEPENDENT VARIABLES

Beginning with equation (1):

First, consider the inner integral:
$∫_{a}^{b} f(x,y)dx$
What relationship does f(x,y)dx have to F(x)? It is the partial derivative F₂₁(x):

The result of the first integration will then be F₂(x). This is my principal assumption. Specifically,
$(8) F(x,y) = ∫∫ f(x,y) dx dy = ∫∫ {∂^2 F/(∂y∂x)}dx dy$
I'm not sure how I feel about that. The assumption is sufficient to make the sums work out. And to use the Fundamental Theorem, we must already know the relationship between f and F. I don't see a way around equality (8). But I already know that F₂₁(x, y,) and F₁₂(x, y) are identical for 'nice' functions, and the FTC is just going to give that information back to me.

That's why I need to work out the concrete terms, determine if it holds, and if so, how. Otherwise I'm thinking in a circle. I will use the notation of partial derivatives, but I did not begin with this assumption. I tried several ways of indexing the sums, and in the end found that "indexing" was the same thing as labeling partial derivatives. The FTC makes this process clear.

I. Integrate with respect to x
$(4)' F_2(x,y) = ∂F(x,y)/∂y = ∫{a -b} f(x,y)dx = Lim{δ --> 0 } Σ{k=1 to n} f(ξ _k, y)(x_k – x_{k-1})$

Choose the ξ_k so that
$(5)' f(ξ_k, y)(x_k - x_{k-1}) = F_2(x_k, y) – F_2(x_{k-1}, y)$
This is a directional derivative. There is no slope in the y direction because I'm not varying y. Hence the equality. This is the crux. Not-varying one of two independent variables is exactly the same thing as taking a partial derivative. The x intervals sum and cancel to a discrete value, independently of the y intervals. Multiplication and addition are commutative and associative, so the sums and products of these values can be arranged in any order. Taking the integrals separately is also exactly the same thing as saying the integration can be done in any order. x and y never mix in the infinite sums. That's not a theorem; that's what we're doing. This is what I mean about Fubini's theorem, whatever the hell it is. Integrating the independent variables separately is how we insure there is only one answer. Onward!

Now we have
$(6)' ∫{a-b} f(x,y)dx = f(ξ_1, y)(x_1−x_0) + f(ξ_2, y)(x_2−x_1)+ . . . +f (ξ_n, y)(x_n − x_{n-1}) = [F_2(x_1, y) −F_2(x_0,y)] + [F_2(x_2, y)−F_2(x_1, y)] + . . . + [F_2(x_n, y)−F_2(x_{n-1}, y)]$

Rearranging the terms,
$(7)' F_2(x_n, y)− F_2(x_{n-1}, y) +F_2(x_{n-1}, y) − F_2(x_{n-2}, y) + . . .+ F_2(x_2, y) − F_2(x_1, y) + F_2(x_1, y) − F_2(x_0, y)$
__________________________________________________________________

II. Integrate with respect to y
I will integrate the general form:
$F(x,y)= ∫{c-d} F_2(x,y)dy$
For the purposes of integration, this is the same as
$F(α,y)= ∫{c-d} F_2(α,y)dy$
and can be applied to (7)'. The process is now the same as above.
$(4)'' ∫∫{c-d} F_2(x,y)dy = Lim{δ −>0} Σ{k=1-n} F_2(x, ζ_k)(y_k – y_{k-1})$
Choose the ζ_kso that
$(5)'' F_2(x,ζ_k)(y_k − y_{k-1}) = F(x, y_k) – F(x, y_{k-1})$

That's still a derivative on the left! (5)'' can alternately read,
$(∂F/∂y)(x,ζ_k)(y_k −y_{k-1}) = F(x, y_k) – F(x, y_{k-1})$
Thus we have
$(6)'' ∫{c-d} F_2(x,y)dy = F_2(x, ζ_1)(y_1−y_0) + F_2,(x, ζ_2)(y_2−y_1) + . . . + F_2(x, ζ_n)(y_n − y_{n-1}) = [F(x, y_1) −F(x, y_0)] + [F(x, y_2) − F(x, y_1)] + . . . + [F(x, y_n)−F(x, y_{n-1})]$

And the last line can again be rearranged so that all but two terms cancel:
$(7)'' F(x, y_n) − F(x, y_{n-1}) + F(x, y_{n-1}) − F(x, y_{n-2}) + . . . + F(x, y_2) − F(x, y_1) + \;F(x, y_1) − F(x, y_0)$
______________________________________________________________
$\begin{align*} &=\;\;\;\;F(x,y_n)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; -\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; F(x,y_0)\\ &=\;\;\;\;F(x,d)\; – \;F(x,c) \end{align*}$

III. Equivalence
Now compute the result:
$∫{c-d} (F_2(b,y) − F_2(a,y) )dy = [F(b,d) − F(b,c)] − [F(a,d)−F(a,c)] = [F(b,d) − F(a,d)] − [F(b,c)−F(a,c)] = ∫{a-b} [F_1(x,d)−F_1(x,c)]dx$

... with the happy result that giving one order of integration entails the equivalence of the other.

HOW'S MY DRIVING?

I have the feeling double integration is applicable in a variety of circumstances, leaving this approach insufficient for many cases. Or maybe I made mistakes, and it's all wrong. Somebody is letting a 12-year old drive a semi, and both of those guys are me. Tell me how my driving is. Leave me a comment. Correct a mistake. Prattle on about Space-Wogs.

Fuckin' space wogs. Always framping the scrumpets.

{Thanks again to CodeCogs for free LaTeX equation editing and hosting the resulting, tiny ~~bitches~~ images.}

Monday, March 5, 2012

Math Reading List 2010 -

Homework over the last two years is now several thousand pages, and I need to organize some of the information I return to regularly. These are titles which have more specific purpose than general reference. General texts are for subjects new to me. I have in each case selected a single book to use as a self-study course.

The literature on signals is all Coming Up. Oppenheim's signal systems text can be completed without first learning DiffEQ, but I am not in a rush to get to signal processing. Attending the audio panel at Pax Dev, none of the problems described required signal processing to fix.

__________________________________

Click on any title to see related posts

Blogger doesn't get organization. Suggestions welcome!

Core Mathematics

100 Great Problems of Elementary Mathematics: Their History and Solutions
Heinrich Dörrie
Had this book for a while; Last month I decided to work through all 100 problems in depth, from the beginning.
{Amazon page}

Elements
Euclid
The 13 books of the Elements can be done very quickly, and I am still deciding how to use them. After completing book I in a day, I realized I wanted a greater understanding. I'd like to work out all the algebra and vector transforms introduced by each new Proposition. The resulting numerical methods would be a geometer's toolsbox for navigating 3-space. Less efficient than the worst CAD environment, surely; but I believe well worth the practice. I wished to use the blog to help me organize the information, and work out clean, presentable transforms, but typing math into Blogger is a misery.
{Google Ebook}

Disquisitiones Arithmeticae
Carl Friedrich Gauss
{Available as a free Google Ebook}

Theory and Application of Infinite Series
Konrad Knopp
{Amazon Page}

A First Course in Numerical Analysis
Anthony Ralston and Philip Rabinowitz
{Amazon page}

Matrices and Linear Algebra
Hans Schneider and George Philip Barker
(Completed 2010) Checked every theorem and proof; done all problems except the last chapter.
{Amazon page}

Advanced Calculus
David Widder
I use this book as regular reference. Completed the sections on partial differentiation, vectors, differential geometry, and the Stieltjes integral. I know it inside and out; again, I do every problem except where they were repetitive or I failed to find a solution.
Widder is a giant poo and assumes you will do the problem sets by rote, ignorant of the gaping assumptions until he brings them to your attention. The problem sets are poorly arranged.
On the other hand, he is terrifically concise.
2011-

Physics

Théorie analytique de la chaleur
Joseph Fourier
{Free Google ebook}

Applied Vector Analysis
Hwei P. Hsu
Plowing through this book right now (Mar '12). Much is review; I am focused on
-working with geometric primitives directly in vector notation
-solving problems in different notations: algebra, geometry, linear combinations of vectors, vector-valued functions
-clearing up any confusions about the core methods.
-acquiring surface and line integrals
-completing the book as quickly as possible

Principia
Isaac Newton
{Available in a number of editions for free. I have a print copy, but do not recommend the edition}

Ordinary Differential Equations: An ElementaryTextbook for Students of Mathematics, Engineering and the Sciences

Morris Tenenbaum and Harry Pollard

Status: Halfway; up to Simple Harmonic Motion. I alternated between this book and Linear Algebra for about a year. Linear Equations of Order Greater Than One occupied the first half of 2011 before I took a break. I do every problem, except where they are repetitive and I no longer need the practice.

{Amazon Page}

A First Course in Partial Differential Equations
H. F. Weinberger
{Amazon page}

Numerical Methods

Feynman Lectures on Computation
Richard P. Feynman
Read it and tried many of the exercises, but I was not yet familiar enough with the territory. I will revisit it soon.
{Amazon Page}

Introduction to Algorithms
Thomas Cormen, Charles Leiserson, Ronald Rivest.
1st Edition
{Amazon page}

Signals

The Measurement of Power Spectra
R. B. Blackmann and J. W. Tukey
{Amazon page}

An Introduction to Harmonic Analysis
Yitzhak Katznelson
{Amazon page}

The Science of Radio
Paul J. Nahin
I'll have to substitute my own SciLab code for the Electronics Workbench and Matlab exercises. Which means I have to learn SciLab! Rad!
{Amazon page}

An Introduction to Random Vibrations, Spectral and Wavelet Analysis
D.E. Newland
(3rd edition) A civil and mechanical engineering text! Guaranteed not to treat spectral analysis as a subset of aesthetics.
{Amazon page}

Signals and Systems: An Introduction to Analog and Digital Signal Processing
Alan Oppenheim, Alan Wilsky, Ian Young
{My edition is old. It corresponds to the:}
MIT OpenCourseWare Online Course in Digital Signal Processing

The Fourier Transform and its Applications
Brad Osgood
Online video series with accompanying materials through Stanford University. (EE261)

Music

Musimathics (2 vols.) Gareth Loy
Comprehensive reference for music-related mathematical systems, analysis, etc.
{Amazon Page}

Popper's Dreamland

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Thursday, March 8, 2012

Vector surfaces

Tuesday, March 6, 2012

Double integration equivalence

Monday, March 5, 2012

Math Reading List 2010 -

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