**diff options**

author | bryan newbold <bnewbold@snark.mit.edu> | 2009-02-01 08:34:55 -0500 |
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committer | bryan newbold <bnewbold@snark.mit.edu> | 2009-02-01 08:34:55 -0500 |

commit | 7ad3b8aef5ad4492e94d350c0ce32d89797b3cab (patch) | |

tree | 9a18754b8025daa381ad592a632f95532b3a129b | |

parent | 7218768aba3a43f1a62867f35f05cb85a30d7ed2 (diff) | |

download | knowledge-7ad3b8aef5ad4492e94d350c0ce32d89797b3cab.tar.gz knowledge-7ad3b8aef5ad4492e94d350c0ce32d89797b3cab.zip |

cleanup, fixed some math

-rw-r--r-- | books/WantedBooks | 33 | ||||

-rw-r--r-- | books/wanted books | 53 | ||||

-rw-r--r-- | math/algebra | 1 | ||||

-rw-r--r-- | math/books to read | 33 | ||||

-rw-r--r-- | math/good books | 5 | ||||

-rw-r--r-- | math/tensors | 40 | ||||

-rw-r--r-- | physics/general relativity | 3 | ||||

-rw-r--r-- | physics/gravitational waves | 6 | ||||

-rw-r--r-- | physics/special relativity | 15 |

9 files changed, 93 insertions, 96 deletions

diff --git a/books/WantedBooks b/books/WantedBooks deleted file mode 100644 index 51b0361..0000000 --- a/books/WantedBooks +++ /dev/null @@ -1,33 +0,0 @@ -================================= -Wanted Books -================================= - -These are some books i'd love to own a copy of or at least read through -carefully. - -Technical ------------- - - * **Visual Complex Analysis** by Tristan Needham - * **Visualizing Data: Exploring and Explaining Data with the Processing Environment** by Ben Fry - * **General Theory of Relativity** by P. A.M. Dirac - * **Computability and Unsolvability** by Martin Davis - * **Mathematical Foundations of Quantum Mechanics** by John von Neumann - * **Real and Complex Analysis** by Walter Rudin - * **Art of Computer Programming** by Donald E. Knuth - * **Gravitation** by MTW - * **Euclid's Elements** - * **C Programming Language** by Brian W. Kernighan - * **Mathematical Physics** by Robert Geroch - -Novels ------------ - - * **Return From the Stars** by Stanislaw Lem - * **Mortal Engines** by Stanislaw Lem - * **V.** by Thomas Pynchon - -Other -------- - * **Pictures Showing What Happens on Each Page of Thomas Pynchon's Novel Gravity's Rainbow** by Zak Smith - diff --git a/books/wanted books b/books/wanted books new file mode 100644 index 0000000..6c29a2e --- /dev/null +++ b/books/wanted books @@ -0,0 +1,53 @@ +================================= +Wanted Books +================================= + +These are some books i'd love to own a copy of or at least read through +carefully. + +Technical +------------ + + * **Visual Complex Analysis** by Tristan Needham + * **Visualizing Data: Exploring and Explaining Data with the Processing Environment** by Ben Fry + * **General Theory of Relativity** by P. A.M. Dirac + * **Computability and Unsolvability** by Martin Davis + * **Mathematical Foundations of Quantum Mechanics** by John von Neumann + * **Real and Complex Analysis** by Walter Rudin + * **Art of Computer Programming** by Donald E. Knuth + * **Gravitation** by MTW + * **Euclid's Elements** + * **C Programming Language** by Brian W. Kernighan + * **Mathematical Physics** by Robert Geroch + +Math +~~~~~~~~~~~~~~ +(I haven't really looked in to most of these, just sound interesting) + + + * `On formally undecidable propositions of Principa Mathematica and related systems`:title:, by Kurt Godel. + * `Computability and Unsolvability`:title:, by Martin Davis. + * `Mathematical Foundations of Information Theory`:title:, by A.I. Khinchin. + * `Calculus of Variations with Applications to Physics and Engineering`:title:, by Robert Weinstock. + * `Relativity, Thermodynamics, and Cosmology`:title:, by Richard Tolman. + * `Mathematics Applied to Continuum Mechanics`:title:, by Lee Segel. + * `Optimization Theory and Applications`:title:, by Donald Pierre. + * `The Variational Principles of Mechanics`:title:, by Cornelius Lanczos. + * `Tensor Analysis for Physicists`:title:, by J.A. Schonten. + * `Investigations on the Theory of Brownian Movement`:title:, by Albert Einstein. + * `Great Experiments in Physics`:title:, ed. by ???. + * `Curvature and Homology`:title:, by Samuel Goldberd. + * `The Philosophy of Mathematics`:title:, by Stephan Korner. + * `The Various and Ingenious Machines of Agostino Ramelli`:title:, by A. Ramelli (!). + * `Experiments in Topology`:title:, by Stephan Barr. + +Novels +----------- + + * **Return From the Stars** by Stanislaw Lem + * **Mortal Engines** by Stanislaw Lem + +Other +------- + * **Pictures Showing What Happens on Each Page of Thomas Pynchon's Novel Gravity's Rainbow** by Zak Smith + diff --git a/math/algebra b/math/algebra index b337a2e..96197ff 100644 --- a/math/algebra +++ b/math/algebra @@ -5,6 +5,7 @@ Algebra .. note:: Most of the definitions and notation in the section are based on [rudin]_ or [meserve]_ .. list-table:: Closure of binary operators on given sets of numbers + * Operation - :latex:`$+$` - :latex:`$\times$` diff --git a/math/books to read b/math/books to read deleted file mode 100644 index ba75d8e..0000000 --- a/math/books to read +++ /dev/null @@ -1,33 +0,0 @@ -=============================================== -Math books that look interesting -=============================================== - -`On formally undecidable propositions of Principa Mathematica and related systems`:title:, by Kurt Godel. - -`Computability and Unsolvability`:title:, by Martin Davis. - -`Mathematical Foundations of Information Theory`:title:, by A.I. Khinchin. - -`Calculus of Variations with Applications to Physics and Engineering`:title:, by Robert Weinstock. - -`Relativity, Thermodynamics, and Cosmology`:title:, by Richard Tolman. - -`Mathematics Applied to Continuum Mechanics`:title:, by Lee Segel. - -`Optimization Theory and Applications`:title:, by Donald Pierre. - -`The Variational Principles of Mechanics`:title:, by Cornelius Lanczos. - -`Tensor Analysis for Physicists`:title:, by J.A. Schonten. - -`Investigations on the Theory of Brownian Movement`:title:, by Albert Einstein. - -`Great Experiments in Physics`:title:, ed. by ???. - -`Curvature and Homology`:title:, by Samuel Goldberd. - -`The Philosophy of Mathematics`:title:, by Stephan Korner. - -`The Various and Ingenious Machines of Agostino Ramelli`:title:, by A. Ramelli (!). - -`Experiments in Topology`:title:, by Stephan Barr. diff --git a/math/good books b/math/good books deleted file mode 100644 index bc3efe5..0000000 --- a/math/good books +++ /dev/null @@ -1,5 +0,0 @@ -========================================== -Recommended Math Reading -========================================== - -BLANK diff --git a/math/tensors b/math/tensors index 42fa841..e15270a 100644 --- a/math/tensors +++ b/math/tensors @@ -8,20 +8,28 @@ Tensors, Differential Geometry, Manifolds On a manifold, only "short" vectors exist. Longer vectors are in a space tangent to the manifold. -There are points (P), separation vectors (\Delta \vector P), curves ( Q(\zeta) ), tangent vectors ( \delta P / \delta \zeta \equiv \lim_{\Delta \zeta \rightarrow 0} \frac{ \vector{ Q(\zeta+\delta \zeta) - Q(\zeta) } }{\delta \zeta} ) +There are points (:m:`$P$`), separation vectors (:m:`$\Delta \vector P$`), +curves (:m:`$Q(\zeta)$`), tangent vectors (:m:`$\delta P / \delta \zeta \equiv +\lim_{\Delta \zeta \rightarrow 0} \frac{ \vector{ Q(\zeta+\delta \zeta) - +Q(\zeta) } }{\delta \zeta}$`) -Coordinates: \Chi^\alpha (P), where \alpha = 0,1,2,3; Q(\Chi_0, \Chi_1, ...) +Coordinates: :m:`$\Chi^\alpha (P)$`, where :m:`$\alpha = 0,1,2,3$`; +:m:`$Q(\Chi_0, \Chi_1, ...)$` there is an isomorphism between points and coordinates -Coordinate basis: \vector{e_\alpha} \equiv \left( \frac{\partial Q}{\partial \Chi^\alpha} \right) - for instance, on a sphere with angles \omega, \phi: - \vector{e_\phi} = \left( \frac{\partial Q(\phi, \theta)}{\partial \phi}\right)_\theta +Coordinate basis: :m:`$\vector{e_\alpha} \equiv \left( \frac{\partial +Q}{\partial \Chi^\alpha} \right$`) + + for instance, on a sphere with angles :m:`$\omega, \phi$`: + + :m:`$\vector{e_\phi} = \left( \frac{\partial Q(\phi, \theta)}{\partial \phi}\right)_\theta$` Components of a vector: - \vector{A} = \frac{\partial P}{\partial \Chi^\alpha } + + :m:`$\vector{A} = \frac{\partial P}{\partial \Chi^\alpha }$` Directional Derivatives: consider a scalar function defined on a manifold \Psi(P) - \partial_\vector{A} \Psi = A^\alpha \frac{\partial \Psi}{\partial \Chi^\alpha} + :m:`$\partial_\vector{A} \Psi = A^\alpha \frac{\partial \Psi}{\partial \Chi^\alpha}$` Mathematicians like to say that the coordinate bases are actually directional derivatives @@ -32,24 +40,24 @@ A **tensor** :m:`$\bold{T}$` has a number of slots (called it's **rank**), takes as an example for a rank-3 tensor: :m:`$$\bold{T} ( \alpha \vector{A} + \beta \vector{B}, \vector{C}, \vector{D}) = - \alpha \bold{T} (\vector{A}, \vector{C}, \vector{D}) + - \beta \bold{T} (\vector{B}, \vector{C}, \vector{D}) $$` +\alpha \bold{T} (\vector{A}, \vector{C}, \vector{D}) + \beta \bold{T} +(\vector{B}, \vector{C}, \vector{D}) $$` Even a regular vector is a tensor: pass it a second vector and take the inner product (aka dot product) to get a real. -Define the **metric tensor** -:m:`$\bold{g}(\vector{A}, \vector{B}) = \vector{A} \dot \vector{B}$`. The +Define the **metric tensor ** +:m:`$\bold{g}(\vector{A}, \vector{B}) = \vector{A} \cdot \vector{B}$`. The metric tensor is rank two and symetric (the vectors A and B could be swapped without changing the scalar output value) and is the same as the inner product. -:m:`$$\Delta P \dot \Delta P \equiv \Delta P^2 \equiv (length of \Delta P)^2 A \dot B = 1/4[ (A+B)^2 - (A-B)^2 ]$$` +:m:`$$\Delta P \cdot \Delta P \equiv \Delta P^2 \equiv (length of \Delta P)^2 A \cdot B = 1/4[ (A+B)^2 - (A-B)^2 ]$$` Starting with individual vectors, we can construct tensors by taking the product of their inner products with empty slots; for example :m:`$$\vector{A} \crossop \vector{B} \crossop \vector{C} (\_ ,\_ ,\_)$$` -:m:`$$\vector{A} \crossop \vector{B} \crossop \vector{C} (\vector{E}, \vector{F}, \vector{G}) = ( \vector{A} \dot \vector{E})(\vector{B} \dot \vector{F})(\vecotr{C} \dot \vector{G}) $$` +:m:`$$\vector{A} \crossop \vector{B} \crossop \vector{C} (\vector{E}, \vector{F}, \vector{G}) = ( \vector{A} \cdot \vector{E})(\vector{B} \cdot \vector{F})(\vecotr{C} \cdot \vector{G}) $$` Spacetime -------------- @@ -57,10 +65,10 @@ Spacetime Two types of vectors. Timelike: :m:`$\vector{\Delta P}$` - (\vector{\Delta P})^2 = -(\Delta \Tau)^2 + :m:`$(\vector{\Delta P})^2 = -(\Delta \Tau)^2$` -Spacelike: \vector{\Delta Q} - (\vector{\Delta Q})^2 = +(\Delta S)^2 +Spacelike: :m:`$\vector{\Delta Q}$` + :m:`$(\vector{\Delta Q})^2 = +(\Delta S)^2$` Because product of "up" and "down" basis vectors must be a positive Kronecker delta, and timelikes squared come out negative, the time "up" basis must be negative of the time "down" basis vector. diff --git a/physics/general relativity b/physics/general relativity index 7521db5..7fc29eb 100644 --- a/physics/general relativity +++ b/physics/general relativity @@ -8,8 +8,9 @@ General Relativity .. note:: Most of this content is based on a 2002 Caltech course taught by Kip Thorn [PH237]_ -*See also `math/tensors </k/math/tensors>`_* +*See also `math/tensors </k/math/tensors>`__* +(no content) References ---------------- diff --git a/physics/gravitational waves b/physics/gravitational waves index ab7174e..5aa1744 100644 --- a/physics/gravitational waves +++ b/physics/gravitational waves @@ -38,9 +38,11 @@ Fifth term: current quadrupole Energy ---------------- -Quick calculation: for a source with mass M, size L, period P, the quadrupole moment $M_2 \approx M L^2$, h \approx 1/c^2 (Newtonian potential energy) ???? +Quick calculation: for a source with mass M, size L, period P, the quadrupole +moment :m:`$M_2 \approx M L^2$`, :m:`$h \approx 1/c^2` (Newtonian potential +energy) ???? -h on the order of $10^{-22}$ +h on the order of :m:`$10^{-22}$` Propagation ----------------- diff --git a/physics/special relativity b/physics/special relativity index 41bf4b8..37fd3e9 100644 --- a/physics/special relativity +++ b/physics/special relativity @@ -8,7 +8,7 @@ Special Relativity .. note:: Most of this content is based on a 2002 Caltech course taught by Kip Thorn [PH237]_ -*See also `physics/general relativity</k/physics/generalrelativity/>`_* +*See also `physics/general relativity</k/physics/generalrelativity/>`__* As opposed to general relativity, special relativity takes place in a *flat* Minkowski space time: a 4-space with three spatial dimensions and one time @@ -17,22 +17,25 @@ dimension. +----------------+--------------------+ | Index notation | Variable | Type | +----------------+--------------------+ -| `$x^\0`:m: | `$t$`:m: | Time | -| `$x^\1`:m: | `$x$`:m: | Spatial | -| `$x^\2`:m: | `$y$`:m: | Spatial | -| `$x^\3`:m: | `$z$`:m: | Spatial | +| `$x^0$`:m: | `$t$`:m: | Time | +| `$x^1$`:m: | `$x$`:m: | Spatial | +| `$x^2$`:m: | `$y$`:m: | Spatial | +| `$x^3$`:m: | `$z$`:m: | Spatial | +----------------+--------------------+ Separations ------------- -The separation `$(\Delta s)^2`:m: between two events in space time, in a given +The separation `$(\Delta s)^2$`:m: between two events in space time, in a given Lorentzian/inertial frame, is defined as: :m:`$$ (\Delta s)^2 \equiv -(\Delta t)^2 + (\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2 $$` + or + :m:`$$ (\Delta s)^2 \equiv -(\Delta x^0)^2 + \sum_{i,j} \delta_{ij} \Delta x^i \Delta x^j$$` + where :m:`$\delta_{ij}$` is the Kronecker delta (unity or 1 when :m:`$i=j$`; zero otherwise), and the indices i and j are over the spatial dimensions 1,2,3 (corresponding to x,y,z). It can be shown that this separation |