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authorbryan newbold <bnewbold@snark.mit.edu>2009-02-01 08:34:55 -0500
committerbryan newbold <bnewbold@snark.mit.edu>2009-02-01 08:34:55 -0500
commit7ad3b8aef5ad4492e94d350c0ce32d89797b3cab (patch)
tree9a18754b8025daa381ad592a632f95532b3a129b
parent7218768aba3a43f1a62867f35f05cb85a30d7ed2 (diff)
downloadknowledge-7ad3b8aef5ad4492e94d350c0ce32d89797b3cab.tar.gz
knowledge-7ad3b8aef5ad4492e94d350c0ce32d89797b3cab.zip
cleanup, fixed some math
-rw-r--r--books/WantedBooks33
-rw-r--r--books/wanted books53
-rw-r--r--math/algebra1
-rw-r--r--math/books to read33
-rw-r--r--math/good books5
-rw-r--r--math/tensors40
-rw-r--r--physics/general relativity3
-rw-r--r--physics/gravitational waves6
-rw-r--r--physics/special relativity15
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