Merge branch 'master' of ssh://bnewbold@animus.robocracy.org/srv/git/knowledge

4 files changed, 43 insertions, 65 deletions

diff --git a/math/algebra b/math/algebra
index b337a2e..3e39ddb 100644
--- a/math/algebra
+++ b/math/algebra
@@ -5,42 +5,50 @@ 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:`$\times$`
-     - :latex:`$-$`
-     - :latex:`$\divide$`
-     - :latex:`$^$`
-     - :latex:`$\sqrt{\text{ }}$`
-    * Positive Integers
+
+   * - Operation name
+     - addition
+     - product
+     - subtraction
+     - division
+     - power
+     - root
+   * - Operation symbol
+     - :latex:`$a + b$`
+     - :latex:`$a\times b$`
+     - :latex:`$a-b$`
+     - :latex:`$\frac{a}{b}$`
+     - :latex:`$a^b$`
+     - :latex:`$\sqrt{\text{a}}$`
+   * - Positive Integers
      - Y
      - Y
      - N
      - N
      - Y
      - N
-    * Positive rationals
+   * - Positive rationals
      - Y
      - Y
      - N
      - Y
      - Y
      - N
-    * Rationals (and zero)
+   * - Rationals (and zero)
      - Y
      - Y
      - Y
      - Y
      - Y
      - N
-    * Reals wrt positive integers
+   * - Reals wrt positive integers
      - Y
      - Y
      - Y
      - Y
      - Y
      - Y
-    * Complex numbers
+   * - Complex numbers
      - Y
      - Y
      - Y
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..d46810e 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 
+: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.