From 7ad3b8aef5ad4492e94d350c0ce32d89797b3cab Mon Sep 17 00:00:00 2001
From: bryan newbold <bnewbold@snark.mit.edu>
Date: Sun, 1 Feb 2009 08:34:55 -0500
Subject: cleanup, fixed some math

---
 math/tensors | 40 ++++++++++++++++++++++++----------------
 1 file changed, 24 insertions(+), 16 deletions(-)

(limited to 'math/tensors')

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.
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