From a290d583ca3c4dfc39115068f209d64449c93a03 Mon Sep 17 00:00:00 2001
From: bnewbold <bnewbold@ziggy.(none)>
Date: Sun, 24 Jan 2010 03:39:25 -0500
Subject: math fixes

---
 math/tensors.page | 44 +++++++++++++++++++++-----------------------
 1 file changed, 21 insertions(+), 23 deletions(-)

(limited to 'math/tensors.page')

diff --git a/math/tensors.page b/math/tensors.page
index e1d24fc..7ea1848 100644
--- a/math/tensors.page
+++ b/math/tensors.page
@@ -1,5 +1,3 @@
-
-
 Tensors, Differential Geometry, Manifolds
 ============================================
 
@@ -9,66 +7,66 @@ Most of this content is based on a 2002 Caltech course taught by Kip Thorn (PH23
 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 (:m:`$Q(\zeta)$`), tangent vectors (:m:`$\delta P / \delta \zeta \equiv
+curves ($Q(\zeta)$), tangent vectors ($\delta P / \delta \zeta \equiv
 \lim_{\Delta \zeta \rightarrow 0} \frac{ \vector{ Q(\zeta+\delta \zeta) -
-Q(\zeta) } }{\delta \zeta}$`)
+Q(\zeta) } }{\delta \zeta}$)
 
-Coordinates: :m:`$\Chi^\alpha (P)$`, where :m:`$\alpha = 0,1,2,3$`;
-:m:`$Q(\Chi_0, \Chi_1, ...)$`
+Coordinates: $\Chi^\alpha (P)$, where $\alpha = 0,1,2,3$;
+$Q(\Chi_0, \Chi_1, ...)$
     there is an isomorphism between points and coordinates
 
-Coordinate basis: :m:`$\vector{e_\alpha} \equiv \left( \frac{\partial
-Q}{\partial \Chi^\alpha} \right$`)
+Coordinate basis: $\vector{e_\alpha} \equiv \left( \frac{\partial
+Q}{\partial \Chi^\alpha} \right$)
     
-    for instance, on a sphere with angles :m:`$\omega, \phi$`: 
+    for instance, on a sphere with angles $\omega, \phi$: 
     
-    :m:`$\vector{e_\phi} = \left( \frac{\partial Q(\phi, \theta)}{\partial \phi}\right)_\theta$`
+    $\vector{e_\phi} = \left( \frac{\partial Q(\phi, \theta)}{\partial \phi}\right)_\theta$
 
 Components of a vector:
 
-    :m:`$\vector{A} = \frac{\partial P}{\partial \Chi^\alpha }$`
+    $\vector{A} = \frac{\partial P}{\partial \Chi^\alpha }$
 
 Directional Derivatives: consider a scalar function defined on a manifold \Psi(P)
-    :m:`$\partial_\vector{A} \Psi = A^\alpha \frac{\partial \Psi}{\partial \Chi^\alpha}$`
+    $\partial_\vector{A} \Psi = A^\alpha \frac{\partial \Psi}{\partial \Chi^\alpha}$
 
 Mathematicians like to say that the coordinate bases are actually directional derivatives
 
 Tensors
 ------------
 
-A **tensor** :m:`$\bold{T}$` has a number of slots (called it's **rank**), takes a vector in each slot, and returns a real number. It is linear in vectors; 
+A **tensor** $\bold{T}$ has a number of slots (called it's **rank**), takes a vector in each slot, and returns a real number. It is linear in vectors; 
 as an example for a rank-3 tensor:
 
-:m:`$$\bold{T} ( \alpha \vector{A} + \beta \vector{B}, \vector{C}, \vector{D}) =
+$$\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}) $$`
+(\vector{B}, \vector{C}, \vector{D}) $$
 
 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:`$\bold{g}(\vector{A}, \vector{B}) = \vector{A} \cdot \vector{B}$`. The 
+$\bold{g}(\vector{A}, \vector{B}) = \vector{A} \cdot \vector{B}$. 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:`$$\Delta P \cdot \Delta P \equiv \Delta P^2 \equiv (length of \Delta P)^2 A \cdot B = 1/4[ (A+B)^2 - (A-B)^2 ]$$`
+$$\Delta P \cdot \Delta P \equiv \Delta P^2 \equiv (length of \Delta P)^2 A \cdot B = 1/4[ (A+B)^2 - (A-B)^2 ]$$
 
 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:`$$\vector{A} \crossop \vector{B} \crossop \vector{C} (\vector{E}, \vector{F}, \vector{G}) = ( \vector{A} \cdot \vector{E})(\vector{B} \cdot \vector{F})(\vecotr{C} \cdot \vector{G}) $$` 
+$$\vector{A} \crossop \vector{B} \crossop \vector{C} (\_ ,\_ ,\_)$$ 
+$$\vector{A} \crossop \vector{B} \crossop \vector{C} (\vector{E}, \vector{F}, \vector{G}) = ( \vector{A} \cdot \vector{E})(\vector{B} \cdot \vector{F})(\vecotr{C} \cdot \vector{G}) $$ 
 
 Spacetime
 --------------
 
 Two types of vectors.
 
-Timelike: :m:`$\vector{\Delta P}$`
-    :m:`$(\vector{\Delta P})^2 = -(\Delta \Tau)^2$`
+Timelike: $\vector{\Delta P}$
+    $(\vector{\Delta P})^2 = -(\Delta \Tau)^2$
 
-Spacelike: :m:`$\vector{\Delta Q}$`
-    :m:`$(\vector{\Delta Q})^2 = +(\Delta S)^2$`
+Spacelike: $\vector{\Delta Q}$
+    $(\vector{\Delta Q})^2 = +(\Delta S)^2$
 
 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.
-- 
cgit v1.2.3