diff options
Diffstat (limited to 'math')
| -rw-r--r-- | math/algebra | 32 | ||||
| -rw-r--r-- | math/books to read | 33 | ||||
| -rw-r--r-- | math/good books | 5 | ||||
| -rw-r--r-- | math/tensors | 38 |
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. |