From 70fe68492bfdcb09b007195e44c5da55668bcae6 Mon Sep 17 00:00:00 2001 From: Bryan Newbold Date: Sat, 31 Jan 2009 21:44:47 -0800 Subject: updated courses with spring selections --- MIT Courses | 19 ++++++++++--------- 1 file changed, 10 insertions(+), 9 deletions(-) diff --git a/MIT Courses b/MIT Courses index 33dfcbe..31ebaf1 100644 --- a/MIT Courses +++ b/MIT Courses @@ -48,19 +48,18 @@ humanities concentration in philosophy. + **21F.052** - French Film Classics (spring, in english) + **18.100b** - Analysis -* Fall 2008 (planned) - + 8.05 - Quantum Mechanics II - + 18.901 - Topology - + 24.118 - Paradox and Infinity - + 24.209 - Philosophy in Film - + 6.111 - Digital Systems Lab (FPGA Design) +* Fall 2008 + + **8.05** - Quantum Mechanics II + + **24.118** - Paradox and Infinity + + **24.209** - Philosophy in Film + + **6.111** - Digital Systems Lab (FPGA Design) * Spring 2009 (planned) + 8.06 - Quantum Mechanics III - + 8.08 - Statistical Physics II - + 8.962 (cosmo) or 8.942 (GR) + + 8.962 - General Relativity + + 6.945 - Adventures in Symbolic Programming + + 4.602 - Modern Art and Mass Culture + 8.ThU - Thesis - + Pop Culture HASSD Other interesting courses: + 24.215 - Topics in the Philosophy of Science (fall) @@ -75,6 +74,7 @@ Other interesting courses: + 18.565 - Recursion Theory + 18.703 - Modern Algebra (spring) + 18.701/702 - Algebra I/II (fall/spring) + + 18.901 - Topology + 18.950 - Differential Geometry + 8.261J - Introduction to Computational Neuroscience (fall) + 8.292J - Fluid Physics (spring) @@ -82,3 +82,4 @@ Other interesting courses: + 8.594J - Introduction to Neural Networks (grad, fall) + 8.284 - Modern Astrophysics (spring) + 8.942 - Cosmology (grad, spring) + -- cgit v1.2.3 From 7ad3b8aef5ad4492e94d350c0ce32d89797b3cab Mon Sep 17 00:00:00 2001 From: bryan newbold Date: Sun, 1 Feb 2009 08:34:55 -0500 Subject: cleanup, fixed some math --- books/WantedBooks | 33 ---------------------------- books/wanted books | 53 +++++++++++++++++++++++++++++++++++++++++++++ math/algebra | 1 + math/books to read | 33 ---------------------------- math/good books | 5 ----- math/tensors | 40 ++++++++++++++++++++-------------- physics/general relativity | 3 ++- physics/gravitational waves | 6 +++-- physics/special relativity | 15 ++++++++----- 9 files changed, 93 insertions(+), 96 deletions(-) delete mode 100644 books/WantedBooks create mode 100644 books/wanted books delete mode 100644 math/books to read delete mode 100644 math/good books 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 `_* +*See also `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`_* +*See also `physics/general relativity`__* 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 -- cgit v1.2.3 From c2a4fe1d51f1306ad10168bb3d5104f1bb04c097 Mon Sep 17 00:00:00 2001 From: bryan newbold Date: Sun, 1 Feb 2009 08:38:00 -0500 Subject: more fixes --- math/algebra | 12 ++++++------ math/tensors | 2 +- 2 files changed, 7 insertions(+), 7 deletions(-) diff --git a/math/algebra b/math/algebra index 96197ff..d43a59a 100644 --- a/math/algebra +++ b/math/algebra @@ -6,42 +6,42 @@ Algebra .. list-table:: Closure of binary operators on given sets of numbers - * Operation + * - Operation - :latex:`$+$` - :latex:`$\times$` - :latex:`$-$` - :latex:`$\divide$` - :latex:`$^$` - :latex:`$\sqrt{\text{ }}$` - * Positive Integers + * - 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/tensors b/math/tensors index e15270a..d46810e 100644 --- a/math/tensors +++ b/math/tensors @@ -46,7 +46,7 @@ as an example for a rank-3 tensor: 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 ** +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. -- cgit v1.2.3 From d81a634c9053bb191eabbf5e03a1a8ed03a1bde1 Mon Sep 17 00:00:00 2001 From: bryan newbold Date: Sun, 1 Feb 2009 08:41:01 -0500 Subject: algibraic names and symbols --- math/algebra | 19 +++++++++++++------ 1 file changed, 13 insertions(+), 6 deletions(-) diff --git a/math/algebra b/math/algebra index d43a59a..5f15f4a 100644 --- a/math/algebra +++ b/math/algebra @@ -6,13 +6,20 @@ Algebra .. list-table:: Closure of binary operators on given sets of numbers - * - Operation - - :latex:`$+$` - - :latex:`$\times$` - - :latex:`$-$` + * - Operation name + - addition + - product + - subtraction + - division + - power + - root + * - Operation symbol + - :latex:`$a + b$` + - :latex:`$a\times b$` + - :latex:`$a-b$` - :latex:`$\divide$` - - :latex:`$^$` - - :latex:`$\sqrt{\text{ }}$` + - :latex:`$a^b$` + - :latex:`$\sqrt{\text{a}}$` * - Positive Integers - Y - Y -- cgit v1.2.3 From 838e1b2d55f39113fffe0f998cb1011b57747691 Mon Sep 17 00:00:00 2001 From: bryan newbold Date: Sun, 1 Feb 2009 08:41:36 -0500 Subject: algibraic names and symbols again --- math/algebra | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/math/algebra b/math/algebra index 5f15f4a..3e39ddb 100644 --- a/math/algebra +++ b/math/algebra @@ -17,7 +17,7 @@ Algebra - :latex:`$a + b$` - :latex:`$a\times b$` - :latex:`$a-b$` - - :latex:`$\divide$` + - :latex:`$\frac{a}{b}$` - :latex:`$a^b$` - :latex:`$\sqrt{\text{a}}$` * - Positive Integers -- cgit v1.2.3