4 ms·

The most advanced math I had were the 1st and 2nd year multivariable calculus and linear algebra courses in college. I am interested in visualizations/simulati

by dtquad 2y ago

The most advanced math I had were the 1st and 2nd year multivariable calculus and linear algebra courses in college. I am interested in visualizations/simulations of physical systems as a way to learn advanced math. Is there any books or resources that take that approach?

bloqs 2y ago
Brilliant.com
- hutzimabu8 2y ago
  [dead]
ericdfoley 2y ago
Computational Science and Engineering by Gilbert Strang
kubielid 2y ago
Absolutely love the book “Learn Physics with Functional Programming”. Uses Haskell, and teaches it so knowing Haskell is an unnecessary prerequisite, though familiarity with programming in general would be very beneficial, and focuses on 2D and 3D visualisation.
thechao 2y ago
I recently bootstrapped myself from your level to algebraic geometry — enough to read an annotated version of Einstein's paper. If I had to do it again, I'd just skip to the parts that worked: (1) hit up Michael Penn on YT, but work all the problems he works; and, (2) learn GA, I like MacDonald's "Linear and Geometric Algebra". I'm going to be honest: unless you do the homework, you can't learn the math. I learned Penn because it's "real" (mainstream) math & lets you read in the same language; I learned GA because it's designed to proselytize to other mathematicians & deliberately presents serous ideas in very approachable ways.
- citizen_friend 2y ago
  Isn't general relativity framed in differential geometry, not algebraic geometry? All of Einstein's papers on special relativity use high-school algebra and calculus.
  - thechao 2y ago
    Yes, GR uses differential geometry; however, a good (mechanical) understanding of any exterior algebra helps when working with tensors; and, I found GA much easier to understand than algebraic & differential geometry, when starting out. The idea is to familiarize yourself with learning math. The actual math of the GR paper is pretty straight forward: it's the physics which is so mind bending.
    - citizen_friend 2y ago
      I see. Was just confused because you mentioned algebraic geometry and I wasn't sure.
lagrange77 2y ago
Take a look at this playlist [0] by MathTheBeautiful. It says it's about PDEs, but it starts with ODEs. I think most differential equations courses are too focused on symbolic solving techniques. To me, understanding a (physical) system is mainly understanding the differential equation (system) itself, not it's solution. MathTheBeautiful really excels at this approach. Solutions are of course important as well, but i think that's what computers are for. [0] https://www.youtube.com/playlist?list=PLlXfTHzgMRUK56vbQgzCVM9vxjKxc8DCr https://www.youtube.com/playlist?list=PLlXfTHzgMRUK56vbQgzCV...
photochemsyn 2y ago
The next two steps beyond that are probably vector calculus and complex analysis. Check out Herb Gross's classic chalkboard lectures on the topic: https://ocw.mit.edu/courses/res-18-008-calculus-revisited-complex-variables-differential-equations-and-linear-algebra-fall-2011/ https://ocw.mit.edu/courses/res-18-008-calculus-revisited-co... When looking into simulations of physical systems, you'll run into partial differential equations, but be careful about learning resources that don't put numerical methods front and center. The article on Numerical Weather Prediction in the post has a good description: > "Analytical solution of the equations is impossible, so approximate methods must be employed. We consider methods of discretizing the spatial domain to reduce the PDEs to an algebraic system and of advancing the solution in time." Given Python's popularity in scientific computing, a lot of the available materials on the topic are in that language, using libraries like numpy and scipy a lot. I've been playing around with custom ChatGPT here - you can construct a workflow that takes a description of a common equation, generates the LaTex expression for it, translates that to a sympy expression, and then from that generate the numerical method code using numpy and then the code to plot the behavior over a given range in matplotlib. Bonkers, we're living in the future.

by dtquad 2y ago

bloqs 2y ago

Brilliant.com

hutzimabu8 2y ago
[dead]

ericdfoley 2y ago

Computational Science and Engineering by Gilbert Strang

kubielid 2y ago

Absolutely love the book “Learn Physics with Functional Programming”. Uses Haskell, and teaches it so knowing Haskell is an unnecessary prerequisite, though familiarity with programming in general would be very beneficial, and focuses on 2D and 3D visualisation.

thechao 2y ago

I recently bootstrapped myself from your level to algebraic geometry — enough to read an annotated version of Einstein's paper. If I had to do it again, I'd just skip to the parts that worked: (1) hit up Michael Penn on YT, but work all the problems he works; and, (2) learn GA, I like MacDonald's "Linear and Geometric Algebra". I'm going to be honest: unless you do the homework, you can't learn the math. I learned Penn because it's "real" (mainstream) math & lets you read in the same language; I learned GA because it's designed to proselytize to other mathematicians & deliberately presents serous ideas in very approachable ways.

citizen_friend 2y ago
Isn't general relativity framed in differential geometry, not algebraic geometry? All of Einstein's papers on special relativity use high-school algebra and calculus.
- thechao 2y ago
  Yes, GR uses differential geometry; however, a good (mechanical) understanding of any exterior algebra helps when working with tensors; and, I found GA much easier to understand than algebraic & differential geometry, when starting out. The idea is to familiarize yourself with learning math. The actual math of the GR paper is pretty straight forward: it's the physics which is so mind bending.
  - citizen_friend 2y ago
    I see. Was just confused because you mentioned algebraic geometry and I wasn't sure.

lagrange77 2y ago

Take a look at this playlist [0] by MathTheBeautiful. It says it's about PDEs, but it starts with ODEs. I think most differential equations courses are too focused on symbolic solving techniques. To me, understanding a (physical) system is mainly understanding the differential equation (system) itself, not it's solution. MathTheBeautiful really excels at this approach. Solutions are of course important as well, but i think that's what computers are for. [0] https://www.youtube.com/playlist?list=PLlXfTHzgMRUK56vbQgzCVM9vxjKxc8DCr https://www.youtube.com/playlist?list=PLlXfTHzgMRUK56vbQgzCV...

photochemsyn 2y ago

The next two steps beyond that are probably vector calculus and complex analysis. Check out Herb Gross's classic chalkboard lectures on the topic: https://ocw.mit.edu/courses/res-18-008-calculus-revisited-complex-variables-differential-equations-and-linear-algebra-fall-2011/ https://ocw.mit.edu/courses/res-18-008-calculus-revisited-co... When looking into simulations of physical systems, you'll run into partial differential equations, but be careful about learning resources that don't put numerical methods front and center. The article on Numerical Weather Prediction in the post has a good description: > "Analytical solution of the equations is impossible, so approximate methods must be employed. We consider methods of discretizing the spatial domain to reduce the PDEs to an algebraic system and of advancing the solution in time." Given Python's popularity in scientific computing, a lot of the available materials on the topic are in that language, using libraries like numpy and scipy a lot. I've been playing around with custom ChatGPT here - you can construct a workflow that takes a description of a common equation, generates the LaTex expression for it, translates that to a sympy expression, and then from that generate the numerical method code using numpy and then the code to plot the behavior over a given range in matplotlib. Bonkers, we're living in the future.