Difference between revisions of "Useful math"

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This is about useful math in the context of [[atomically precise manufacturing]].
+
This page is about useful math in the wide context of [[atomically precise manufacturing]].
 +
 
 +
 
 +
Specific application areas include:
 +
----
 +
* friction and dissipation
 +
* thermally driven self assembly
 +
----
 +
* quantum chemistry
 +
* molecular modelling
 +
----
 +
* 3d modelling
 +
* differential geometry for larger scale gears
 +
* ...
  
 
== Thermodynamics and statistical physics ==
 
== Thermodynamics and statistical physics ==
Line 26: Line 39:
 
== Math for modelling with atomistic detail ==
 
== Math for modelling with atomistic detail ==
  
−
=== From first principles ===
+
=== From first principles – e.g. for quantum chemistry ===
  
 
The exact solutions of the Schrödinger equation for the hydrogen problem. <br>
 
The exact solutions of the Schrödinger equation for the hydrogen problem. <br>
Line 44: Line 57:
 
Related: Density functional theory.
 
Related: Density functional theory.
  
−
=== Phenomenological models ===
+
=== Phenomenological models – e.g. for molecular modelling ===
  
 
* Lennard Jones potential – and similar ones – good for molecular dynamics simulations
 
* Lennard Jones potential – and similar ones – good for molecular dynamics simulations
Line 55: Line 68:
 
Related: Born–Oppenheimer approximation – and its deceiving pseudo convergence (to check)
 
Related: Born–Oppenheimer approximation – and its deceiving pseudo convergence (to check)
  
−
== Generally useful math tools from Analysis & co ==
+
== Generally useful math tools ==
  
−
* eigenvectors (linear algebra)
+
Hamiltonian mechanics finds heavy use in in quantummechaincs. <br>
−
* vector spaces with functions as base vectors (aka Hilbert spaces)  
+
Interestingly in [[gem-gum]] systems at slightly larger scales things behave very classically. <br>
−
* "integral kernels" – just a fancy word for projections in vector spaces with functions as base vectors – "overlap integrals"
+
Lagrangian mechanics might be useful there.
−
* (The crazy math symbol of an integral with a sum drawn over for quantum systems that contain both continuous band and discrete energy states)  
+
 
−
* commutators and anti-commutators
+
* [https://en.wikipedia.org/wiki/Hamiltonian_mechanics Hamiltonian mechanics]
−
* Creation and annihilation operators
+
* [https://en.wikipedia.org/wiki/Lagrangian_mechanics Lagrangian mechanics]
−
* all sorts of tricks an hackery with matrix math – selfadjungatedness & co
+
Related: [https://en.wikipedia.org/wiki/Stationary_Action_Principle principle of least action] and [https://en.wikipedia.org/wiki/Variational_principle variational principle (and calculus)]
−
* distributions aka generalized functions – including dirac deltas and Heaviside steps – quite a bit of math rules to memorize there
+
 
−
* support function (in the limit a dirac delta) => [[unusual math]]
+
=== Basic math for physics ===
−
* Liouville's theorem (complex analysis)
+
 
−
* Cauchy–Riemann equations – complex differentiability (aka holomorphic function) – Cauchy's integral theorem
+
* Finding zeros: – [https://en.wikipedia.org/wiki/Newton%27s_method Newton's method] – [https://en.wikipedia.org/wiki/Regula_falsi Regula falsi]
−
* Fourier transformations – folds
+
* Integrating differential equations: – [https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods Runge Kutta methods] – [https://en.wikipedia.org/wiki/Leapfrog_integration Leapfrog integration]
−
* Einstein notation
+
----
−
* Complete set of commuting observables
+
* [https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors Eigenvalues and eigenvectors] – (linear algebra)
−
* Clebsch–Gordan coefficients
+
* Vector spaces with functions as base vectors – ([https://en.wikipedia.org/wiki/Hilbert_space Hilbert spaces])  
−
* bra-ket formalism – abstracting math from positional 3D space – treating positional space and impulse equally
+
* "Integral kernels" – [https://en.wikipedia.org/wiki/Integral_transform Integral transform]
 +
* [https://en.wikipedia.org/wiki/Fourier_transform Fourier transformations] – (convolution becomes multiplication)
 +
* ([https://en.wikipedia.org/wiki/Laplace_transform Laplace transformations] – more used in electrical system engineering)  
 +
* [https://en.wikipedia.org/wiki/Convolution Convolution (aka folding)]
 +
* [https://en.wikipedia.org/wiki/Linear_form Linear functional] – [https://en.wikipedia.org/wiki/Adjoint_functors Adjoint_functors]
 +
* https://en.wikipedia.org/wiki/Self-adjoint_operator Self-adjoint operator]
 +
* [https://en.wikipedia.org/wiki/Orbital_overlap Orbital overlap / Overlap integral]
 +
----
 +
* All sorts of tricks an hackery with matrix math – selfadjunctness & co – [https://en.wikipedia.org/wiki/Category:Matrix_theory (Category:Matrix_theory)]
 
----
 
----
−
* Lagrangian and Hamiltonian mechanics – principle of least action – variational calculus
+
* '''[https://en.wikipedia.org/wiki/List_of_equations_in_quantum_mechanics List of equations in quantum mechanics]'''
−
* Nöther theorem – linking conserved quantities to invariance under transformations (aka symmetries) – related: generating functions => [[unusual math]]
+
* <small>([https://en.wikipedia.org/wiki/Heisenberg_picture Heisenberg picture] and [https://en.wikipedia.org/wiki/Matrix_mechanics Matrix mechanics])</small>
−
* (de) Betragsquadrat ~ (en) [https://en.wikipedia.org/wiki/Square_(algebra)#Absolute_square Absolute square]
+
 
----
 
----
−
* Finding zeros: – Newton method – Regula falsi
+
[https://en.wikipedia.org/wiki/Category:Perturbation_theory Category:Perturbation_theory] – particularly:
−
* Integrating differential equations: – Runge Kutta methods – Leap frog methods
+
* [https://en.wikipedia.org/wiki/Perturbation_theory_(quantum_mechanics) Perturbation_theory_(quantum_mechanics)] which employs the
−
* Implicit differentiation
+
* [https://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process Gram–Schmidt_process] in Hilbert space (? IIRC)
−
* Finding extrema with side conditions: "Lagrange multipliers"
+
'''[https://en.wikipedia.org/wiki/Category:Condensed_matter_physics Category:Condensed_matter_physics]'''
−
* (Reversely calculated) gradient descent in multi-dimensional scalar fields: ...
+
 
 +
=== Useful for 3D modelling and robotics ===
 +
 
 +
* screws (math object), wrenches (math objects), dual numbers – useful for robotics (forwards & backwards kinematic) and differential geometry for gear flanks
 +
* math tools for surface based commuter graphics (manly dealing with triangulations)
 +
* math tools for volume based computer graphics:
 +
* functional-representation (F-Rep), implicit surfaces, algebraic varieties, distance fields (magnitude of gradient is 1 for these)
 +
* ray-marching algorithms
 +
 
 +
=== Some multi-purpouse misc math ===
 +
 
 +
* [https://en.wikipedia.org/wiki/Gradient_descent Gradient descent]
 +
* (Reversely calculated) gradient descent in multi-dimensional scalar fields: [https://en.wikipedia.org/wiki/Conjugate_gradient_method Conjugate gradient method]
 +
* [https://en.wikipedia.org/wiki/Lagrange_multiplier Lagrange multipliers] – finding extrema under geometric side constraints
 +
* [https://en.wikipedia.org/wiki/Implicit_function#Implicit_differentiation Implicit differentiation]
 +
 
 +
'''Generalized functions (1D):'''
 +
* [https://en.wikipedia.org/wiki/Dirac_delta_function Dirac delta function] … derivative of Heaviside function
 +
* [https://en.wikipedia.org/wiki/Heaviside_step_function Heaviside step function] … antiderivative of Dirac delta function
 +
 
 +
=== Basic higher dimensional higher order derivatives ===
 +
 +
* [https://en.wikipedia.org/wiki/Gradient Gradient] … First order derivative of a scalar field
 +
* [https://en.wikipedia.org/wiki/Directional_derivative Directional derivative] … gradient projected into direction vector at specific point
 +
* [https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant Jacobian matrix and determinant]
 +
* [https://en.wikipedia.org/wiki/Hessian_matrix Hessian matrix]
 +
 
 +
'''For a taylor series of a scalar field:'''
 +
* First derivative: Gadient
 +
* Second derivative: Hessian Matrix
 +
* Third derivative: unnamed three index tensor
 +
 
 +
'''For a taylor series of a vector field:'''
 +
* First derivative of a scalar field: Jacobian matrix
 +
* Second derivative: unnamed three index tensor
 +
 
 +
'''For a taylor series of a tensor fields of arbitrary rank (all derivatives):'''
 +
* See: [[Beautiful differentiation (Conal Elliott)]]
 +
 
 +
'''Other uses of the Jacobian matrix:'''
 +
* For a transformation of a coordinate system into an other coordinate system. <br> For locally linear transformations – unless generalized derivatives are allowed (like multidimensional analogs to Dirac-deltas)
 +
* For [https://en.wikipedia.org/wiki/Propagation_of_uncertainty propagation of uncertainty] in scientific measurements near sigma limit
 +
 
 +
=== Useful for analysis of selfassembly and dissipation ===
 +
 
 +
* [https://en.wikipedia.org/wiki/Arrhenius_equation Arrhenius equation] – "a formula for the temperature dependence of reaction rates"
 +
* [https://en.wikipedia.org/wiki/Onsager_reciprocal_relations Onsager reciprocal relations] – modelling transport phenomena – [[statistical physics]]
 +
* '''[https://en.wikipedia.org/wiki/Fluctuation-dissipation_theorem Fluctuation-dissipation theorem]''' – links drag to Brownian motion – [[friction]] <br>– The paper "[[Evaluating the Friction of Rotary Joints in Molecular Machines (paper)]]" uses a simplified result from this.
 +
* [https://en.wikipedia.org/wiki/Langevin_equation Langevin equation] – for modelling brownian motion – [[statistical physics]] <br>– [https://en.wikipedia.org/wiki/Einstein_relation_(kinetic_theory) Einstein relation (kinetic theory)] – diffusion coefficient from microscopic mobility
 +
 
 +
=== Important for non-qunatum mechanical molecular dynamics simulations ===
 +
 
 +
* [https://en.wikipedia.org/wiki/Molecular_dynamics Molecular dynamics]
 +
* [https://en.wikipedia.org/wiki/Lennard-Jones_potential Lennard-Jones potential]
 +
Tools to set up the right initial distribution of particle motions:
 +
* '''[https://en.wikipedia.org/wiki/Equipartition_theorem Equipartition theorem]'''
 +
* [https://en.wikipedia.org/wiki/Thermodynamic_beta Thermodynamic beta (k<sub>B</sub>T)] in the Boltzmann factor in the [https://en.wikipedia.org/wiki/Boltzmann_distribution Boltzmann distribution]
 +
* For fermions like electrons: [https://en.wikipedia.org/wiki/Fermi%E2%80%93Dirac_statistics Fermi–Dirac statistics]
 +
* For bosons like phonons (and photons): [https://en.wikipedia.org/wiki/Bose%E2%80%93Einstein_statistics Bose–Einstein statistics]
 +
* [https://en.wikipedia.org/wiki/Maxwell%E2%80%93Boltzmann_statistics Maxwell–Boltzmann statistics] & [https://en.wikipedia.org/wiki/Maxwell%E2%80%93Boltzmann_distribution Maxwell–Boltzmann distribution]
 +
 
 +
=== Thermodynamics ===
 +
 
 +
[https://en.wikipedia.org/wiki/Statistical_ensemble_(mathematical_physics) Statistical ensemble (mathematical physics)]
 +
[https://en.wikipedia.org/wiki/Statistical_ensemble_(mathematical_physics)#Correcting_overcounting_in_phase_space (overcounting)]
 +
[https://en.wikipedia.org/wiki/Category:Statistical_ensembles (list of ensembles)]:
 +
* [https://en.wikipedia.org/wiki/Canonical_ensemble Canonical ensemble] – NVE – heat bath
 +
* [https://en.wikipedia.org/wiki/Microcanonical_ensemble Microcanonical ensemble] – NVT – isolated
 +
----
 +
Changes in the rate of a chemical reaction against temperature. (chemical kinetics)
 +
* [https://en.wikipedia.org/wiki/Eyring_equation Eyring equation] – from first principles – ([https://en.wikipedia.org/wiki/Transition_state_theory Transition state theory])
 +
* [https://en.wikipedia.org/wiki/Arrhenius_equation Arrhenius equation] – empirical / phenomenological
 +
 
 +
=== For more precise quantum mechanical calculations ===
 +
 
 +
* [https://en.wikipedia.org/wiki/Square_(algebra)#Absolute_square Absolute square] – to the the density from the wave function
 +
* [https://en.wikipedia.org/wiki/Bra%E2%80%93ket_notation Bra-ket notation] – abstracting math from positional 3D space – treating positional space and impulse equally
 +
----
 +
* [https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation Schrödinger equation] – and exact exact solutions – and iterative methods
 +
* ([https://en.wikipedia.org/wiki/Helium_atom Helium atom] as the simplemost three body case and first case where there is electron shielding)
 +
* Approximations: [https://en.wikipedia.org/wiki/Slater-type_orbital Slater type orbital] and [https://en.wikipedia.org/wiki/Gaussian_orbital Gaussian_orbital]
 +
* "overlap integrals" – e.g. [https://en.wikipedia.org/wiki/Orbital_overlap Orbital overlap] – projections in vector spaces with functions as base vectors
 +
* <small>(The crazy math symbol of an integral with a sum drawn over for quantum systems that contain both continuous band and discrete energy states)</small>
 +
* [https://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process Gram–Schmidt process] – for getting a reasonable orthonormal basis as a starting point
 +
----
 +
* '''[https://en.wikipedia.org/wiki/Hartree%E2%80%93Fock_method Hartree–Fock method]'''
 +
* '''[https://en.wikipedia.org/wiki/Density_functional_theory Density functional theory]'''
 +
----
 +
* [https://en.wikipedia.org/wiki/Complete_set_of_commuting_observables Complete set of commuting observables] – "the measurement of one observable has no effect on the result of measuring another observable in the set"
 +
* commutators and anti-commutators – [https://en.wikipedia.org/wiki/Commutator#Ring_theory Commutator ~> Ring theory]
 +
----
 +
* [https://en.wikipedia.org/wiki/Clebsch%E2%80%93Gordan_coefficients Clebsch–Gordan coefficients] – for coupling angular momenta<br> – [https://pdg.lbl.gov/2019/reviews/rpp2019-rev-clebsch-gordan-coefs.pdf a good table] and [https://youtu.be/UPyf9ntr-B8 a good video explanation how to use it]
 +
----
 +
* [https://en.wikipedia.org/wiki/Density_matrix Density matrix]
 +
 
 +
=== Maybe more relevant for high energy free particle physics ===
 +
 
 +
* [https://en.wikipedia.org/wiki/Green%27s_function Green's function] – needed for scattering problems
 +
* [https://en.wikipedia.org/wiki/Liouville%27s_theorem_(Hamiltonian) Liouville's theorem (Hamiltonian)] – on incompessibility of phase space <br>– Liouville's theorem puts limits on focusing particle beams after they left solid state <br>– To cheat and reduce the phase space of a free floating particle beam at least some indirect interaction with solid state matter (for removal of excess phase space) is necessary [http://www.iap.tuwien.ac.at/www/atomic/surface/capillaries (Capillary guiding)]
 +
* Canonical transformations
 +
* [https://en.wikipedia.org/wiki/Canonical_coordinates Canonical coordinates] – (in [https://en.wikipedia.org/wiki/Hamiltonian_mechanics Hamiltonian mechanics])
 +
* [https://en.wikipedia.org/wiki/Generalized_coordinates Generalized coordinates] – (in [https://en.wikipedia.org/wiki/Lagrangian_mechanics Lagrangian mechanics])
 +
----
 +
* [https://en.wikipedia.org/wiki/Distribution_(mathematics) Distributions] (one class of [https://en.wikipedia.org/wiki/Generalized_function generalized functions]) – including [https://en.wikipedia.org/wiki/Dirac_delta_function Dirac deltas] and [https://en.wikipedia.org/wiki/Heaviside_step_function Heaviside steps] – quite a bit of math rules to memorize there
 +
* [https://en.wikipedia.org/wiki/Support_(mathematics)#Compact_support support function] [https://de.wikipedia.org/wiki/Stetige_Funktion_mit_kompaktem_Tr%C3%A4ger (de)]
 +
* Support functions => Test functions => [https://en.wikipedia.org/wiki/Bump_function Bump_function] – (in the limit a Dirac delta) ~ [[unusual math]]
 +
----
 +
* [https://en.wikipedia.org/wiki/Liouville%27s_theorem_(complex_analysis) Liouville's theorem (complex analysis)]
 +
* [https://en.wikipedia.org/wiki/Cauchy%E2%80%93Riemann_equations Cauchy–Riemann equations] – complex differentiability; holomorphic; analytic; ...
 +
* Cauchy's integral theorem
 +
----
 +
* [https://en.wikipedia.org/wiki/Ladder operator Ladder_operator]
 +
* [https://en.wikipedia.org/wiki/Creation_and_annihilation_operators Creation and annihilation operators] – ([https://en.wikipedia.org/wiki/Coherent_state Coherent state])
 +
----
 +
* [https://en.wikipedia.org/wiki/Einstein_notation Einstein notation]
 +
----
 +
* '''[https://en.wikipedia.org/wiki/Noether%27s_theorem Nöther's theorem]''' – linking conserved quantities to invariance under transformations (aka symmetries) – related: generating functions => [[unusual math]]
 +
 
 +
== Most fundamental concepts ==
 +
 
 +
* causation vs correlation
 +
* necessity vs sufficiency (if and only if aka iff)
 +
* convergence ... <small>''(one or two closely associated topics are missing here ... which ones? ...)''</smalL>
  
 
== Useful algorithms in computer graphics ==
 
== Useful algorithms in computer graphics ==
  
 
* GJK algorithm (collision detection)
 
* GJK algorithm (collision detection)
−
* ...
+
* Raymarching and more general "Walk on Spheres" <br> (also useful.for physical simulations)
 +
* Octree and more advanced subdivision algorithms
 +
* Marching cubes (for ugly triangulations – somehow this is everyone's favorite though)
  
−
== Notes ==
+
== Useful math for larger scale gear-train design ==
 +
 
 +
Differential geometry for generation of conjugate profiles in generalized gear-sets <br>
 +
(only solved for general axial alignments cycloid gear profiles as of yet 2021) <br>
 +
Associated math includes:
 +
* [https://en.wikipedia.org/wiki/Pl%C3%BCcker_coordinates Plücker coordinates]
 +
* [https://en.wikipedia.org/wiki/Screw_theory screw theory] – [https://en.wikipedia.org/wiki/Screw_axis screw axis]
 +
* [https://en.wikipedia.org/wiki/Dual_number dual numbers]
 +
 
 +
== Potentially extremely useful computer science ==
 +
 
 +
'''Automatic differentiation ...'''
 +
* generalized to arbitrary dimensionality (Jaconian matrix is first deerivative of vector field, Hessian matrix is second derivative of a scalar field, following are higher tensors)
 +
* generalized to arbitrary degree (Basically a taylor series)
 +
See: "Beautiful differentiation" by Conal Elliott (March 2009) Appeared in ICFP 2009 [http://conal.net/papers/beautiful-differentiation/ (link)]
 +
 
 +
'''Generalized code interpretation for vastly more resuability ...'''
 +
* generalizing lambda calculus to a category theoretic interpretation that allows for reuse of exactly the same code in vastly different compilation targets
 +
See: "Compiling to categories" by Conal Elliott (February 2017) Appeared at ICFP 2017 [http://conal.net/papers/compiling-to-categories/ (link)]
 +
 
 +
----
 +
 
 +
The revolution to "'''[[content addressed]]'''" systems. <br>
 +
Maybe the most powerful weapon against '''dependency hell''' and all its various workaround hacks.
 +
 
 +
== Notes / Misc ==
  
 
* Not to confuse "Holomorphic function" and "Holonomic constraints"
 
* Not to confuse "Holomorphic function" and "Holonomic constraints"
 +
 +
* Wikipedia [[https://en.wikipedia.org//wiki/Functional_derivative Functional derivative]]
 +
 +
== Related ==
 +
 +
* [[Atomic orbitals]]
 +
* [[Hartree-Fock method]]
 +
----
 +
* [[Pages with math]]
 +
----
 +
* [[Exotic math]]
 +
 +
 +
[[Category:Pages with math]]
 +
[[category:Information]]

Latest revision as of 12:27, 19 September 2023

This page is about useful math in the wide context of atomically precise manufacturing.


Specific application areas include:

  • friction and dissipation
  • thermally driven self assembly
  • quantum chemistry
  • molecular modelling
  • 3d modelling
  • differential geometry for larger scale gears
  • ...

Thermodynamics and statistical physics

Summing up over all the possible microstate configurations of a system.
Thereby deriving a partitioning function – (some exotic math involved in there)
From this partitioning function then thermodynamic laws can be re-derived and explained.
These thermodynamic laws can be (and historically have been) formerly phemomenologically derived.
Meaning derived from their effects not their causes.

Related:

  • Thermodynamic potentials and associated statistical ensembles
  • Transformation between the potentials – Legendre Transformation
  • Conjugated pairs of valuables (extrinsic and intrinsic) – a pairs product always gives the physical unit of energy

General note on solid state physics

Prevalent are long chains of simplifications by approximations that pile up and up and up.
Changing the application area of the models hugely may requires reevaluation of all these approximation steps.
Given that the chains of approximation are not formalized on computers (state 2021) this is difficult error prone and tedious.

Also: Following all the derivations from the lowermost assumptions
it becomes very evident that energy is a relative concept. (Not talking about relativity theory here).

Math for modelling with atomistic detail

From first principles – e.g. for quantum chemistry

The exact solutions of the Schrödinger equation for the hydrogen problem.
Using the property of it being a "separable partial differential equation"

  • Laguerre polynomials for the radial part
  • Spherical harmonics for the angular parts

The major reason why exact solutions are way off for other elements than hydrogen
(and the less relevant highly charged one electron ions) is the shielding effect of the inner electrons.
To get good approximations for orbitals it is necessary to do iterative self-consistent-field methods.
The exact hydrogen solutions can serve as a good initial guess starting point.

Also Useful in getting good starting points:

  • the Grahm Schmidt orthogonalization method
  • composing Gaussian distributions as base functions for orbitals
  • the Hartree-Fock method – helps filling up states consistent with pauli exclusion rules – antideterminant for fermionic states

Related: Density functional theory.

Phenomenological models – e.g. for molecular modelling

  • Lennard Jones potential – and similar ones – good for molecular dynamics simulations
  • Hund's rule of maximum multiplicity – not particularly useful in the context of chemically bond atoms

Misc

Derivation of London dispersion forces from first principles by
integrating over virtual electron states (related: virtual particles, feynman graphs) ...
Related: Born–Oppenheimer approximation – and its deceiving pseudo convergence (to check)

Generally useful math tools

Hamiltonian mechanics finds heavy use in in quantummechaincs.
Interestingly in gem-gum systems at slightly larger scales things behave very classically.
Lagrangian mechanics might be useful there.

Related: principle of least action and variational principle (and calculus)

Basic math for physics

Category:Perturbation_theory – particularly:

Category:Condensed_matter_physics

Useful for 3D modelling and robotics

  • screws (math object), wrenches (math objects), dual numbers – useful for robotics (forwards & backwards kinematic) and differential geometry for gear flanks
  • math tools for surface based commuter graphics (manly dealing with triangulations)
  • math tools for volume based computer graphics:
  • functional-representation (F-Rep), implicit surfaces, algebraic varieties, distance fields (magnitude of gradient is 1 for these)
  • ray-marching algorithms

Some multi-purpouse misc math

Generalized functions (1D):

Basic higher dimensional higher order derivatives

For a taylor series of a scalar field:

  • First derivative: Gadient
  • Second derivative: Hessian Matrix
  • Third derivative: unnamed three index tensor

For a taylor series of a vector field:

  • First derivative of a scalar field: Jacobian matrix
  • Second derivative: unnamed three index tensor

For a taylor series of a tensor fields of arbitrary rank (all derivatives):

Other uses of the Jacobian matrix:

  • For a transformation of a coordinate system into an other coordinate system.
    For locally linear transformations – unless generalized derivatives are allowed (like multidimensional analogs to Dirac-deltas)
  • For propagation of uncertainty in scientific measurements near sigma limit

Useful for analysis of selfassembly and dissipation

Important for non-qunatum mechanical molecular dynamics simulations

Tools to set up the right initial distribution of particle motions:

Thermodynamics

Statistical ensemble (mathematical physics) (overcounting) (list of ensembles):

Changes in the rate of a chemical reaction against temperature. (chemical kinetics)

For more precise quantum mechanical calculations

  • Absolute square – to the the density from the wave function
  • Bra-ket notation – abstracting math from positional 3D space – treating positional space and impulse equally
  • Schrödinger equation – and exact exact solutions – and iterative methods
  • (Helium atom as the simplemost three body case and first case where there is electron shielding)
  • Approximations: Slater type orbital and Gaussian_orbital
  • "overlap integrals" – e.g. Orbital overlap – projections in vector spaces with functions as base vectors
  • (The crazy math symbol of an integral with a sum drawn over for quantum systems that contain both continuous band and discrete energy states)
  • Gram–Schmidt process – for getting a reasonable orthonormal basis as a starting point

Maybe more relevant for high energy free particle physics

  • Nöther's theorem – linking conserved quantities to invariance under transformations (aka symmetries) – related: generating functions => unusual math

Most fundamental concepts

  • causation vs correlation
  • necessity vs sufficiency (if and only if aka iff)
  • convergence ... (one or two closely associated topics are missing here ... which ones? ...)

Useful algorithms in computer graphics

  • GJK algorithm (collision detection)
  • Raymarching and more general "Walk on Spheres"
    (also useful.for physical simulations)
  • Octree and more advanced subdivision algorithms
  • Marching cubes (for ugly triangulations – somehow this is everyone's favorite though)

Useful math for larger scale gear-train design

Differential geometry for generation of conjugate profiles in generalized gear-sets
(only solved for general axial alignments cycloid gear profiles as of yet 2021)
Associated math includes:

Potentially extremely useful computer science

Automatic differentiation ...

  • generalized to arbitrary dimensionality (Jaconian matrix is first deerivative of vector field, Hessian matrix is second derivative of a scalar field, following are higher tensors)
  • generalized to arbitrary degree (Basically a taylor series)

See: "Beautiful differentiation" by Conal Elliott (March 2009) Appeared in ICFP 2009 (link)

Generalized code interpretation for vastly more resuability ...

  • generalizing lambda calculus to a category theoretic interpretation that allows for reuse of exactly the same code in vastly different compilation targets

See: "Compiling to categories" by Conal Elliott (February 2017) Appeared at ICFP 2017 (link)

The revolution to "content addressed" systems.
Maybe the most powerful weapon against dependency hell and all its various workaround hacks.

Notes / Misc

  • Not to confuse "Holomorphic function" and "Holonomic constraints"

Related

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