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  • Differential equations
    ==Discrete models: how to set up differential equations== Below, we produce discrete models and then differential equations from verbal descriptions.
    64 KB (11,426 words) - 14:21, 24 November 2018
  • Modeling with discrete exterior calculus
    ...odel such a simple thing as a circular wave... We will be using [[discrete differential forms]] to model elementary [[ODEs]] and [[PDEs]] in dimensions 1 and 2 wit ...emaining as general as possible, we utilize discrete exterior calculus and differential forms. Our main conclusion is that the isotropy of heat on a grid is heavil
    31 KB (5,254 words) - 17:57, 21 July 2012
  • Calculus exercises: part IV
    ...let$ Verify that the function $y=cx^{2}$ is a solution of the differential equation: ...let$ Verify that the function $y=cx^{2}$ is a solution of the differential equation:
    13 KB (2,128 words) - 02:28, 5 September 2017
  • Partial differential equations
    The result of the substitution is the following equation: ==The heat equation with respect to difference quotients==
    53 KB (9,682 words) - 23:19, 18 November 2018
  • Applications of differential calculus
    ...als of degree below $5$). What about a ''numerical solution''? Solving the equation numerically means finding a sequence of numbers $d_n$ such that $d_n\to c$ ...Intermediate Value Theorem}--> as an iterated search for a solution of the equation $f(x)=0$. We constructed a sequence of nested intervals<!--\index{nested in
    59 KB (10,063 words) - 04:59, 21 February 2019
  • Calculus I, the discrete version
    '''Definition.''' The ''standard differential'', denoted by $dx$, is the edge function equal to $h$ on every edge, i.e., We can use the standard differential to construct an edge function from any node function by simple multiplicati
    64 KB (11,521 words) - 19:48, 22 June 2017
  • Differential equations: exercises
    *Verify that $-2x^{2}y+y^{2}=1$ is an implicit solution of the differential equation $2xy+(x^2-y)\frac{dy}{dx}=0$. Find one explicit solution. ...ich material is forgotten is proportional to $A(t)$. Set up a differential equation for $A(t)$.
    6 KB (988 words) - 18:05, 29 December 2016
  • Test
    #A modern view of calculus: [[Why do we need differential forms?|differential forms]] ===Continuous differential forms===
    16 KB (2,139 words) - 23:01, 9 February 2015
  • Applied Topology and Geometry
    #A modern view of calculus: [[Why do we need differential forms?|differential forms]] ===Continuous differential forms===
    16 KB (2,088 words) - 16:37, 29 November 2014
  • Differentiation
    ...the intermediate variable, whether it is the difference $\Delta x$ or the differential $dx$, is “cancelled”: ...' $dx$ and $dy$ as two new variables -- related to each other by the above equation?
    82 KB (14,116 words) - 19:50, 6 December 2018
  • PDEs
    ...s, $d_x$, to be taken into account. The result is a ''partial differential equation'' (PDE). ...esult of the substitution is a PDE of second degree called the ''diffusion equation'' of forms:
    44 KB (7,469 words) - 18:12, 30 November 2015
  • Diffusion
    Below we derive a PDE for discrete differential forms. Then, we will gradually develop a broad approach to the problem, whi ...esult of the substitution is a PDE of second degree called the ''diffusion equation'' of forms:
    39 KB (6,850 words) - 15:29, 17 July 2015
  • Diffusion equation
    Just as advection, heat transfer<!--\index{heat equation}--> exhibits a dispersal pattern. We can see, however, that without a flow ...s, $d_x$, to be taken into account. The result is a ''partial differential equation'' (PDE).
    35 KB (5,917 words) - 12:51, 30 June 2016
  • ODEs
    We start with a few simple ''ordinary differential equations (ODEs)'' with respect to the exterior derivative $d$ that have ex This equation has a solution, i.e., a $0$-form that satisfies the equation everywhere, if and only if $G$ is exact. In that case,
    47 KB (8,415 words) - 15:46, 1 December 2015
  • Discrete Calculus. An Introduction
    ...ext is ''discrete'' calculus. In particular, this is about the calculus of differential forms. The ''continuous'' counterpart is developed first because, typically Introduction: [[Why do we need differential forms?]]
    4 KB (466 words) - 19:07, 8 July 2014
  • Several variables
    Thus, the ''general solution'' of this system of differential equations is: ...rivatives are equal to $0$. To find the location of the ridge we solve the equation
    76 KB (13,017 words) - 20:26, 23 February 2019
  • Differential equations: course
    This is a one-semester course in differential equations and applications. model -> derivation of the DE -> analysis -> discretization -> [[difference equation]] ->
    2 KB (231 words) - 21:57, 2 November 2011
  • Stability Analysis
    ...We prove that, in-general, for the scheme that we have given for the WAVE-EQUATION ON A SQUARE GRID, the scheme will be STABLE if ...on for the wave-height (note that this is simply a discrete time-dependent differential two-form on our cell-complex of squares). We argue via the discretized cas
    12 KB (2,051 words) - 03:51, 11 August 2012
  • Calculus Illustrated
    ...ube channel]. The channel also shows some physics simulations for Volume 5 Differential Equations. More recent lectures are here: [https://www.dropbox.com/sh/kv370 ==Volume 2: Differential Calculus==
    16 KB (1,933 words) - 19:50, 28 June 2021
  • Links
    *Integration of differential forms #[[Integration of differential forms of degree 0 and 1]]
    6 KB (998 words) - 12:40, 31 August 2015
  • Isotropy in numerical PDEs
    '''[[Heat equation]]''' '''[[Wave equation]]'''
    10 KB (1,593 words) - 13:20, 8 April 2013
  • Freshman's introduction to discrete calculus
    ==The differential== '''Definition.''' The ''differential of a node function'' $f$ at edge $[A,B]$ is defined to be the number
    42 KB (7,443 words) - 14:18, 1 August 2016
  • Modelling with discrete vector fields and forms
    ...escribe motion in dimension $1$. It is given by an [[ordinary differential equation]] (ODE): ...continuous and $x:I \to R$ is differentiable on an open interval $I$. The equation has to be satisfied for all $t \in I$.
    9 KB (1,561 words) - 16:06, 27 August 2015
  • Navier–Stokes equations
    These are [[partial differential equations]] that describe the motion of fluid. ==Switch to differential forms==
    5 KB (742 words) - 03:32, 30 August 2012
  • Peter Saveliev's Academic Portfolio
    *MTH 335 [[Differential Equations -- Fall 2011]] .../Teaching/Fall10/m691/syllabus.html Differential geometry] (new course), [[differential forms: course#Lectures|TeX lecture notes]]
    25 KB (3,536 words) - 14:28, 17 January 2017
  • Elementary PDEs
    ...s, $d_x$, to be taken into account. The result is a ''partial differential equation'' (PDE). The result of the substitution is a PDE of second degree called the ''heat equation'' of cochains:
    16 KB (2,843 words) - 21:41, 23 March 2016
  • Elementary ODEs
    We start with ''ordinary differential equations (ODEs) of cochains'' with respect to their exterior derivative $d This equation has a solution, i.e., a $0$-cochain that satisfies the equation for every $1$-cochain $a$:
    16 KB (2,913 words) - 22:40, 15 July 2016
  • Maxwell equations
    ...n of the equations. Maxwell equations have long been written in terms of [[differential forms]] so I just need to interpret them as discrete ones, in order to deri The "differential forms" of the equations, i.e., the [[PDEs]]:
    6 KB (922 words) - 00:30, 9 April 2016
  • Discrete calculus article
    ...alculus has two entry points, differential calculus and integral calculus. Differential calculus concerns incremental rates of change and the slopes of piece-wise Discrete differential calculus is the study of the definition, properties, and applications of th
    27 KB (4,329 words) - 16:02, 1 September 2019
  • Differential equations: midterm
    These are exercises for [[Differential equations: course]] #Solve the differential equation: $y^{\prime}=-6xy.$
    1 KB (173 words) - 19:03, 1 November 2011
  • Applications of ODEs
    We derive from the last equation the following: Solving this equation for $\Delta p_n$, we choose the negative sign for the square root:
    50 KB (8,692 words) - 14:29, 24 November 2018
  • Computational science training: 2012
    *[[Differential equations: course]], just the basic ideas; *[[Differential forms: course]], at least the discrete part.
    5 KB (732 words) - 17:42, 8 April 2013
  • Systems of ODEs
    We solve the equation: We solve the equation:
    63 KB (10,958 words) - 14:27, 24 November 2018
  • Advanced Topology: exercises
    #Show that the set of differential forms is a vector space. #Form the axiomatic definition of differential form, prove that $\Omega^1(p)=span\{dx,dy\}.$
    9 KB (1,487 words) - 18:18, 9 May 2013
  • Wave equation from Maxwell equations
    The "differential forms" of the [[Maxwell equations]]: ==Maxwell equations of differential forms==
    4 KB (655 words) - 14:51, 13 July 2012
  • Continuous vs discrete differential forms
    ...lization of a [[cubical complex]] $K$ (or [[cell complex]]) and [[discrete differential forms]], i.e., [[cochains]], over $K$. So far, we have observed only that t We have thought of continuous [[differential forms]], $\varphi \in \Omega ^k ({\bf R}^n)$, mainly as ''integrands''. Als
    9 KB (1,483 words) - 13:54, 13 April 2013
  • Discrete differential geometry
    .... We follow a broad approach to these issues by developing, in addition to differential calculus, its discrete version as well as other possible calculi. We borrow *Calculus of [[discrete differential forms]]
    3 KB (361 words) - 14:41, 19 July 2013
  • Current students' projects
    *What parts of calculus, ODEs, PDEs, differential geometry, etc have discrete counterparts? ...ysics, represent each quantity as a [[discrete differential forms|discrete differential form]] of appropriate degree determined by its nature. These equations are
    4 KB (527 words) - 14:04, 25 August 2013
  • Parametric curves
    Recall some motivation for differential calculus, from Calc 1. ...where $f$ is the location. Then $|| f ||^2$ = constant, differentiate the equation. $( (|| f ||)^2 )′ = 0$, so $2< f', f' > = 0$, thus $< f', f > = 0$. Henc
    34 KB (5,665 words) - 15:12, 13 November 2012
  • Modern Thermodynamics by John Denker
    The terminology is different sometimes... "The term “inexact differential” is sometimes used in this connection, but that term is a misnomer, or at I agree that "differential" is out of place. "Inexact" might be an idiom but with a meaningful mathema
    8 KB (1,251 words) - 03:54, 29 March 2011
  • Integration
    We can think of the definition as an equation for functions: This equation has infinitely many solutions when $f$ is integrable. Furthermore, accordin
    69 KB (11,727 words) - 03:34, 30 January 2019
  • Differential equations: final exam
    This is for [[Differential equations: course]] #Verify that the function $y=cx^{2}$ is a solution of the differential equation: $xy^{\prime}=2y.$ Are there any others?
    1 KB (233 words) - 01:42, 13 December 2011
  • Integer-valued calculus
    Now we recast this construction in the language of ''differential forms''. Differential forms come from the integral theorems of $2$-dimensional calculus. First, '
    27 KB (3,824 words) - 19:07, 26 January 2019
  • Differential forms: exams
    Tests for [[Differential forms: course]]. (a) Compute the exterior derivative of the following discrete differential 1-form in R²:
    2 KB (348 words) - 22:38, 6 October 2012
  • Wave equation
    ...ith these two forces being equal, we have derived the ''wave equation'' of differential forms: If $k$ and $m$ are constant forms (and $R$ is a field), the wave equation takes a familiar shape:
    10 KB (1,775 words) - 02:40, 9 April 2016
  • Differential Equations Demystified by Krantz
    Used the book for [[Differential Equations -- Fall 2011]], see [[Differential equations: course]]. Ch. 1. What is a [[differential equation]]?
    810 bytes (103 words) - 15:21, 16 November 2011
  • Differential calculus of parametric curves
    ...'' The distance to some point remains the same, say, $1$, and the implicit equation is We substitute $x=\cos t$ and $y=\sin t$ into the equation and use the ''Pythagorean Theorem'' to prove that this is indeed the unit c
    130 KB (22,842 words) - 13:52, 24 November 2018
  • Topology Illustrated -- Index
    *[[differential form|differential form]] *[[diffusion equation|diffusion equation]]
    16 KB (1,773 words) - 00:41, 17 February 2016
  • Series
    ...ear every time and we are left with just one to be cancelled from the last equation. We assume that $x\ne a$ throughout. ...l then they are equal to $x=a$. Now these two terms are cancelled from our equation producing:
    113 KB (19,100 words) - 23:07, 3 January 2019
  • Laminar Flow Over a Flat Plate With MATLAB
    ...erential equation]] that was simplified from a complex system of [[partial differential equations]], and the final result is ...trying to interpret the behavior only from the solution to a differential equation.
    1 KB (185 words) - 16:48, 20 February 2011

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