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  • ==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
  • ...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
  • ...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
  • 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
  • ...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
  • '''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
  • *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
  • #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
  • #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
  • ...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
  • ...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
  • 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
  • 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
  • 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
  • ...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
  • 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
  • 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
  • ...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
  • ...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
  • *Integration of differential forms #[[Integration of differential forms of degree 0 and 1]]
    6 KB (998 words) - 12:40, 31 August 2015
  • '''[[Heat equation]]''' '''[[Wave equation]]'''
    10 KB (1,593 words) - 13:20, 8 April 2013
  • ==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
  • ...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
  • These are [[partial differential equations]] that describe the motion of fluid. ==Switch to differential forms==
    5 KB (742 words) - 03:32, 30 August 2012
  • *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
  • ...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
  • 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
  • ...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
  • ...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
  • These are exercises for [[Differential equations: course]] #Solve the differential equation: $y^{\prime}=-6xy.$
    1 KB (173 words) - 19:03, 1 November 2011
  • 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
  • *[[Differential equations: course]], just the basic ideas; *[[Differential forms: course]], at least the discrete part.
    5 KB (732 words) - 17:42, 8 April 2013
  • We solve the equation: We solve the equation:
    63 KB (10,958 words) - 14:27, 24 November 2018
  • #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
  • The "differential forms" of the [[Maxwell equations]]: ==Maxwell equations of differential forms==
    4 KB (655 words) - 14:51, 13 July 2012
  • ...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
  • *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
  • .... 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • ...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
  • 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
  • ...'' 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
  • *[[differential form|differential form]] *[[diffusion equation|diffusion equation]]
    16 KB (1,773 words) - 00:41, 17 February 2016
  • ...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
  • This is called a [[differential equation]]. Here $k$ is the growth rate. Previous discussion suggests the differential equation:
    8 KB (1,201 words) - 15:45, 2 May 2011
  • an ordinary differential equation (ODE) of order $1$ with right-hand side function $P$ is: If we can't find $y$ explicitly, we think of the differential equation as a formula by which the slope of the tangent line to the graph of $y$ can
    21 KB (3,664 words) - 02:02, 18 July 2018
  • ...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
  • where $ω = f(x)dx$ is a ''[[differential form]]'' (of degree 1). The point is that the integral is now a function th Differential forms are also seen in [[calculus]] in "[[integration by substitution]]":
    2 KB (343 words) - 14:10, 7 October 2012
  • Higher order differential forms are ''multi''-linear functions but this topic lies outside the scope ...now from calculus is that the left-hand side is ''not'' a fraction but the equation can be rewritten as if it is:
    16 KB (2,753 words) - 13:55, 16 March 2016
  • #Are the following operators from the vector space of all differential functions to itself linear? (a) $Q(f)=f$, (b) $T(f)=f+2$, (c) $D(f)=f'$, (d ...n{equation*}A=\left[\begin{array}{cc}1 & 1 \\ 1 & 1\end{array}\right].\end{equation*}
    2 KB (330 words) - 02:21, 7 May 2013
  • ==Differential forms on curves== Now, what do [[differential forms]] look like when they are defined on manifold?
    10 KB (1,588 words) - 17:11, 27 August 2015
  • ==Chapter 9: Introduction to Differential Equations== 9.1 Solving [[ODEs|Differential Equations]]
    6 KB (634 words) - 16:38, 1 March 2013
  • The equation here holds for ''every'' $x$. In particular, if we pick $x=0$, then, immedi The issue is complex for continuous differential forms. In order to detect -- cohomologically -- the hole in ${\bf R}^2-\{(0
    17 KB (2,592 words) - 14:38, 14 April 2013
  • ...licitly in many areas of mathematics. For example, the solution set of the equation $f(x) = b$ is $f^{-1}(\{b\})$. In particular, the kernel of a linear operat *Suppose $f:X\to {\bf R}$ is continuous. Show that the solution set of any equation with respect to $f$, i.e., $\{x\in X: f(x)=r\}$ for some $r\in {\bf R}$, is
    42 KB (7,138 words) - 19:08, 28 November 2015
  • ...e motion in the Euclidean space. It is given by an [[ordinary differential equation]] (ODE): With that in mind, the equation looks the same:
    2 KB (377 words) - 17:13, 27 August 2015
  • *[[wave equation]], *[[heat equation]].
    6 KB (794 words) - 00:56, 1 June 2012
  • ...ou could recognize this fact from the direction field\ of the differential equation.
    6 KB (823 words) - 20:23, 13 June 2011
  • ...f calculus, it is also called a (discrete) ''differential form''<!--\index{differential forms}--> of degree $1$. This idea is the basis of the modern approach to c ...ince the second component is a vertex, it's case 2. We substitute the last equation:
    36 KB (6,395 words) - 14:09, 1 December 2015
  • #redirect[[Differential forms]] First, $0$-forms. Suppose $F$ is a continuously differential function of $3$ variables. Then,
    8 KB (1,539 words) - 18:17, 22 August 2015
  • ==Differential forms== [[image:wave equation Excel simplest.png|center]]
    4 KB (651 words) - 23:33, 30 October 2015
  • #redirect[[differential forms]] '''What do differential forms look like?'''
    5 KB (959 words) - 13:15, 12 August 2015
  • ...A$ and $B$, finding the vector $C$ such that $B+C=A$ amounts to solving an equation, just as with numbers: ...change. That is why the change and the rate of change is a vector. But the equation looks exactly the same... even though each letter may contain unlimited amo
    113 KB (19,680 words) - 00:08, 23 February 2019
  • *6.1 Differential forms **7.3.10 [[PDEs#Solutions of the wave equation|Solutions of the wave equation]]
    4 KB (545 words) - 13:18, 29 October 2015
  • This homomorphism is called the ''[[boundary operator]]'' or ''differential''. The equation is illustrated by this (non-commutative) diagram:
    5 KB (652 words) - 13:43, 4 August 2013
  • ...issipativity in the plane and The dissipativity of the generalized Lienard equation. It is generalized in a different direction in [[A Lefschetz-type coincide
    3 KB (469 words) - 16:12, 26 March 2011
  • *P. Saveliev, ''The dissipativity of the generalized Lienard equation''. Differential Equations, 28 (1992) 6, 794-800. *P. Saveliev, ''The dissipativity of systems of ordinary differential equations in the plane''. Moscow University Mathematics Bulletin, 43 (1988)
    4 KB (583 words) - 11:43, 5 July 2016
  • *[[Introduction to differential forms: course]] UC *[[Ordinary differential equation: course]] UC
    3 KB (374 words) - 15:43, 20 April 2012
  • Differential equation
    240 bytes (35 words) - 19:34, 17 August 2011
  • 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
    32 KB (5,426 words) - 21:57, 5 August 2016
  • ...e motion in the Euclidean space. It is given by an ''ordinary differential equation'' (ODE):
    322 bytes (58 words) - 18:41, 23 November 2012
  • Here $F\cdot dr$ is a differential form of dimension $1$. '''Corollary.''' The [[Cauchy-Riemann equation]] $g_x=f_y$ is a necessary condition for a vector field $F=(f,g)$ to be con
    4 KB (778 words) - 16:47, 16 July 2014
  • Note: solutions of a “doubly periodic” differential equation, $(x',y')=f(x,y)$ with $f(x+T,y)=f(x,y+S)=f(x,y)$, can be thought of as pat
    26 KB (4,538 words) - 23:15, 26 November 2015
  • The left side of the equation is an integral over a region $G = [ a, b ]$ of dimension $1$. The right-han It is written in terms of [[differential forms]] and the [[exterior derivative]]:
    16 KB (2,752 words) - 14:18, 28 December 2012
  • *a non-autonomous ordinary differential equation: $M$ is the space, $F$ is the time, and $N$ is the space-time; and
    19 KB (3,563 words) - 15:20, 9 December 2012
  • ...cy -- with an element of a narrower, nice class of functions. For example, differential functions are approximated with polynomials (Taylor) or trigonometric polyn ...n of $A$ into X and $\operatorname{Id}_A$ is the identity map on $A$. This equation allows us to establish a relation between the homology groups of $A$ and $X
    51 KB (9,162 words) - 15:33, 1 December 2015
  • This equation has two instances under the fundamental correspondence. We get the ''differential identities'' for free!
    6 KB (879 words) - 13:00, 17 April 2013
  • '''Analysis:''' The set of [[differential function]]s $C^1({\bf R})$ and we want to find the functions closest to $\s Multiply the equation by $u_i$ using the inner product:
    10 KB (1,688 words) - 17:59, 13 October 2011
  • The last part comes from $\frac{du}{dx}=3x^{2}$ (also see [[differential forms]]). The first equation is in fact the ''change of variables''.
    7 KB (1,114 words) - 18:15, 21 July 2011