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• The gradient
  ...'' reversible if we limit ourselves to smooth (i.e., infinitely many times differentiable) functions. The problem is more profound with vector fields as we shall see We recognize this as a discrete $1$-form. Now, the question above becomes: is it possible to produce this pattern of
  74 KB (13,039 words) - 14:05, 24 November 2018
• Differential calculus of parametric curves
  ...): $c_{k}=[t_{k-1},t_{k}]$. It is then a $1$-form, the difference of a $0$-form $F$. When this limit exists, the parametric curve $F$ is called ''differentiable'' at $t=s$.
  130 KB (22,842 words) - 13:52, 24 November 2018
• Differentiation
  ...this one has a parameter, the location $x$. That is why with the input a differentiable function $f$, the output of this limits is another function $f'$. What this ...s equal to the sum of their derivatives; i.e., for any two functions $f,g$ differentiable at $x$, we have at $x$:
  82 KB (14,116 words) - 19:50, 6 December 2018
• Metric tensor
  '''Proposition.''' Given a differentiable function $f:{\bf R}\to {\bf R}$, for any differentiable $r:{\bf R}\to {\bf R}$ with $r'>0$,
  42 KB (7,131 words) - 17:31, 30 November 2015
• Forms in Euclidean spaces
  The simplest example of a differential form is a $1$-form over the real line: [[image: 1-form plotted.png|center]]
  44 KB (7,778 words) - 23:32, 24 April 2015
• Functions of several variables: derivatives and integrals
  First let's look at the point-slope form of ''linear functions'': In that case, the function $f$ is called ''differentiable'' at $X=A$. Then vector $M$ is called the ''gradient'' or the ''derivative'
  42 KB (6,904 words) - 15:15, 30 October 2017
• Geometry
  '''Proposition.''' Given a differentiable function $f:{\bf R}\to {\bf R}$, for any differentiable $r:{\bf R}\to {\bf R}$ with $r'>0$,
  41 KB (6,928 words) - 17:31, 26 October 2015
• Metric complexes
  Note: we have used this notation for $f(a)=\langle f,a \rangle$, where $f$ a form. ...is an example of a ''line integral''<!--\index{line integral}--> of a $1$-form $\rho$ over a $1$-chain $a$ in complex $K$ equipped with a metric tensor:
  35 KB (5,871 words) - 22:43, 7 April 2016
• Multilinear algebra
  The set of all possible directions at point $A\in {\bf R}^n$ form a vector space of the same dimension. It is $V_A$, a copy of ${\bf R}^n$, a '''Exercise.''' Finish the sentence “$P_k$ consists of all vectors of the form $v + u, \ u,v\in V^k$, such that...”.
  49 KB (8,852 words) - 00:30, 29 May 2015
• Vector fields
  ...the flow is shown as the thickness of the arrow. This is a real-valued $1$-form. ...mbers can be combined into a ''vector''. The result is a vector-valued $1$-form. $\square$
  91 KB (16,253 words) - 04:52, 9 January 2019
• Differential equations
  It is then a $0$-form. For example, here are a few solutions of the equation: ...of the velocity during the time interval $[t_{i-1},t_i]$. It is then a $1$-form. We set up a recursive equation:
  64 KB (11,426 words) - 14:21, 24 November 2018
• Differential calculus
  '''Theorem.''' Suppose a function $y=f(x)$ is differentiable at $x=c$. ...is its exact value? According ''Fermat's Theorem'', since the function is differentiable, the point, $c$, has to satisfy $f'(c)=0$. Find the derivative:
  84 KB (14,321 words) - 00:49, 7 December 2018
• Exterior derivative
  ==What may be the meaning of the derivative of a differential form?== ...function is a $0$-[[differential forms|form]] but its derivative is a $1$-form. </center>
  12 KB (2,089 words) - 18:16, 22 August 2015
• Series
  '''Theorem (Best linear approximation).''' Suppose $f$ is differentiable at $x=a$ and ...eorem (Best quadratic approximation).''' Suppose $f$ is twice continuously differentiable at $x=a$ and
  113 KB (19,100 words) - 23:07, 3 January 2019
• The derivative
  ...mer, “cutting”, line, once we have the two points, we know the point-slope form of the line, as presented in Chapter 2. Once they realized that the radius and such a line form a $90$ degrees angle, the problem was solved by the ancient Greeks with jus
  75 KB (13,000 words) - 15:12, 7 December 2018
• Vector functions
  ...f'(a)$ is a linear map and $v_1, v_2$ are linearly independent when $f$ is differentiable. *$f$ is differentiable at $x = a$ and
  28 KB (4,769 words) - 19:42, 18 August 2011
• Functions of several variables
  The standard, slope-intercept, form of the equation of a line in the $xy$-plane is: A similar, also in some sense ''slope-intercept'', form of the equation of a plane in ${\bf R}^3$ is:
  97 KB (17,654 words) - 13:59, 24 November 2018
• Partial differential equations
  ...om $a$. This is a discrete $1$-form with respect to location $x$ and a $0$-form with respect to time $t$. ...' of the wall $A$ at a given time over the same time period. This is a $0$-form.
  53 KB (9,682 words) - 23:19, 18 November 2018
• Integration of differential forms of degree 2
  ...$k$ vectors in $ \in T_aM$ and finds a real number in ${\bf R}$, and it is differentiable with respect to $a$ and linear with respect of each of the vectors $v$. As an example, in the familiar $2$-form $dxdy$, $dx$ and $dy$ are the two vectors.
  15 KB (2,341 words) - 20:53, 13 March 2013
• Vector and complex variables
  The four coefficients of $x,y$ form the first table: ...re, of course, infinitely many solutions. An additional restriction in the form of another linear equation may reduce the number to one... or none. The var
  113 KB (18,750 words) - 02:33, 10 December 2018
• Discrete forms and cochains
  ...intervals or, moreover, of $1$-''chains''. We can see this idea in the new form of the additivity property: ...s. The form on the left is what we call the ''exterior derivative'' of the form on the right.
  34 KB (5,619 words) - 16:00, 30 November 2015
• Topology Illustrated -- Index
  *[[cubical form|cubical form]] *[[differentiable form|differentiable form]]
  16 KB (1,773 words) - 00:41, 17 February 2016
• Modelling with discrete vector fields and forms
  where $f:R \to R$ is continuous and $x:I \to R$ is differentiable on an open interval $I$. The equation has to be satisfied for all $t \in I$ ...ut $dx$ its [[exterior derivative]], a $1$-form, and $dt$ the constant $1$-form.
  9 KB (1,561 words) - 16:06, 27 August 2015
• Calculus exercises: advanced
  *Solve the following system, expressing the solution in vector form: $$\begin{array}{lll}x_2-x_3=1,\\x_1+x_2+x_3=2.\end{array}$$ **c. If the partial derivatives exist then the function is differentiable.
  14 KB (2,538 words) - 18:35, 14 October 2017
• Applications of differential calculus
  These two sequences of numbers form a sequence of intervals: Thus to compare two functions $f$ and $g$, at infinity or at a point, we form a fraction from them and evaluate the limit of the ratio:
  59 KB (10,063 words) - 04:59, 21 February 2019
• Discrete differential forms
  ...$f$ is [[continuous]]. But if $b > 1 + \frac{a}{2} \pi$, $f$ is nowhere [[differentiable]]. (For another example, consider the issue of [[lengths of curves]] in the | $0$-form
  10 KB (1,471 words) - 12:50, 12 August 2015
• Mean Value Theorem
  ...rem ([[Mean Value Theorem]]).''' Let $[ a, b ] \subset D(h)$, where $h$ is differentiable on $( a, b )$ and $h$ is continuous on $[ a, b ]$. Then ...f which isn't true. This is what it does and does not say about a function differentiable at $X=C$:
  7 KB (1,171 words) - 20:28, 10 July 2018
• Integration of differential forms of degree 0 and 1
  '''Definition:''' Given a $0$-form $\varphi$, the ''integral of the form $\varphi$ over oriented $0$-manifold'' $M=\{p\}$ is: *$0$-form $=$ number assigned to each vertex.
  12 KB (1,906 words) - 17:44, 31 December 2012
• Tangent bundles and differential forms
  Recall, a continuous $1$-form in ${\bf R}^3$ is a function ...$(dx,dy,dz)$. The function is [[linear operator|linear]] on $dx,dy,dz$, [[differentiable]] on $x,y,z$.
  6 KB (1,177 words) - 15:53, 5 November 2012
• Convergence of the discrete to the continuous
  *right then down: we acquire a $0$-form $g$ by sampling function $f$, and then we acquire $dg$ by taking the differ ...he derivative $f'$ of $f$, and then we find the exterior derivative (a $1$-form) $dg$ by integrating $f'$ on the segments:
  21 KB (3,664 words) - 02:02, 18 July 2018
• Calculus I, the discrete version
  ...e domain and a new edge for each node. Together, these new nodes and edges form a new copy of the domain. [[image:form and its dual.png| center]]
  64 KB (11,521 words) - 19:48, 22 June 2017
• Continuous vs discrete differential forms
  ...hat, continuous and discrete forms are very similar in the sense that they form vector spaces that behave similarly. ...on we need to be able to match two very different entities -- a continuous form and a discrete one -- one by one. Let's review what they are.
  9 KB (1,483 words) - 13:54, 13 April 2013
• Systems of ODEs
  provided $x=x(t),\ y=y(t)$ are differentiable at $t$ and $u=f(x,y),\ v=g(x,y)$ are continuous at $(x(t),y(t))$. ...ir of functions $x=x(t)$ and $y=y(t)$ (a parametric curve) with either one differentiable on an open interval $I$ such that for every $t$ in $I$ we have:
  63 KB (10,958 words) - 14:27, 24 November 2018
• Closed and exact forms
  ...(Topological property of exterior derivative).''' If $\varphi$ is a $C^2$-form in $\Omega ^k({\bf R}^n)$, then $d_{k+1}(d_k \varphi)=0$ (we can also write ...uct]] and $A=A(x^1,...,x^n)$ is a coefficient function, twice continuously differentiable. Then, using the definition of exterior derivative and its formula for $0$-
  9 KB (1,423 words) - 20:53, 13 March 2013
• Transformations
  The four coefficients of $x,y$ form the first table: Now form a matrix:
  46 KB (7,625 words) - 13:08, 26 February 2018
• Manifolds model a curved universe
  ...al "generically". They are usually required to be [[continuous]] or even [[differentiable]]. Recall $y = f(x)$ is a $0$-form and $dy=f'(x)dx$ is a $1$-form.
  10 KB (1,588 words) - 17:11, 27 August 2015
• Extrema of functions of several variables
  ...[[First derivative test]]).''' Suppose $a$ is a local extreme point of a [[differentiable function]] $f$. Then which form a line.
  9 KB (1,511 words) - 16:07, 17 August 2011
• Differentiation without limits: part 1
  : for any [[differentiable function]] $f\cdot g$ wrt $x$. : for differentiable $f$ and $g$.
  6 KB (1,004 words) - 16:00, 2 May 2011
• Integration
  ...aled versions of the old ones. Let's recast this statement in the integral form. ...ched versions of the old ones. Let's recast this statement in the integral form:
  69 KB (11,727 words) - 03:34, 30 January 2019
• Properties of the exterior derivative
  where $A,B$ are continuously differentiable functions and $dX,dY$ are basis elements of $ \Omega^k,\Omega^m$ respective '''Theorem.''' $d(d\varphi) = 0$ for any $C^2$-form $\varphi \in \Omega^k$.
  8 KB (1,539 words) - 18:17, 22 August 2015
• Tangent bundle
  ...which is a [[differential form]] of degree $1$, for the same purpose. This form is ''defined'' on the tangent vectors to the domain manifold. We now invest ...n. Let's parametrize $C$ as $p \colon [a,b] \rightarrow C$, where $p$ is [[differentiable]].
  5 KB (859 words) - 02:33, 22 January 2013
• First derivative test
  ...st derivative test in dim n).''' Suppose a is a local extreme point of a [[differentiable function]] f. Then which form a line.
  3 KB (438 words) - 19:02, 7 August 2010
• Functions of several variables OLD
  ...restriction of $f$ to $L, D(g) = L$. Now every element $v$ in $L$ has the form The idea comes from the fact that if you zoom in on the graph of a [[differentiable function]], it looks like a straight line.
  34 KB (5,636 words) - 23:52, 7 October 2017
• Euler and Lefschetz numbers
  Recall that we have $n$ commodities freely traded with possible prices that form a ''price vector'' $p=(p_1,...,p_n),$ at each moment of time. Then all posi under some norm so that the new prices form the $n$-simplex $\sigma$.
  41 KB (7,169 words) - 14:00, 1 December 2015
• Calculus on graphs
  ...by $C^0(G)$ and $C^1(G)$ respectively (not to be confused with the set of differentiable functions). ...cochains are called ''discrete differential forms''<!--\index{differential form}-->, to be discussed later.
  16 KB (2,578 words) - 00:14, 18 February 2016
• Linear approximations
  Assume $f$ is [[differentiable]] at $x=a$. In that case, when you zoom in on the point, the tangent line w [[Point-slope]] form:
  2 KB (384 words) - 15:44, 2 May 2011
• More about manifolds
  ...half-disk but, when it's glued to its antipodal point, the two half-disks form a whole disk. Let's classify the level sets of a twice [[differentiable]] function.
  9 KB (1,542 words) - 19:58, 21 January 2014
• Affine approximation
  The idea comes from the fact that if you zoom in on the graph of a [[differentiable function]], it looks like a straight line. Let $f: {\bf R}^n {\rightarrow} {\bf R}^m$. What is the form of $T$?
  7 KB (1,162 words) - 03:25, 22 August 2011
• Modelling motion on manifolds
  ...o {\bf R}^n$ is a continuous [[vector field]] and $x:I \to {\bf R}^n$ is a differentiable [[parametric curve]] defined on an open interval $I$. and a vector-valued $0$-form.
  2 KB (377 words) - 17:13, 27 August 2015
• A new look at continuity
  '''Exercise.''' In addition to being continuous, $f(x)=x^2$ is also differentiable. How does that help with the accuracy issue? Hint: there is a simple depend ...to algebra, it will explain how these pieces fit together ''globally'' to form components<!--\index{path-component}-->, holes<!--\index{holes}-->, voids<!
  17 KB (2,946 words) - 04:51, 25 November 2015
• Applications of integral calculus
  Therefore, the balance equation has this form: Now, something a bit more specific. What if these points form the graph of a function $y=f(x)$ defined at the nodes of a partition of $[a
  103 KB (18,460 words) - 01:01, 13 February 2019
• Calculus exercises: part III
  *Find the reduced row echelon form of the following system of linear equations. What kind of set is its soluti ...${\bf R}^2$ that are orthogonal to $<-1,3>$. Write the set in the standard form of a line through the origin.
  46 KB (8,035 words) - 13:50, 15 March 2018
• Vector integrals
  ...exists (it does when $F$ is [[continuous]] and the curve is continuously [[differentiable]]), then we define the ''line integral'' as this limit (here $F \cdot V ds$ is a [[differential form]]).
  15 KB (2,545 words) - 19:47, 20 August 2011
• Linear algebra: test 2
  #Suppose $V$ is the space of differentiable at $0$ functions of two variables. Suppose $A:V\longrightarrow \mathbf{R}^{ Form a matrix from these columns.
  4 KB (583 words) - 01:13, 12 October 2011
• Matrices as functions
  *2. $\frac{d}{dx} \colon C^1({\bf R}) \rightarrow ?$, (from the [[differentiable]] functions to what?) ...row {\bf R}$: $f'(a)=3$, or $\frac{dy}{dx}=3$, or $dy=3dx$ ([[differential form]]).
  13 KB (2,187 words) - 22:17, 9 September 2011
• Geometry in calculus
  This looks very much like the definition of a $2$-form except it's ''symmetric not antisymmetric''! if $C$ is parametrized by a differentiable function $p:[a,b] \rightarrow {\bf R}^n$. We can also compute the [[curvatu
  9 KB (1,604 words) - 18:08, 27 August 2015
• Homology as an equivalence relation
  ...and $B$ are ''homologous'' if there is a surface $S$ such that $A$ and $B$ form its boundary. *$F$ is differentiable on $X$,
  21 KB (3,530 words) - 19:54, 23 June 2015
• Homology classes
  ...and $B$ are ''homologous'' if there is a surface $S$ such that $A$ and $B$ form its boundary. *$F$ is differentiable on $X$,
  21 KB (3,581 words) - 15:51, 28 November 2015
• Homotopy
  ...this surface. The homotopy above is piece-wise linear and the one below is differentiable: '''Exercise.''' Prove that pointed spaces and pointed maps form a category.
  46 KB (7,846 words) - 02:47, 30 November 2015
• Compact spaces
  ...$\alpha$. choose a single element of $\gamma$ that contains it. These sets form a finite subcover of $\gamma$. $\blacksquare$ Hint: To understand the concept, limit the set to differentiable functions with the derivatives between, say, $-1$ and $1$.
  19 KB (3,207 words) - 13:06, 29 November 2015
• Derivative as a limit
  The [[point-slope form]] of the line: We say that $f$ isn't ''[[differentiable]]''.
  5 KB (857 words) - 13:57, 25 May 2011
• Vector calculus: review
  ...s of a linear space. (b) Show that the vectors $(1,0,0), (1,1,0), (1,1,1)$ form a basis of ${\bf R}^3$. #Let $F$ be a differentiable parametric curve. If $F^{\prime}(t)$ is perpendicular to $F(t)$ for all $t,
  7 KB (1,394 words) - 02:36, 22 August 2011