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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
• 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
• 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
• 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

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