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• The gradient
  ...d on the nodes of the partition. Furthermore, every continuous function is integrable and, therefore, is somebody's derivative. In this sense, the arrow can be r What we know so far is that we can compute the rate of change of such a function of two variables in the two main directions. The result is given by a vecto
  74 KB (13,039 words) - 14:05, 24 November 2018
• Integration
  Suppose a function $f$ is defined on an open interval $I$. Then a function $F$ defined on $I$ that satisfies $F' = f(x)$ for all $x$ is called an ''an This equation has infinitely many solutions when $f$ is integrable. Furthermore, according to the ''Anti-differentiation Theorem'', if $F$ is
  69 KB (11,727 words) - 03:34, 30 January 2019
• The integral
  ...nfirm the formula with nothing but a spreadsheet. We plot the graph of the function: ...e development of algebra, the Cartesian coordinate system, and the idea of function (Chapters 2, 3, and 4).
  66 KB (11,473 words) - 21:36, 19 January 2019
• Integrals of functions of several variables
  *the height of the bar in this rectangle equal to the value of the function and with the ones outside the domain replaced with $0$s, and Suppose a function $y = f(X)=f(x,y)$ defined at the tertiary nodes of the partition of the rec
  73 KB (13,324 words) - 14:06, 24 November 2018
• Several variables
  ...formulas can now be solved in order to be able to model the location as a function of time. The result is these recursive formulas for the ''Riemann sums'': ...00$ and $0$ respectively. Below, the velocity is computed as a Riemann sum function of the previous column, with the same formula:
  76 KB (13,017 words) - 20:26, 23 February 2019
• Vector and complex variables
  ...real-valued functions of two variables. Consider $u=f(x,y)=2x-3y$, such a function: Consider another such function: $v=g(x,y)=x+5y$ is also a real-valued function of two variables:
  113 KB (18,750 words) - 02:33, 10 December 2018
• Differential calculus of parametric curves
  A parametric curve is such a function: ...the latter vector, $OX$. In either case, this is just a combination of two function of the same independent variable.
  130 KB (22,842 words) - 13:52, 24 November 2018
• Applications of integral calculus
  ...preadsheet, $\sum_i f(c_i)\cdot.1$, and them subtract the data for the new function, $\sum_i g(c_i)\cdot.1$. Furthermore, we have ...he following. We ''recognize'' this expression as the Riemann sum of a new function, $f-g$:
  103 KB (18,460 words) - 01:01, 13 February 2019
• Differential equations
  ...re, the ''difference'' of a function $y$ defined at the primary nodes is a function defined at the secondary nodes of the partition: We can also think of this sequence as a function defined at the nodes of the partition:
  64 KB (11,426 words) - 14:21, 24 November 2018
• Series
  ...t is called its best linear approximation and its happens to be the linear function the graph of which is the tangent line at the point. The replacement is jus However, there is a more basic approximation: a constant function, $y=C(x)$.
  113 KB (19,100 words) - 23:07, 3 January 2019
• Discrete functions
  ...we have a function, $f$, representing the position, $y$, of your car as a function of time, $x$: <!--150-->[[Image:graph of function 3.png| center]]
  15 KB (2,532 words) - 12:21, 11 July 2016
• Discrete forms and cochains
  ...$ is often thought of as a function the input of which is any integrable ''function'' $f$ while the output is a real number. This idea is revealed by the usual ...the limit of the Riemann sums of $f$. The student then discovers that this function is ''linear'':
  34 KB (5,619 words) - 16:00, 30 November 2015
• Calculus exercises: part II
  ...ccuracy at least $.001$. We are to approximate with Taylor polynomials the function $f(x)=x^{1/2}$ around the point $a=4$. We estimate this function on the interval $[4,4.01]$. Find $K_1$ such that
  15 KB (2,591 words) - 17:15, 8 March 2018
• Forms in Euclidean spaces
  where $f$ is a function of $x\in {\bf R}$ multiplied by the second variable called $dx\in {\bf R}$. *first we plot the curve (green) which is the restriction of our function $\varphi$ to a fixed value of $dx$;
  44 KB (7,778 words) - 23:32, 24 April 2015
• Calculus exercises: advanced
  *Prove that the function $f(X)=||X||$ is or is not linear. *Here is a plot of a few level curves of a function $F(x,y)$ with a minimizer at $(1,0)$ and a maximizer at $(-1,0)$. Sketch a
  14 KB (2,538 words) - 18:35, 14 October 2017
• Integration in dimension n
  <center>let $f: n$-dim box $\rightarrow {\bf R}$ be scalar function, then</center> and call $f$ an [[integrable function]] on $[ a, b ]$.
  33 KB (5,415 words) - 05:58, 20 August 2011
• Why do we need differential forms?
  <center>''a function on the right and its derivative is on the left''. </center> ...one variable is simply a series of numbers. Just look at this "graph" of a function in Excel:
  8 KB (1,319 words) - 22:58, 9 February 2015
• Convergence of the discrete to the continuous
  The diagram commutes. Indeed, given a function $f:{\bf R}\to {\bf R}$, we can proceed in two ways: *right then down: we acquire a $0$-form $g$ by sampling function $f$, and then we acquire $dg$ by taking the differences of the values of $g
  21 KB (3,664 words) - 02:02, 18 July 2018
• Closedness and exactness of 1-forms
  This is how we solve the "exactness problem". Given a continuous function $f$, it is exact if it's the derivative of someone: ...] of $f$. We can construct $F$ using nothing but continuity. Indeed $f$ is integrable on $[a,b]$ and its [[Riemann integral]] exists on all intervals within $[a,
  8 KB (1,421 words) - 13:41, 10 April 2013
• Exterior derivative
  What we are used to is that the derivative of a function is also a function. But here we'll rely on the following: <center>a function is a $0$-[[differential forms|form]] but its derivative is a $1$-form. </ce
  12 KB (2,089 words) - 18:16, 22 August 2015
• Riemann integral
  *let $f: n-dim$ box $→ ℝ$ be scalar function, then and call $f$ an [[integrable function]] on $[ a, b ]$.
  2 KB (361 words) - 16:47, 30 September 2013
• Integral: definition
  $$ \text{Upper half of the circle } = \text{ the graph of this function }$$ We will approximate by "sampling" our function at several values of $x$, $n$ of them.
  9 KB (1,406 words) - 19:50, 20 July 2011
• Images as functions of two variables
  .... The union of these squares gives us the [[domain]] of the function. This function has only two possible values. ...2⁸ levels of gray. These numbers together form the ''gray scale function'' of the image. An example is on the right.
  3 KB (498 words) - 03:05, 29 August 2010
• Real analysis: final 1
  ...ble function $f:\mathbf{R}^{N}\rightarrow\mathbf{R.}$ Give an example of a function $f:\mathbf{R}^{2}\rightarrow\mathbf{R}$ such that both partial derivatives #Give example of such a function $f:\mathbf{R}^{2}\rightarrow\mathbf{R}$ that $f$ is not continuous at $(0,0
  3 KB (562 words) - 20:29, 13 June 2011
• Calculus as a part of topology
  The early calculus is about the derivative, i.e., the rate of change of a function. This doesn't seem like a part of Topology, Algebra, and Geometry. However, We will approximate by "sampling" our function at several values of $x$, $n$ of them.
  13 KB (2,233 words) - 14:41, 20 February 2015
• The algebra of oriented cells
  ...or any given integrable function $f:[A,B]\to {\bf R}$, we can define a new function: Then, according to the above identities, this function is ''linear''<!--\index{linearity}-->! Indeed, we've shown:
  36 KB (6,395 words) - 14:09, 1 December 2015
• More on vector spaces
  ==Function spaces== *$f$ represents the function as a whole, and
  14 KB (2,471 words) - 21:48, 5 September 2011
• Discrete parametric curves
  ...omain, the $t$-axis, with the same nodes... Then, the $2$-dimensional node function (a curve) is Given motion with the position function $F : {\bf R} \to {\bf R}^n$ during a time interval $[ a, b ]$, we have two
  32 KB (5,426 words) - 21:57, 5 August 2016
• Integral theorems of vector calculus
  ...ion of two variables]]. This function is constant on each pixel, so it's [[integrable]]. Then be a function with [[continuous]] [[derivative]], and $R$ a simple region. Then
  16 KB (2,752 words) - 14:18, 28 December 2012
• Independence of path
  It is sufficient for the force to have a [[potential function]]: '''Theorem.''' If $F$ has a potential function, then it is conservative:
  5 KB (883 words) - 20:41, 20 August 2011
• Real analysis: test 2
  #Give example of such a function $f:\mathbf{R}^{2}\rightarrow\mathbf{R}$ that $f$ is not continuous at $(0,0 #Describe Newton's method. Give an example of a function for which the method does not apply.
  2 KB (275 words) - 20:30, 13 June 2011
• Fourier coefficients
  ...= e^{inx}: n \in Z\}$ is an [[orthonormal basis]] for the space of square-integrable functions of $[−π, π]$. This space is an [[inner product space]]: ...can be expressed in terms of the Fourier coefficients $\hat{f}(n)$ of the function $f$:
  1 KB (188 words) - 23:36, 21 May 2013
• Volume as an integral
  and call f an [[integrable function]] on [ a, b ].
  3 KB (513 words) - 02:51, 17 August 2010
• Fundamental Theorem of Calculus
  For an [[integrable]] function $f$,
  605 bytes (107 words) - 20:19, 28 August 2011
• Area integral
  and call $f$ an [[integrable function]] on $[ a, b ]$.
  3 KB (414 words) - 03:45, 22 August 2011