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• Calculus exercises: part I
  It's just a limit. But we recognize that this is the derivative of some function. We compare the expression to the formula in the definitio Now, find the derivative of $x\cdot e^{x}$ with PR: above set
  49 KB (8,436 words) - 17:14, 8 March 2018
• Differentiation
  Furthermore, the derivative is defined as a limit. Unlike the limits we saw prior to derivatives, this ...unctions differentiable at a point is differentiable at that point and its derivative is equal to the sum of their derivatives; i.e., for any two functions $f,g$
  82 KB (14,116 words) - 19:50, 6 December 2018
• Calculus of chain maps
  ...nes on ${\mathbb R}_x$ are called ''primal chain functions''. The exterior derivative is also given: ==The derivative of a $0$-chain map is a $1$-chain map==
  41 KB (7,344 words) - 12:52, 25 July 2016
• Applications of integral calculus
  ...-g(c_{i})$. Then, its area is $(f(c_{i})-g(c_{i})) \Delta x_2$. Hence, the total area of the rectangles is: *we represent the total quantity $Q$ as the sum of its values $Q_i$ over simpler, or smaller, regio
  103 KB (18,460 words) - 01:01, 13 February 2019
• Differential calculus of parametric curves
  ==The instantaneous rate of change: derivative== ...es of change and, after the limit, the two components of the vector of the derivative of the curve.
  130 KB (22,842 words) - 13:52, 24 November 2018
• Metric tensor
  ...ay the cells are attached to each other affects the matrix of the exterior derivative: ...ere we start to need geometry is when we move from the first to the second derivative.
  42 KB (7,131 words) - 17:31, 30 November 2015
• Calculus I, the discrete version
  <!--150-->[[image:first derivative and Monotonicity.png| center]] ==The derivative==
  64 KB (11,521 words) - 19:48, 22 June 2017
• Introduction to discrete calculus
  ==The derivative of a $0$-cochain is a $1$-cochain== Even though we are familiar with the derivative of a function as the ''rate of change'', the ''change'' will be initially o
  40 KB (6,983 words) - 19:24, 23 July 2016
• Geometry
  ...ay the cells are attached to each other affects the matrix of the exterior derivative: ...ere we start to need geometry is when we move from the first to the second derivative.
  41 KB (6,928 words) - 17:31, 26 October 2015
• Calculus on cubical complexes
  *the derivative of a function is the ''rate of change'', while *the exterior derivative of a $0$-form is the ''change''.
  36 KB (6,218 words) - 16:26, 30 November 2015
• Discrete forms
  *the derivative of a function is the ''rate of change'', while *the exterior derivative of a $0$-form is the ''change''.
  35 KB (6,055 words) - 13:23, 24 August 2015
• Freshman's introduction to discrete calculus
  <!--150-->[[image:first derivative and Monotonicity.png| center]] ==The derivative==
  42 KB (7,443 words) - 14:18, 1 August 2016
• ODEs
  ...le ''ordinary differential equations (ODEs)'' with respect to the exterior derivative $d$ that have explicit solutions. *the first derivative $f'$ instead of the exterior derivative, and
  47 KB (8,415 words) - 15:46, 1 December 2015
• Several variables
  ...e time is in the first column progressing from $0$ every $.05$. The second derivative is in the next, $0$ and $-32$, respectively. In the next column, the initia A ''pattern'' is clear: growth by $2$. We have the total of six (linear) functions!
  76 KB (13,017 words) - 20:26, 23 February 2019
• Metric complexes
  ...o each other affects the matrix of the boundary operator (and the exterior derivative): that is closed: $A_N=A_0$. The ''total curvature'' of curve $C$ is the sum of the curvatures at the vertices (exce
  35 KB (5,871 words) - 22:43, 7 April 2016
• Tangent bundles
  We now combine all the tangent spaces into one total tangent space. It contains all possible directions in each location: each t The total work over a path in the complex is the ''line integral''<!--\index{line int
  16 KB (2,753 words) - 13:55, 16 March 2016
• The integral
  \text{the derivative }&\text{ the integral}\quad\\ ==The total value of a function: the Riemann sum==
  66 KB (11,473 words) - 21:36, 19 January 2019
• Applications of differential calculus
  Warning: the method fails when it reaches a point where the derivative is equal to (or even close to) $0$. The most important use of the latter notation is in the definition of the derivative:
  59 KB (10,063 words) - 04:59, 21 February 2019
• Functions of several variables OLD
  To define [[derivative]]s we need limits and for [[limits]] we need to understand better the [[top These two cases lead to the concept of [[partial derivative]]s.
  34 KB (5,636 words) - 23:52, 7 October 2017
• PDEs
  ...portion $h$, dependent on the presumed length of the time interval, of the total amount is shared. ...of the ODE of population growth and decay -- with respect to the exterior derivative $d_t$ over time. This time, however, for each cell there are four adjacent
  44 KB (7,469 words) - 18:12, 30 November 2015
• Partial differential equations
  ...ote that the total amount of heat in the rod remains the same (seen under “total” in the spreadsheet shown above). ...ortion, $k$, dependent on the presumed length of the time interval, of the total amount is exchanged.
  53 KB (9,682 words) - 23:19, 18 November 2018
• Functions of several variables: derivatives and integrals
  ==The derivative of a function of several variables== '''Definition.''' The ''partial derivative of $z=f(X)=f(x_1,...,x_n)$ with respect $x_k$ at'' $X=A=(a_1,...,a_n)$ are
  42 KB (6,904 words) - 15:15, 30 October 2017
• Combinatorial cell maps
  This is a vertical flip; there are also the horizontal and diagonal flips, a total of $4$. Only these four axes allow condition (A) to be satisfied. $\square$ is the (total) ''chain map''<!--\index{chain map}--> generated by $f$.
  31 KB (5,330 words) - 22:14, 14 March 2016
• Calculus Illustrated
  *Chapter 3. The derivative 4 The limit of the difference quotient: the derivative
  16 KB (1,933 words) - 19:50, 28 June 2021
• Diffusion equation
  ...portion $h$, dependent on the presumed length of the time interval, of the total amount is exchanged. The two images below are the initial state (a single i ...of the ODE of population growth and decay -- with respect to the exterior derivative $d_t$ over time. For each cell there are four adjacent cells and four tempe
  35 KB (5,917 words) - 12:51, 30 June 2016
• Vector fields
  Assuming a fixed mass, the total force gives us our acceleration. We to compute: What is the total flow along this “staircase”? We simply add the values located on these
  91 KB (16,253 words) - 04:52, 9 January 2019
• Calculus exercises: part III
  and its derivative equals Find the directional derivative $D_v f(1,0,1)$, where
  46 KB (8,035 words) - 13:50, 15 March 2018
• Integration
  But we don't recognize $\sin (x^{2})$ as the derivative of any function we know... *the derivative of the “inside” function is present as a factor.
  69 KB (11,727 words) - 03:34, 30 January 2019
• More metric complexes
  The expression can be understood as the total flux of $a$ across a region of area $1$ on $a^\star$: ...ve” is always trivial. Instead, one can define the “Hodge-dualized” second derivative computed following these four steps:
  21 KB (3,445 words) - 13:53, 19 February 2016
• Geometric complexes
  The expression can be understood as: the total flux of $a$ across a region of area $1$ on $a^\star$. ...ve” is always trivial. Instead, one can define the “Hodge-dualized” second derivative computed following these four steps:
  20 KB (3,354 words) - 17:37, 30 November 2015
• Transformations
  ...r pound. How much of each do you need to have $6$ pounds of blend with the total price of $\$14$? ...the Kenyan coffee and let $y$ be the weight of Colombian coffee. Then the total price of the blend is $\$ 14$. Therefore, we have a system:
  46 KB (7,625 words) - 13:08, 26 February 2018
• Material derivative
  The ''material derivative'' is the [[rate of change]] of some physical quantity ([[heat]], or [[momen ...d the physical quantity is the temperature of the fluid. Then the material derivative describes the temperature evolution of a certain fluid parcel in time, as i
  2 KB (388 words) - 21:31, 17 July 2012
• Diffusion
  *$d_t$ is the exterior derivative with respect to time (just the difference since the dimension is $1$); and *$d_x$ is the exterior derivative with respect to location.
  39 KB (6,850 words) - 15:29, 17 July 2015
• Differential equations
  ...opulation $\Delta y$ is proportional to $y$... and $T-y$, where $T$ is the total possible population: In the meantime, the derivative, if any, would satisfy the following:
  64 KB (11,426 words) - 14:21, 24 November 2018
• Vector and complex variables
  ...r pound. How much of each do you need to have $6$ pounds of blend with the total price of $\$14$? ...the Kenyan coffee and let $y$ be the weight of Colombian coffee. Then the total price of the blend is $\$ 14$. Therefore, we have a system:
  113 KB (18,750 words) - 02:33, 10 December 2018
• Discrete forms and cochains
  <center>''a function on the right and its derivative is on the left''. </center> ...as differential forms. 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
• Affine approximation
  ...linear map evaluated at $x-a$. This linear map $L_a$ is called the ''total derivative of $f$ at $x = a$''. Then, the total derivative
  7 KB (1,162 words) - 03:25, 22 August 2011
• Forms
  We collect the tangent spaces into the (dimension $1$) ''total tangent space'' of $K$: Then $\varphi$ is a function on the total tangent space,
  13 KB (2,459 words) - 03:27, 25 June 2015
• Discrete calculus article
  ..., however, can take the squaring function as an input. This means that the derivative takes all the information of the squaring function—such as that two is se ...me. For example, travelling a steady 50&nbsp;mph for 3 hours results in a total distance of 150 miles. In the diagram on the left, when constant velocity
  27 KB (4,329 words) - 16:02, 1 September 2019
• Calculus I -- Fall 2017
  *Goals: good understanding of limits, the derivative and the integral, fluent differentiation. i.e., the total score is the following weighted average of the five scores:
  13 KB (2,075 words) - 13:35, 27 November 2017
• Maxwell equations
  *$J$: ''total current density'' (including both free and bound current); *$d$ and $d^*$ are the [[exterior derivative]] of the primal and the dual complex respectively (they are [[adjoint]] ope
  6 KB (922 words) - 00:30, 9 April 2016
• Applied Calculus -- Spring 2015
  *Grade Breakdown: TOTAL = .05×A + .40×(Q + H) + .20×M + .35×F **2.1 The Derivative and the Slope of a Graph
  9 KB (1,141 words) - 16:08, 26 April 2015
• Integral theorems of vector calculus
  '''Theorem.''' Given a [[vector field]] $F = ( p, q )$ with continuous [[derivative]]. Then '''Theorem.''' Suppose vector field $F = ( p, q )$ has continuous derivative. Then
  16 KB (2,752 words) - 14:18, 28 December 2012
• Elementary ODEs
  ...differential equations (ODEs) of cochains'' with respect to their exterior derivative $d$. We choose a few simple examples that have explicit solutions. *the first derivative $f'$ instead of the exterior derivative, and
  16 KB (2,913 words) - 22:40, 15 July 2016
• Elementary PDEs
  ...of the ODE of population growth and decay -- with respect to the exterior derivative $d_t$ over time. For each cell there are two adjacent cells and two tempera *$d_t$ is the exterior derivative with respect to time; and
  16 KB (2,843 words) - 21:41, 23 March 2016
• Gradient
  How do we understand the [[derivative]] of [[functions of several variables]]? *The [[directional derivative]] $\nabla_e f(a)$ is a number for each $e$, $||e||=1$, there are infinitely
  6 KB (962 words) - 15:45, 17 August 2011
• Discrete parametric curves
  *derivative, ...at any point: the slope] of the tangent line is equal to the value of the derivative of $f$ at the point.
  32 KB (5,426 words) - 21:57, 5 August 2016
• Integral: introduction
  This is what we have learned about the [[derivative]]: *Geometrically, the derivative is about ''[[slope]]s''.
  3 KB (466 words) - 17:26, 20 July 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 T ==The derivative==
  13 KB (2,233 words) - 14:41, 20 February 2015
• Calculus I -- Spring 2017 -- midterm
  Name:_________________________ $\qquad$ 10 problems, 100 points total ...The graph of a function $f(x)$ is given below. Estimate the values of the derivative $f'(x)$ for $x=0,4,$ and $6$. (Just the answer)
  2 KB (263 words) - 21:54, 6 March 2017

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