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• Tangent line
  34 bytes (5 words) - 13:38, 19 October 2012
• Line and surface integrals
  39 bytes (7 words) - 16:24, 19 March 2017
• Line integral
  29 bytes (3 words) - 00:18, 7 October 2012
• Line integrals
  ==Line integrals: flux== providing the mass of a curve of variable density. The third is the ''line integral along an oriented curve'':
  12 KB (2,194 words) - 14:37, 5 December 2017

Page text matches

• Functions
  ...two sets. We chose to speak of locations spaced over an infinite straight line associated with the ''integers'', '''denoted''' by: [[image:markings on a straight line.png| center]]
  151 KB (25,679 words) - 17:09, 20 February 2019
• The derivative
  ...ght see a curved line even after zooming in but it is a virtually straight line. ...''touches'' the curve and is called a ''tangent'' (the Greek for “touch”) line.
  75 KB (13,000 words) - 15:12, 7 December 2018
• Functions of several variables
  ...mn has surfaces but not all of them because some of them fail the vertical line test -- such as the sphere -- and can't be represented by graphs of functio ...previously. Indeed, recall that the ''Tangent Problem'' asks for a tangent line to a curve at a given point. It has been solved for parametric curves in Ch
  97 KB (17,654 words) - 13:59, 24 November 2018
• Classes of functions
  ...are in the same column; and the graph has all dots on the same horizontal line. The value of $y=F(x)$ doesn't vary as $x$ varies; it is ''constant''. The graph of a constant ''numerical'' function is a horizontal line:
  143 KB (24,052 words) - 13:11, 23 February 2019
• Operations on functions
  '''Example (equations).''' Numerical sets are subsets of the real line ${\bf R}$ and some of them came from solving these equations, as their ''so ...al price of $\$14$? The result is two linear relations each representing a line on the $xy$-plane:
  142 KB (23,566 words) - 02:01, 23 February 2019
• Vector fields
  *2. A line segment joining a planet and the Sun sweeps out equal areas during equal in [[image:additivity of line sums.png| center]]
  91 KB (16,253 words) - 04:52, 9 January 2019
• Precalculus
  *1. a line is drawn, the $x$-''axis''; *2. one of the two direction on the line is chosen as ''positive'', then the other is ''negative'';
  100 KB (16,148 words) - 20:04, 18 January 2017
• Differential calculus of parametric curves
  Recall that the ''Tangent Problem'' asks for a tangent line to a curve at a given point. It has been solved for the graphs of numerical '''Example (straight line).''' Linear motion is represented as follows:
  130 KB (22,842 words) - 13:52, 24 November 2018
• Applications of differential calculus
  In other words, we find the intersection of the tangent line with the $x$-axis: ...ut to $2\pi,\ 100\pi$, etc. If we choose $x_0=\pi /2$ exactly, the tangent line is horizontal, failure... $\square$
  59 KB (10,063 words) - 04:59, 21 February 2019
• Functions in higher dimensions
  ...nal cases when $P$ and $Q$ lie on the same vertical or the same horizontal line (and the triangle “degenerate” into a segment) are treated separately. ...ce Formula for the plane gives us the distance measured ''along a straight line'' as if we are walking through a field. But what if we are walking through
  113 KB (19,680 words) - 00:08, 23 February 2019
• Calculus of sequences
  ==The real number line== We visualize these as markings on a straight line, according to the order of the planks:
  113 KB (18,425 words) - 13:42, 8 February 2019
• Calculus exercises: part III
  Find a tangent line to the curve parametrized by $f$ at the point $t=2$. ...Therefore, it suffices to simply use $f'(2)$ as a direction vector for the line. Further,
  46 KB (8,035 words) - 13:50, 15 March 2018
• Manifolds
  ...vation that the light from a star passing the sun deviates from a straight line may be considered as evidence in support of this idea: ...arrive to the idea of a locally Euclidean space. The main examples are the line and the circle.
  51 KB (8,919 words) - 01:58, 30 November 2015
• Applications of integral calculus
  ...demonstrated for the special case when such a region is cut by a vertical line: This case is illustrated below (the line is $x=b$):
  103 KB (18,460 words) - 01:01, 13 February 2019
• Systems of ODEs
  *the horizontal line $y=\alpha/\beta$ is crossed vertically by the solutions. *the vertical line $x=\gamma/\delta$ is crossed horizontally by the solutions.
  63 KB (10,958 words) - 14:27, 24 November 2018
• Vector and complex variables
  In fact, either equation is a representations of a line on the plane. Then the solution $(x,y)=(4,2)$ is the point of their interse A ''flip about the line $x=y$'' that appeared in the context of finding the graph of the inverse fu
  113 KB (18,750 words) - 02:33, 10 December 2018
• More on vector spaces
  '''Theorem:''' A straight line in ${\bf R}^2$ is a subspace if and only if it contains $0$. Suppose $V$ is a vectors space, what is a ''line in a vector space''?
  14 KB (2,471 words) - 21:48, 5 September 2011
• Calculus exercises: part I
  *The graph of a function $f$ is given below. Find an equation of the line secant to the graph at $(0,-1)$. *The secant line of the sign function are shown below. What do they tell you about the diffe
  49 KB (8,436 words) - 17:14, 8 March 2018
• Several variables
  ...he curve isn't the graph of any function of one variable as the ''Vertical Line Test'' is violated. $\square$ ...f change'' (also known as the difference quotient, the slope of the secant line, etc.):
  76 KB (13,017 words) - 20:26, 23 February 2019
• Derivative as a limit
  ''Derivative'' is the [[slope]] of the [[tangent line]]. But what ''is'' the tangent line? (Silly answer: Line the slope of which is equal to the derivative).
  5 KB (857 words) - 13:57, 25 May 2011
• Discrete parametric curves
  is also a line but the motion starts at $b \in {\bf R}^n$. [[Image:tangent line examples.jpg|center]]
  32 KB (5,426 words) - 21:57, 5 August 2016
• Preview of calculus: part 1
  ; Answer : We use a tangent line. Why? The tangent line touches the point A and no where else in the curve. If we zoom in, they vir
  10 KB (1,532 words) - 00:07, 2 May 2011
• Convergence of the discrete to the continuous
  ...f the differential equation as a formula by which the slope of the tangent line to the graph of $y$ can be computed at any point on the curve, once the loc ...y the slope of the tangent line of $y$). We take a step along that tangent line up to the next point. We choose a value for the horizontal component of thi
  21 KB (3,664 words) - 02:02, 18 July 2018
• Applications of ODEs
  ...metric curves'' defined at the nodes of the standard partition of the real line: ...he string is found as the negative of the projection of the gravity on the line from $0$ to the current location $X=<x,z>$:
  50 KB (8,692 words) - 14:29, 24 November 2018
• Vector integrals
  ...e $e_1, e_2$ are the (fixed) basis vectors of ${\bf R}^2$. So, if $L$ is a line perpendicular to the square (or a surface), then the orientation is a basis ...]] (on the right) is not good for computing the arc length, but fine for [[line integrals]]: displacement, work, etc. Same applies to [[surface integral]]s
  15 KB (2,545 words) - 19:47, 20 August 2011
• Differential equations
  ...nt line at $(t,y(t))$. This slope is the same along any given ''vertical'' line: ...ent line at $(t,y)$. This slope is the same along any given ''horizontal'' line:
  64 KB (11,426 words) - 14:21, 24 November 2018
• Functions of several variables OLD
  ...to $Q$ for any given $P, Q {\in} {\bf R}^n$. The idea is to use a straight line. In ${\bf R}^m$, we simply replace $|\cdot|$ with $||\cdot||$ in the last line:
  34 KB (5,636 words) - 23:52, 7 October 2017
• Limits and continuity
  *3. we think of the function as a ''transformation'' of the real line. ...from Chapter 3: numerical functions are transformations of the real number line... and vice versa.
  107 KB (18,743 words) - 17:00, 10 February 2019
• Linear algebra
  S may be a line... S = (-(3/2)&alpha;, &alpha;, -(1/2)&alpha;) represents a straight line in '''R³'''.
  27 KB (4,667 words) - 01:07, 19 February 2011
• Linear algebra of Euclidean space
  $S$ may be a line... ...ter>$S = (-(3/2) \alpha , \alpha , -(1/2) \alpha )$ represents a straight line in ${\bf R}^3$.</center>
  26 KB (3,993 words) - 19:48, 26 August 2011
• Differential calculus
  ...he most horizontal. In other words, this is where the slope of the tangent line is zero. But that's the derivative of our function. From the ''Power Formul ...words, the monotonicity is determined by the sign of slope of the tangent line. We conclude that on interval $(0,25)$, the derivative is positive and on $
  84 KB (14,321 words) - 00:49, 7 December 2018
• Differentiation
  ...erval, we compute the difference quotients along the two intervals (second line) and place the results at the corresponding edge: ...we carry out the same operation and place the result in the middle (third line).
  82 KB (14,116 words) - 19:50, 6 December 2018
• Calculus I, the discrete version
  We divide the $x$-axis (i.e., the real line ${\bf R}$) into discrete pieces. The ration is also known as the ''slope'' of the line.
  64 KB (11,521 words) - 19:48, 22 June 2017
• Preview of calculus
  ; Answer : We use a tangent line. Why? The tangent line touches the point A and no where else in the curve. If we zoom in, they vir
  4 KB (703 words) - 14:34, 9 September 2016
• Transformations
  That's why there is a whole line of points $X$ with $FX=0$. To find it, we solve this equation: ...ngular stretch-shrink but this time it is between the two ends of the same line. To see clearer, consider what happens to a square:
  46 KB (7,625 words) - 13:08, 26 February 2018
• Forms in Euclidean spaces
  The simplest example of a differential form is a $1$-form over the real line: ...'s consider its ''discrete'' counterpart. A discrete $1$-form for the real line is, by definition, a collection $\phi$ of linear maps on tangent spaces:
  44 KB (7,778 words) - 23:32, 24 April 2015
• Integer-valued calculus
  ...have a ''hole'' (or tunnel) in it? Is it possible to travel in a straight line and arrive at the starting point from the opposite direction? Like this: ...'simply-connected''. So, if we know all gradients of all functions and all line integrals of all vector fields, we can tell if there is a hole in the regio
  27 KB (3,824 words) - 19:07, 26 January 2019
• Derivative reflects behavior of the function
  [[Image:meanValueTheoremExample.png|right|Movement along a Straight Line]] *The line that connects the end points used to be horizontal and now is has become in
  8 KB (1,470 words) - 00:39, 16 July 2011
• Calculus on cubical complexes
  ...he simplest setting, we deal with the intervals in the complex of the real line ${\mathbb R}$. Then the form assigns a number to each interval to indicate One should recognize the second line as a line integral:
  36 KB (6,218 words) - 16:26, 30 November 2015
• Precalculus exercises
  ...s of the $x$-coordinates of the intersections between the parabola and the line below: [[image:area between parabola and horizontal line.png| center]]
  17 KB (2,933 words) - 19:37, 30 July 2018
• Series
  ...t is justified by the fact that when you zoom in on the point, the tangent line will merge with the graph: ...of the curve that pass through the point; from those we choose the tangent line.
  113 KB (19,100 words) - 23:07, 3 January 2019
• Line integrals
  ==Line integrals: flux== providing the mass of a curve of variable density. The third is the ''line integral along an oriented curve'':
  12 KB (2,194 words) - 14:37, 5 December 2017
• Linear functions in Euclidean space
  *$0$, a line, or ${\bf R}^2$. *${\rm Im}(f) =$ a line, for example $f(x_1, x_2) = (x_1, 0)$ is the [[projection]].
  23 KB (3,893 words) - 04:43, 15 February 2013
• Freshman's introduction to discrete calculus
  ...h other at the nodes to form a continuous curve. However, it is a straight line? The ration is also known as the ''slope'' of the line.
  42 KB (7,443 words) - 14:18, 1 August 2016
• Homotopy
  This is called the ''straight-line homotopy''. ...ply connected because every loop can be deformed to a point via a straight line homotopy:
  46 KB (7,846 words) - 02:47, 30 November 2015
• Discrete forms
  ...he simplest setting, we deal with the intervals in the complex in the real line ${\mathbb R}$. Then the form assigns a number to each interval to indicate One should recognize the second line as a line integral:
  35 KB (6,055 words) - 13:23, 24 August 2015
• Mobius band
  What happens if we cut the Mobius band along this line? The result is a thinner band with a double twist. '''Problem.''' What happens if we cut the [[Mobius band]] along the middle line?
  3 KB (510 words) - 16:22, 17 March 2014
• Calculus by Rogawski
  2.1 [[Limit]]s, Rates of Change, and [[Tangent Line]]s ==Chapter 16: Line and Surface Integrals==
  6 KB (634 words) - 16:38, 1 March 2013
• Discrete calculus article
  ...'differentiation''. Given a function defined at several points of the real line, the difference quotient at that point is a way of encoding the small-scale ...hat is, if the [[Graph of a function|graph]] of the function is a straight line), then the function can be written as $y=mx + b$, where $x$ is the independ
  27 KB (4,329 words) - 16:02, 1 September 2019
• Parametric curves
  [[Image:tangent line examples.jpg|center]] ...[[tangent line]] to the circle at any point: the [[slope]] of the tangent line is equal to the value of the [[derivative]] of $f$ at the point.
  34 KB (5,665 words) - 15:12, 13 November 2012

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