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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
• 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
• 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
• 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
• 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
• 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
• 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
• Limits: transition from discrete to continuous
  Suppose a real number $x$ is given. We construct a line segment of length $1$ on the plane. Then *$\cos x$ is the projection of the segment on the horizontal line,
  51 KB (9,271 words) - 20:02, 8 September 2016
• 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
• Topology Illustrated -- Index
  *[[line integral|line integral]] *[[line with two origins|line with two origins]]
  16 KB (1,773 words) - 00:41, 17 February 2016
• Vector calculus: midterm
  ##There is only one natural parametrization of a straight line. ...ic formula for the 3D-line through the point $(2,3,4)$ and parallel to the line through the points $(1,1,1)$ and $(-1,-2,-3)$.
  4 KB (674 words) - 02:48, 22 August 2011
• Calculus I -- Fall 2017
  ...line segment (the path that lies on the hill) will be labeled as A and the line that lies on flat ground will be labeled as B. As for the x variable that **Number systems. Distance formula. Slope of a line. Standard equations of lines.
  13 KB (2,075 words) - 13:35, 27 November 2017
• Physics
  ...e above formula still applies but, as we add them together, we produce a ''line integral'': ...varphi =fdx+gdy$ we construct a $0$-form $\psi$ with $d\psi =\varphi$ as a line integral. We fix a point $a \in {\bf R}^n$ and define $\psi$ as a function
  4 KB (778 words) - 16:47, 16 July 2014
• Quotients of vector spaces
  Then each equivalence class is a line: ...which appear in a 1-1 correspondence with the points on the other diagonal line.
  6 KB (1,115 words) - 16:03, 27 August 2015
• Velocity
  [[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.
  4 KB (662 words) - 15:17, 13 November 2012
• Internal structure of a vector space: part 1
  '''Answer:''' It's a line. Prove ${\rm span \hspace{3pt}} S = L$, the line diagonal through $0$.
  10 KB (1,614 words) - 17:13, 22 May 2012
• Homology groups of graphs
  Then each equivalence class is a line: ...appear to be in a 1-1 correspondence with the points of the other diagonal line $y=-x$.
  28 KB (4,685 words) - 17:25, 28 November 2015
• Cubical calculus
  ...he simplest setting, we deal with the intervals in the complex of the real line ${\mathbb R}$. Then the cochain assigns a number to each interval to indica One should recognize the second line as a line integral:
  25 KB (4,238 words) - 02:30, 6 April 2016
• Calculus 3: final
  8. Evaluate the line integral $\int_{C}x^{2}y^{3}dx-y\sqrt{x}dy$, where $C$ is parametrized by 9. (a) Explain what it means for a line integral $\int_{C}\mathbf{F}\cdot d\mathbf{r}$ to be independent of path. (
  4 KB (652 words) - 15:22, 9 March 2014
• Fantasy math
  The bottom line: the numerical/computational aspect should be built in! ...s. [[Vectors]]. The [[dot product]]. The [[cross product]]. Equations of [[line|lines]] and [[plane]]s. [[Vector functions]] and space curves. Derivatives
  8 KB (1,196 words) - 13:02, 24 August 2015
• The gradient
  ...the graph of a function of two variables and the flow seems to follow the line fastest descent; maybe our vector field is the gradient of this function? W ...h ''linear functions''. In other words, what if we travel along a straight line on a flat, not necessarily horizontal, surface (maybe a roof)? After this s
  74 KB (13,039 words) - 14:05, 24 November 2018
• Geometry of Euclidean space
  ...all vectors perpendicular to $x$? Let's call this set $S$. What is $S$? A line: ...ll vectors perpendicular to hyperplane $S$, then $Q = {\rm span}\{v \}$, a line through $0$, a $1$-dimensional subspace.</center>
  21 KB (3,396 words) - 20:31, 10 August 2011
• The integral
  \text{the line touching the curve at a point }&\text{ the area enclosed by the curve }\ \ ...continuous. We know that the area of a trapezoid is the length of the mid-line times the height. Then we have:
  66 KB (11,473 words) - 21:36, 19 January 2019
• Closedness and exactness of 1-forms
  ...varphi =fdx+gdy$ we construct a $0$-form $\psi$ with $d\psi =\varphi$ as a line integral. We fix a point $a \in {\bf R}^n$ and define $\psi$ as a function [[image:line integral for PL.png|center]]
  8 KB (1,421 words) - 13:41, 10 April 2013
• Vector calculus: exercises
  Find a tangent line to the curve parameterized by $f$ at the point $t=2$. ...Therefore, it suffices to simply use $f'(2)$ as a direction vector for the line. Further
  16 KB (2,457 words) - 02:17, 22 August 2011
• Continuous functions
  '''Exercise.''' To what is the Mobius band with the center line cut out homeomorphic? [[Image:point-point line.png|center]]
  42 KB (7,138 words) - 19:08, 28 November 2015
• Vector calculus: review
  #In an effort to find the line in which the planes $ 2x -y- z=2 $ and $-4x+2y+2z=1$ intersect, a student #Parametrically describe the line segment with endpoints $(-1,-1,-1)$ and $(1,1,1).$
  7 KB (1,394 words) - 02:36, 22 August 2011
• Derivative
  [[Image:tangent line examples.jpg|right]] ...circle]]. Then we [[differentiation|differentiate]] and find the [[tangent line]] to the circle at any point:
  4 KB (659 words) - 01:47, 30 August 2010
• What shape of sword is best for cutting?
  [[image:rotated straight line.png| center]] The first example is a ''straight line'' (curvature $0$):
  14 KB (2,504 words) - 14:59, 17 September 2019
• Differentials
  ...\frac{dy}{dx}\bigg| _a$ = the derivative at $a$ = the slope of the tangent line through $(a,f(a))$ = $\frac{rise}{run}$ </center> The slope of the [[secant line]] is $m = \frac{\Delta y}{\Delta x}$, or
  1 KB (243 words) - 18:13, 22 August 2015
• Integral theorems of vector calculus
  ...now. Then, in a similar way, an [[area integral]] can be expressed as a [[line integral]], somehow: ...} FTC) \hspace{3pt} (interpret \hspace{3pt} as \hspace{3pt} a \hspace{3pt} line \hspace{3pt} integral)} \\
  16 KB (2,752 words) - 14:18, 28 December 2012
• Affine approximation
  As the picture shows, a curve is approximated by a straight line while a [[surface]] by a plane. ...in on the graph of a [[differentiable function]], it looks like a straight line.
  7 KB (1,162 words) - 03:25, 22 August 2011
• Limits: part 1
  **on the line through $A$ that is perpendicular to the diagonal, Estimate the [[tangent line]] of a function given [[numerical representation of functions|numerically]]
  10 KB (1,609 words) - 16:13, 2 May 2011
• Applications of derivative: optimization
  Find a point on the line $y=4x + 7$ closest to the origin. [[Image:Origin.png|none|Closest distance to origin for line $y = 4x + 7$.]]
  6 KB (891 words) - 02:15, 17 July 2011
• Linear approximations
  We know that the [[tangent line]] "approximates" the [[graphs of functions|graph]] of $y=f(x)$ around $x=a$ ...case, when you zoom in on the point, the tangent line will (but any other line won't) merge with the graph. This is the geometric meaning of ''best approx
  2 KB (384 words) - 15:44, 2 May 2011
• Applied Calculus -- Spring 2015
  **0.1 The Real Number Line and Order **0.2 Absolute Value and Distance on the Real Number Line
  9 KB (1,141 words) - 16:08, 26 April 2015
• Manifolds model a curved universe
  ...ment 1: with light from a star passing the sun and deviating from straight line. by using the graph of $f$. But for $dy$, we use the tangent line instead:
  10 KB (1,588 words) - 17:11, 27 August 2015
• More about manifolds
  Here $f^{-1}(A)$ is all these points, a whole line! Why? Because: there is no change of $F$ in this direction. ...tersection of these planes is "[[transversal]]", so that it's a line. This line approximates the intersection of the graph with the plane. It turns out to
  9 KB (1,542 words) - 19:58, 21 January 2014
• Preview of calculus: part 3
  ...'s the [[natural base]] exponent. It has a special property: the [[tangent line]] at the y-intercept has 45 degree [[slope]], below: ...tal Line Test.''' A function is one-to-one if and only if every horizontal line has at most one intersection with its graph.
  17 KB (2,498 words) - 15:06, 19 March 2011
• Implicit differentiation
  ...to f^{\prime} \to $ tangent. This is not a graph because of the [[vertical line test]]. What do we do? To find it, I need the [[derivative]], but there is '''Example, Circle.''' Find the [[tangent line]] for the [[circle]] with radius 1 at the point $(\frac{\sqrt{2}}{2},\frac{
  9 KB (1,445 words) - 15:50, 2 May 2011
• Quality of soldering
  The area inside the red contour (i.e., inside your red line) is 140,582 pixels. So the proportion of gray inside the red line is
  3 KB (532 words) - 16:22, 4 March 2011
• Derivative as a function
  *Differentiable : There is a tangent line. [[Image:Cusp.png|Which line is the tangent?]]
  9 KB (1,437 words) - 14:05, 7 October 2012
• Quotients
  ...$x$-axis. Hence, the quotient set corresponds, in this sense, to the real line. Algebraically, we write: ...te a circle<!--\index{circle}-->. An insightful way is to make it from the line. One just winds the helix, which is ${\bf R}$ topologically, around the cir
  26 KB (4,538 words) - 23:15, 26 November 2015
• Discrete forms and cochains
  ...$f$ -- a discrete function $g$ defined at predetermined points of the real line. What is its derivative? ...aluated at one of the intervals rather than a point -- is the slope of the line that connects the end-points of the interval.
  34 KB (5,619 words) - 16:00, 30 November 2015
• Slope
  Pick any two distinct points on the line and compute That's the ''slope'' of the line.
  242 bytes (41 words) - 13:41, 19 October 2012
• Calculus Illustrated
  2 The real number line 3 Numerical functions are transformations of the real number line
  16 KB (1,933 words) - 19:50, 28 June 2021
• Tangent bundles
  The total work over a path in the complex is the ''line integral''<!--\index{line integral}--> of $\varphi$ over a $1$-chain $a$ in $K$. It is simply the sum ...system, $(dx,dy)$ and the best affine approximation (given by the tangent line) becomes a linear function in this system:
  16 KB (2,753 words) - 13:55, 16 March 2016
• Calculus with Analytic Geometry III -- Spring 2014
  ...s in dimension 2 and 3: parametric curves, functions of several variables, line, surface, and volume integrals, some vector calculus ''Chapter 16: Line and Surface Integrals''
  5 KB (621 words) - 14:57, 5 May 2014
• Vector functions
  &= ( 1 + 2b, 3 + b ) + t ( -1, -2 ) {\rm \hspace{3pt} (straight \hspace{3pt} line).} In [[Calc 1]], the answer is "[[tangent line]]'" for functions of one variable. Same for [[parametric curves]]. For [[fu
  28 KB (4,769 words) - 19:42, 18 August 2011
• Linear operators: part 4
  *${\bf R}^3 \rightarrow {\bf R}^1$, or any line. "The [[horizontal line test|horizontal plane test]]", i.e., more than one intersection implies the
  13 KB (2,067 words) - 01:11, 12 September 2011
• Measuring the volume of prostate cancer tumor
  <p>Next, the tumor. The dotted line is made solid using MS Paint. The you run Pixcavator. The [[contour]] has t ...rms this number. Of course, the error can be easily cut down by making the line 1/2 thinner… To be fair the error should be divided by 2.</p>
  6 KB (1,020 words) - 15:01, 21 May 2011
• Metric complexes
  The arc-length is an example of a ''line integral''<!--\index{line integral}--> of a $1$-form $\rho$ over a $1$-chain $a$ in complex $K$ equip What's left? The complex $K$ must be a complex representation of the infinite line or the circle:
  35 KB (5,871 words) - 22:43, 7 April 2016
• Metric tensor
  The arc-length is an example of a ''line integral''<!--\index{line integral}--> of a $1$-form $\rho$ over a $1$-chain $a$ in complex $K$ equip What's left? The complex $K$ has to be a complex representation of the line or the circle:
  42 KB (7,131 words) - 17:31, 30 November 2015
• Polar coordinates
  1) line r = 0: F( 0, θ ) = ( 0, 0 );
  1 KB (167 words) - 02:30, 18 August 2010
• Homeomorphism
  [[Image:point-point line.png|center]] ...(a,c)$ and $(b,d)$, so its slope is $m = \frac{d-c}{b-a}\ne 0$. Hence, the line is given by
  13 KB (2,168 words) - 13:09, 7 August 2014
• Midtearm
  ...Find the line passing through the point $(-1,2)$ and perpendicular to the line $y=-x-2016.$
  2 KB (308 words) - 15:23, 2 March 2016
• Cubical complex
  Dimension $0$ (no change in the first line): Dimension $1$ (no change in the first line):
  29 KB (4,800 words) - 13:41, 1 December 2015
• Calculus I -- Fall 2016
  **Number systems. Distance formula. Slope of a line. Standard equations of lines. ...nition of the derivative at a point and on an interval. Slope of a tangent line. Derivatives of polynomials. Derivatives of trigonometric functions. Deriva
  11 KB (1,671 words) - 23:11, 13 December 2016
• Solving systems of linear equations
  The line is important! It reminds you: ''augmented matrix is not a matrix''. ...set is $S = \{ x = a+tv \colon t \in {\bf R} \}$. By definition, $s$ is a line:
  5 KB (802 words) - 01:38, 6 September 2011
• Elementary Linear Algebra -- Spring 2018 -- midterm
  $\bullet$ '''2.''' In an effort to find the line in which the planes $ 2x -y- z=2 $ and $-4x+2y+2z=1$ intersect, a student $\bullet$ '''5.''' Find the vector equation of the line parallel to both $xy$- and $xz$- coordinate planes and passing through $(2,
  2 KB (308 words) - 23:06, 14 March 2018
• Cubical complexes
  Dimension $0$ (no change in the first line): Dimension $1$ (no change in the first line):
  20 KB (3,319 words) - 14:18, 18 February 2016
• Calculus I -- Fall 2016 -- final
  $\bullet$ '''5.''' Find an equation of the line tangent to the curve $x^{2}+y^{2}=x$ passing through the point $(0,0)$. $\bullet$ '''7.''' Find the point on the line $y=\pi$ that is closest to the origin.
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• Stereo vision
  We need only to consider only the horizontal line through P,Q to find the distance to the object with the red dot: ...riangulation|entirely different]] in [[topology]]): the object lies on the line from the focus of the camera and its mark on the image. Here the big red do
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• Extrema of functions of several variables
  In the $n$-dim case, the [[tangent line]]s are replaced with [[tangent plane]]s, or, more precisely, tangent [[hype which form a line.
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• Data made Euclidean
  ...sense that if you throw three point on the plane the probability that they line up is zero. They are also called “geometrically independent”. ...ew values of $r$ construct the Vietoris-Rips complex for: four points on a line, or at the corners of a square, five points at the corners of a pentagon, f
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• Calculus I -- Fall 2018
  **Number systems. Distance formula. Slope of a line. Standard equations of lines. ...nition of the derivative at a point and on an interval. Slope of a tangent line. Derivatives of polynomials. Derivatives of trigonometric functions. Deriva
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• Topology
  It may be possible then to travel in a straight line and arrive at the starting point from the opposite direction. In fact, the ...equire us to add to the existing list, currently containing the point, the line, and the plane, a new item: space, or, more accurately: a 3-dimensional spa
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• Functions of several variables: derivatives and integrals
  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:
  42 KB (6,904 words) - 15:15, 30 October 2017
• Differential Forms: A Complement to Vector Calculus by Weintraub
  The integral of a 1-form over a curve ([[line integrals]]) The [[fundamental theorem of calculus]] and its analog for line integrals
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• Calculus I -- Spring 2017
  **Number systems. Distance formula. Slope of a line. Standard equations of lines. ...nition of the derivative at a point and on an interval. Slope of a tangent line. Derivatives of polynomials. Derivatives of trigonometric functions. Deriva
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• Using derivative to find extreme values
  Recall, $f^{\prime}(c)$ is the [[limit]] of [[slope]]s of the [[secant line]]s. Take any (secant) line through $(x, f(x))$ and $(c,f(c)), x < c$. Then
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• Geometry
  The arc-length is an example of a ''line integral'' of a $1$-form $\rho$ over a $1$-chain $a$ in complex $K$ equippe What's left? The complex $K$ has to be a complex representation of the line or the circle:
  41 KB (6,928 words) - 17:31, 26 October 2015
• Diffusion
  The answer is, of course, Yes. If we assume that the liquid flows along the line (the pipe $a^\star$) that connects the centers of the rooms, then what matt ...borhood is the Hodge dual of this point. We recognize this expression as a line integral:
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• Calculus I -- Spring 2017 -- final
  $\bullet$ '''3.''' Use implicit differentiation to find an equation of the line tangent to the curve $x^{1/2}+xy=2$ passing through the point $(1,1)$. $\bullet$ '''4.''' Find the point on the line $y=1-2x$ that is closest to the origin.
  1 KB (233 words) - 03:04, 2 May 2017
• Integration
  In the graph, $dx$ is the run and $dy$ is the rise of the tangent line. They are called the ''differentials'' of $x$ and $y$ respectively. The dep We also define the integral over the whole real line $(-\infty,\infty)$ in terms of the ones over rays, as the sum of two corres
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• Homotopy and homotopy equivalence
  This is called the ''straight-line homotopy''. ...ply connected because every loop can be deformed to a point via a straight line homotopy:
  45 KB (7,738 words) - 15:18, 24 October 2015
• Directional derivative
  ...ws the given direction. The point $a$ and the vector $e$ together define a line (a $1$-dimensional [[affine subspace]]) $L$: which is the slope of the [[tangent line]] to the curve at this point.
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• Partial derivatives
  and the slope of the tangent line is $\frac{\partial f}{\partial x}_1(a)$, which is the derivative of $z = f and the slope of the tangent line is $\frac{\partial f}{\partial x_2}(a)$.
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• Calculus by Stewart
  [[Vector fields]]. [[Line integrals]]. [[The fundamental theorem for line integrals]]. [[Green's theorem]]. [[Curl]] and [[divergence]]. [[Parametric
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• Integration of differential forms of degree 0 and 1
  Let's review [[line integral]]s ([[Calculus 3: course|calc 3]]) first. The line integral along $C$ (with respect to this parametrization), is, as an exampl
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• Image analysis software
  | graphical applications (GUI) and a set of command-line utilities ...ou to easily develop custom image analysis macros without writing a single line of code
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• Discrete differential forms
  What if we choose, for $g'$, [[secant line]]s instead of [[tangent line]]s?
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• Precalculus with Scientific Applications -- Spring 2016 -- midterm
  ...Find the line passing through the point $(-1,2)$ and perpendicular to the line $y=-x-2016.$
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• Lengths of curves
  If r is the size of the pixel, a line segment of length a will be represented by roughly $a/r$ pixels… but only ...] of a curve is ''defined'' and computed via approximation of the curve by line segments. The difference is that the end points of these segments lie ''exa
  13 KB (2,113 words) - 20:21, 7 February 2013
• Calculus exercises: part II
  *A chord of a circle is a straight line segment whose end-points lie on the circle. Find the average length of a ch *Given a parametric curve $x=\sin t,\ y=t^2$. Find the line(s) tangent to the curve at the origin.
  15 KB (2,591 words) - 17:15, 8 March 2018
• Combinatorial cell maps
  '''Example.''' This projection of a triangle on a line segment is not a cell map: ...system, $(dx,dy)$ and the best affine approximation (given by the tangent line) becomes a linear function in this system:
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• Discrete functions
  ...$f$ -- a discrete function $g$ defined at predetermined points of the real line. What is its derivative? ...valuated at one of the intervals rather than points -- is the slope of the line that connects the endpoints of the segment.
  15 KB (2,532 words) - 12:21, 11 July 2016
• Products
  ...dance” in robotics by constructing such a “safe” configuration space. On a line segment, this configuration space of three robots will be a cube with three '''Exercise.''' Find the safe configuration space for two robots on a line, the circle, the tripod. What if they are connected by a rod? a rope?
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• Linear operators: part 1
  ...is the projection, $X = {\bf R}^2$, $Y=x$-axis, $f^{-1}(c)$ is a vertical line, $f^{-1}(B)$ is the strip: The fist line is just the two functions "wired together" but the second is more revealing
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• Calculus of discrete differential forms
  So, on $\alpha$, using [[antisymmetry]] to get the third line, Of course we recognize this as a [[line integral]]. We can read this as follows:
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• Configuration spaces
  ...dance" in robotics by constructing such a "safe" configuration space. On a line segment, this configuration space of three robots will be a cube with three '''Exercise.''' Find the safe configuration space for two robots on a line, the circle, the tripod. What if they are connected by a rod? a rope?
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• Anticancer property of gallic acid
  ...ticancer property of gallic acid in A549, a human lung adenocarcinoma cell line, and possible mechanisms] by Dharmendra K. Maurya, Nivedita Nandakumar, and ...icancer properties were explored in A549, a human lung adenocarcinoma cell line... Colony [[counting]] was performed with [[cellAnalyst]] software (AssaySo
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• Publications that use Pixcavator
  ...class=SpellE>gallic</span> acid in A549, a human lung adenocarcinoma cell line, and possible mechanisms, </i><b style='mso-bidi-font-weight: normal'>Journ ...30&Location=U2&doc=GetTRDoc.pdf Preliminary stages for integrated text and line drawing information extraction]'' from Air Force Research Laboratory.
  17 KB (2,350 words) - 14:40, 4 November 2011
• Major and minor axes
  The major axis of an object is the line segment that connects two farthest points in the object. See [[Diameter]]. The minor axis is the line segment that is perpendicular to the major axis with the length minimal sat
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• Calculus with Analytic Geometry III -- Spring 2015
  ''Chapter 16: Line and Surface Integrals'' 16.2 Line Integrals,
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• Constructions
  ...$x$-axis. Hence, the quotient set corresponds, in this sense, to the real line. Algebraically, we need to find a map: ...te a circle<!--\index{circle}-->. An insightful way is to make it from the line. One just winds the helix, which is ${\bf R}$ topologically, around the cir
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• Calculus 3: test 1
  3. Find the plane through the origin perpendicular to the line from $(1,0,0)$ and $(0,1,1)$. 5. Find the line tangent to the curve
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• Seurat's painting
  ...ge. Much less noise and, in addition to the sail, the leaves and the shore line are captured too. So, the bottom line is you can't rely entirely on contrast/persistence and ignore [[measuring o
  3 KB (413 words) - 18:32, 7 January 2011
• Maps of polyhedra
  ...izations $|g|,|h|:X \to Y$ of these simplicial maps, they will be straight-line homotopic: ...Why can't we simply use for $G$ a simplicial approximation of the straight-line homotopy $F$?
  51 KB (9,162 words) - 15:33, 1 December 2015
• Chain complexes
  Dimension $0$ (no change in the first line): Dimension $1$ (no change in the first line):
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• Continuous vs discrete differential forms
  It's like the second line is a grainy image of the real life in the first line. How is this image made and how much information does it preserve.
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• Calculus III -- Fall 2017 -- final
  ...d the directional derivative of the function above in the direction of the line $y=x$. $\bullet$ '''4.''' Sketch the vector field given below and estimate its line integral along the boundary of the square oriented counterclockwise (multip
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• Simplicial complexes
  ...nse that if you throw three points on the plane, the probability that they line up is zero. They are also called “geometrically independent”. ...w values of $r$, construct the Vietoris-Rips complex for: four points on a line, or at the corners of a square, five points at the corners of a pentagon, f
  30 KB (5,021 words) - 13:42, 1 December 2015
• Non-Euclidean topology on the plane
  *The standard basis of the Euclidean topology on the line consists of open intervals.
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• Where do differential forms come from?
  A [[line integral]] is an integral of a form of degree $1$. In dimension $2$, it loo
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• Homology classes
  '''Exercise.''' Give formulas of continuous functions: (a) of the line that creates a loop, (b) of the plane that creates a loop, (c) of the plane
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• Calculus 3: midterm
  #Determine whether these points lie on a straight line: \[A(0,-5,5),B(1,-2,4),C(3,4,2).\]
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• Projective plane
  In particular, the $1$-dimensional projective space is the ''projective line''. What is it? [[Image:projective line.png|center]]
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• Limits at infinity: part 1
  Remember this is a straight line.
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• Computing homology
  ...nstruct a [[cubical complex]] for the '''disjoint union of a circle and an line segment''' and compute its homology, $X$.
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• Calculus exercises: part IV
  $\bullet$ Suppose point $T$ goes along the line $x=1$ while dragging point $P$ on the $xy$-plane by a string $PT$ of length
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• The algebra of cells
  <!--150-->[[image:line from tiles.png|center]]
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• Examples of differential forms
  ...s coordinate system, the best affine approximation (given by the [[tangent line]]) becomes simply a [[linear function]].
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• Affine function
  ...lculus]] the term "affine" is never used. Everything with graph a straight line is a linear function.
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• ODEs
  ...system, $(dx,dy)$ and the best affine approximation (given by the tangent line) becomes a linear function:
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• Why do we need differential forms?
  ...d as a collection of points, then the "derivative" is the [[slope]] of the line that connects a pair of adjacent ones.
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• Homology in Calculus
  #only on the real line.
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• Calculus III -- Fall 2017 -- midterm
  $\bullet$ '''3.''' Find the equation of the line tangent to the curve
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• Quotient sets
  ...t of intersection with the $x$-axis. Hence, the quotient space is the real line. Algebraically,
  3 KB (464 words) - 19:36, 31 October 2012
• Linear algebra: exercises
  ...$ that are orthogonal to $(-1,3)$. Write the set in the standard form of a line through the origin.
  5 KB (833 words) - 13:36, 14 March 2018
• Contrast
  ...[gray scale function]] of the image. Then the tip marked with a horizontal line corresponds to an object. Finally, the height marked in red is the contrast
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• Calculus with Analytic Geometry III -- Spring 2012
  ...s in dimension 2 and 3: parametric curves, functions of several variables, line, surface, and volume integrals, some vector calculus
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• The algebra of cubical cells
  <!--150-->[[image:line from tiles.png| center]]
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• Examples of homology of cubical complexes
  ...nstruct a [[cubical complex]] for the '''disjoint union of a circle and an line segment''' and compute its homology, $X$.
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• Algebra of discrete differential forms
  ...s oriented, as always, along the axes. Then we can integrate $Adx$, as a [[line integral]], along this curve going clockwise one cell at a time:
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• Orientation
  ...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
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• Internal structure of a vector space: part 2
  *1. $V = {\rm span}\{w_1\}$ (a line).
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• Image Processing Handbook by Russ
  ...tic [[edge detection|edge following]] suffers from several problems... [A] line of pixels may be broken or incomplete... or may branch... [T]he inability o
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• Calculus III -- Spring 2015 -- final exam
  ...of radius $1$ centered at $0$ as a parametric curve. (2) Find the tangent line to this circle at the point $(\sqrt{2}/2,\sqrt{2}/2)$. (3) Compute the flux
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• Quotients of complexes
  ...hain complex]] of $K$ can be defined and its homology computed. The second line indicates that $b$ and $d$ are cycles and the first indicates that they are
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• Linear algebra: midterm
  ...ctor space of all continuous functions. Represent in the standard form the line in $V$ that passes through $\sin$ and $\cos$.
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• Difference approximation of derivative
  ...ent for the smooth [[gradient]] does not produce the desired result: the [[line integral]] over 1x1 square of the gradient of a function isn't zero. i.e. t
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• Cartesian coordinate system
  the number line [[Image:number line.jpg]]
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• The topology of data
  ...r than a chosen distance from the center to a single point. For example, a line turns into a [[circle]], a [[plane]] into a [[sphere]], etc (see [[Quotient
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• Exterior derivative
  Unless, of course, this is a straight line.
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• Algebraic operations with forms and cohomology
  So, evaluated on $\alpha$, using antisymmetry to get the third line,
  8 KB (1,289 words) - 15:11, 9 October 2012
• PDEs
  ...borhood is the Hodge dual of this point. We recognize this expression as a line integral:
  44 KB (7,469 words) - 18:12, 30 November 2015
• Internal structure of a vector space: part 3
  *span of $v_1$ is a line,
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• Image Simplification
  Other characteristics of the objects can be computed as [[line integral]]s. Those include but are not limited to perimeters, [[moments]] (
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• Calculus III -- Spring 2015 -- midterm
  $\bullet$ '''5.''' Find the line tangent to the curve $r(t)=<t^{5},t^{4},t^{3}>$ at the point $(1,1,1)$.
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• Rates of change
  [[Image:HoriontalMotion.png|frame|none|Motion along straight line]]
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• Linear algebra: test 1
  Reduced row echelon form; dimension is one, this is a line.
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• Differential equations: exercises
  *Suppose point $T$ goes along the line $x=1$ while dragging point $P$ on the $xy$-plane by a string $PT$ of length
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• Cell decomposition of images
  [[image:line from pixels.png|center]]
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• Topology I -- Fall 2013 -- final exam
  ##the real line ${\bf R}$ equipped with the "ray topology": $\{(p, \infty ) \colon p \in {\
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• First derivative test
  which form a line.
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• Average contrast
  ...[gray scale function]] of the image. Then the tip marked with a horizontal line corresponds to an object. Finally, the average height of the red lines is t
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• Particle statistics with calibration
  ...ta processing with [[Image statistics|Excel]]. First I located the square, line #3248. Its area is 211, so:
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• Introduction to Topology by Gamelin and Greene
  3 The [[real line]]
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• Image search
  ...ch for a watch with a secondary dial returned many watches without. Bottom line, nothing new after a whole year.
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• Calculus II -- Fall 2012 -- final exam
  9. Given a parametric curve $x=\sin t,y=t^2$. Find the line(s) tangent to the curve at the origin.
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• Saliency
  ...[gray scale function]] of the image. Then the tip marked with a horizontal line corresponds to an object. Finally, the volume marked in red is the saliency
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• Linear algebra in elementary calculus
  Now the ''punch line''...
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• Differential equations: final exam
  #Suppose point $T$ goes along the line $x=1$ while dragging point $P$ on the $xy$-plane by a string $PT$ of length
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• Cell maps
  '''Example.''' This projection of a triangle on a line segment is not a cell map:
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• Using derivative to study concavity
  In fact, we need to answer: What happens to [[tangent line]]s?
  5 KB (854 words) - 01:30, 16 July 2011
• Forms
  The total work over a path in the complex is the ''line integral'' of $\varphi$ over a $1$-chain $a$ in $K$. It is simply the sum o
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• Breast carcinoma detection
  ...egion boundaries. Water placed on any pixel enclosed by a common watershed line flows downhill to a common local intensity minimum. Pixels draining to a co
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• Path-connectedness
  ...to $Q$ for any given $P, Q {\in} {\bf R}^n$. The idea is to use a straight line.
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• Is a continuous function always differentiable?
  Why: Graphs of some functions aren't "smooth", so there is no [[tangent line]].
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• Image segmentation
  ...ontinuous... Automatic edge following suffers from several problems... [A] line of pixels may be broken or incomplete... or may branch... [T]he inability o
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• Calculus II -- Fall 2018 -- midterm
  ...$ '''2.''' Find the area enclosed by the curves $y=x^2$, $y=-x^3$, and the line $x=1$.
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• Separation axioms
  The real line ${\bf R}$ equipped with the ray topology $\{(p, \infty ) \colon p \in {\bf
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• Linear operators: part 3
  What if we ''rotate about the line'' along $(1,1,1)$?
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• Diffusion equation
  ...borhood is the Hodge dual of this point. We recognize this expression as a line integral:
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• Chain complexes of cell complexes
  defined on the standard [[cubical complex]] $Z$ of the real line by
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• Using derivative to study monotonicity
  where $f^{\prime}(c)$ is the slope of the [[secant line]].
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• Fundamental group
  ...of the proof is to "lift" the loops in the circle to the ones in the real line:
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• Calculus 1: final exam
  5. Find the point on the line $y=2-2x$ that is closest to the origin.
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• Pattern recognition in computer vision
  <p>To take this line of thought all the way to the end, we have to ask the question: what if we
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• Is a differentiable function always continuous?
  ...ere is a break in the graph ([[discontinuity]]), there can be no [[tangent line]].
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• Independence of path
  ''Proof.'' Simply compute the line integral for a specific [[parametrization]]] $p$ of $C$:
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• Calculus II -- Spring 2018 -- final exam
  ...) Plot this entire parametric curve: $x=\sin t,\ y=\sin^2 t$. (b) Find the line(s) tangent to the curve at the origin.
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• Sequences and their limits
  ...ically, we see how the sequence accumulates toward a particular horizontal line: ...und $0$. Moreover, plotting the points $(1/n,\sin 1/n)$ reveals a straight line with slope $1$:
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• Mean Value Theorem
  *the secant line that connects the end-points that used to be horizontal has become inclined
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• Diffusion with various geometry
  If we assume that the liquid flows along the line (the pipe $a^*$) that connects the centers of the rooms, then what matters
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• Chains vs cochains
  One should recognize this as a [[line integral]]:
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• Vector calculus: exam 2
  7. Find the tangent line to the curve $f(t)=(t,t^2,t^3)$ at the point $(1,1,1)$.
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• Grayscale images - implementation
  |<font color=#0000FF><u>GetMinLine()</u></font>||Retrieves min line integral.
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• Calculus 1: test 1
  ...$a=1$ from the definition (i.e., as a limit). (b) Find the equation of the line tangent to the graph of $y=f(x)$ at the point corresponding to $a=1$.
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• Peter Saveliev's Academic Portfolio
  The bottom line is: the numerical aspect should be [[Fantasy math|built in]] the math that
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• Is every set either open or closed?
  '''Why:''' Consider a half-interval within the real line... Done!
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• Character recognition
  ...on of the location of [[center of mass]] of the letter with respect to the line, its [[size]], and its [[Topological Features of Images|topology]] allows y
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• Calculus I -- Fall 2017 -- midterm
  ...''3.''' The graph of a function $f$ is given below. Find the equation of a line tangent to the graph at $(0,-1)$.
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• Short overview
  defined on the standard [[cubical complex]] $Z$ of the real line by
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• Modeling with discrete exterior calculus
  ...have been added for ease of reading. In Maple, it must be evaluated as one line of code.
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• Discrete Calculus: Applied Analysis on Graphs for Computational Science by Grady and Polimeni
  ...ent]] for the smooth [[gradient]] does not produce the desired result: the line integral over $1 \times 1$ square of the [[gradient]] of a function isn't z
  11 KB (1,663 words) - 16:03, 26 November 2012
• Vector calculus: final 2
  ...n of the plane which passes through the origin and is perpendicular to the line $x=0,y=t,z=2t$.
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• Hacker News
  Bottom line, unable to come up with reasonable rules that work, they end up doing weird
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• Calculus 2: final
  #Find an equation of the line tangent to the parametric curve $x=4\cos θ,y=3\sin θ$ at the point corres
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• Polyhedron
  Should be familiar! Curves locally look like a straight line.
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• Is infinity a number?
  *you can add a point to the [[real line]] and treat this point as the "location" of infinity.
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• Clustering
  <p>To take this line of thought all the way to the end, we have to ask the question: what if we
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• Calculus I -- Spring 2017 -- midterm
  $\bullet$ '''10.''' Use implicit differentiation to find an equation of the line tangent to the curve $x^{2}+y^{2}=4$ passing through the point $(0,-2)$.
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• Tangent
  #REDIRECT[[tangent line]]
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• Dual space
  A two-line version of the proof: $V$ with basis of $n$ elements $\simeq {\bf R}^n$. Th
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• 3D image analysis
  The software, [[CHomP]], is used via command line in order to compute the three [[Betti numbers]] of a bitmap image or text f
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• Visual image search engines
  ...ch for a watch with a secondary dial returned many watches without. Bottom line, nothing new after a whole year.
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• Home of Math
  ''Drop me a line if you have any questions or would like to [[how to contribute|contribute]]
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• Calculus 2: test 3
  #Plot the curve $r=2\cos(3\theta)$. Find the line(s) through the origin tangent to the curve.
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• Precalculus with Scientific Applications -- Spring 2016
  * Modeling the Real World. Real Numbers and Their Properties. The Real Number Line and Order. Integer Exponents. Rational Exponents and Radicals. Algebraic Ex
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• Is the image of an open set under a continuous function open?
  Why: Consider the [[projection]] of a vertical line in the plane onto the $x$-axis, or simply a [[constant function]] from ${\b
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• College Algebra -- Fall 2016 -- midterm
  $\bullet$ '''2.''' Find the equation of the line passing through the points $(-1,1)$ and $(-1,5)$.
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• Calculus Two by Flanigan and Kazdan
  *10.1 [[Line integrals]] 530
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• Tangent bundle
  ...alc 1]] we know that can we use the [[derivative]] to find the ''[[tangent line]]'' for the graph of the function. In higher dimensions we talk instead of
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• Euclidean space made discrete
  [[image:line from pixels.png|center]]
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• A graph, non-tree representation of the topology of a gray scale image by Saveliev
  ...traces all level sets in the image. This is the case with the “fast level line transform” [17] (step 4a). Therefore its complexity is O(N²),
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• Wave equation
  ...borhood is the Hodge dual of this point. We recognize this expression as a line integral:
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• Homology as an equivalence relation
  '''Exercise.''' Give formulas of continuous functions: (a) of the line that creates a loop, (b) of the plane that creates a loop, (c) of the plane
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• Calculus 3: course
  *[[Parametric curves]] and [[line integrals]]
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• Precalculus with Scientific Applications -- Spring 2016 -- final exam
  $\bullet$ '''14.''' Find the equation of the line starting at the point $(1,2,3)$ in the direction of the vector $<1,1,1>$.
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• JPlex examples
  The last line will print out results that look like this:
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• Compact spaces
  First, the real line ${\bf R}=(-\infty,\infty)$ is not compact: it suffices to consider the cove
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• Calculus exercises: advanced
  ...n of the plane which passes through the origin and is perpendicular to the line $x=0,y=t,z=2t$. #There is only one natural parametrization of a straight line.
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• Tangent plane
  In the n-dim case, the [[tangent line]]s to the graph of a function are replaced with ''tangent planes'', or, mor
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• Euler and Lefschetz numbers
  ...forest, the response isn't unique (one can more deviate from the straight line) and the simple theorem above won't apply.
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• Advanced Calculus I -- Fall 2016
  ...ns of several variables. Theory of definite integrals, multiple integrals, line and surface integrals, improper integrals, infinite series. PR: MTH231 and
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