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• Geometry
  Of course, it is sufficient to know the sign of the derivative to distinguish between increasing and decreasing behavior. Therefore, this behavior depends only o ...eveloping, the sign of the exterior derivative will tell us the difference between increasing and decreasing behavior. But the derivative only uses the topolo
  41 KB (6,928 words) - 17:31, 26 October 2015
• Precalculus exercises
  *Find the exact values 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
• Norm
  ...d by how much. To compare, we can measure lengths of vectors and the angle between them. [[Image:length and angle.jpg|250px]]
  3 KB (391 words) - 14:22, 26 July 2012
• Line integrals
  ...the velocity field (it is explained in this section). The main difference between the first and the rest is that the parametric curve isn't the ''integrand'' where $\alpha$ is the angle between $V$ and $D$.
  12 KB (2,194 words) - 14:37, 5 December 2017
• Inner product spaces: part 2
  The angle between $u$ and $v$ is $\alpha$ as: ${\rm cos \hspace{3pt}} \alpha = \frac{< u , v> Best in what sense? We mean that the "area" between the two graphs, $\sin$ and $g$ is minimized. In other words, minimize
  10 KB (1,688 words) - 17:59, 13 October 2011
• Parametric curves
  ...tyle\lim_{x \rightarrow a} f(x) = L$. To illustrate, you don't have to use vectors... ...last condition encodes the idea of approximation: not only the difference between $f$ and its approximation is "small" but it's small even relative to the ch
  34 KB (5,665 words) - 15:12, 13 November 2012
• Calculus III -- Fall 2017
  '''MTH 231 - Calculus with Analytic Geometry III.''' Vectors, curves, and surfaces in space. Derivatives and integrals of functions of m ...he surface on the building. Also, the intersection point of all the normal vectors will be the focus of the mirror/window.
  10 KB (1,596 words) - 13:34, 27 November 2017
• Inner product
  ...screte calculus]] we need to be able compute lengths of vectors and angles between them. An inner product is how one adds geometry to a [[vector space]]. ...er product'' on $V$ is a function that associates a number to each pair of vectors in $V$:
  4 KB (749 words) - 20:12, 1 May 2013
• Heat transfer
  The amount of heat exchanged between two rooms is proportional to: #and, inversely, to the distance between the [[center of mass|centers of mass]] of the rooms.
  2 KB (311 words) - 13:17, 28 August 2015
• Calculus with Analytic Geometry III -- Spring 2014
  '''MTH 231 - Calculus with Analytic Geometry III.''' Vectors, curves, and surfaces in space. Derivatives and integrals of functions of m 12.1 Vectors in the Plane,
  5 KB (621 words) - 14:57, 5 May 2014
• Calculus with Analytic Geometry III -- Spring 2015
  '''MTH 231 - Calculus with Analytic Geometry III.''' Vectors, curves, and surfaces in space. Derivatives and integrals of functions of m 12.1 Vectors in the Plane,
  6 KB (805 words) - 13:38, 6 May 2015
• Topology Illustrated -- Index
  *[[angle between vectors|angle between vectors]]
  16 KB (1,773 words) - 00:41, 17 February 2016
• Calculus by Rogawski
  6.1 Area Between Two Curves 12.1 [[Vectors]] in the Plane
  6 KB (634 words) - 16:38, 1 March 2013
• What shape of sword is best for cutting?
  (Note that the idea of decomposition of vectors is routinely used in physics, e.g., in the study of motion of an object on ...It is done with the help of the rotation matrix (shown at the top) for an angle of rotation $\alpha$:
  14 KB (2,504 words) - 14:59, 17 September 2019
• Euler and Lefschetz numbers
  ...heir “counterparts” in $K$ are also homologous. What is the correspondence between the finer cycles in $K'$ and the coarser cycles in $K$? ...using a polar coordinate parametrization, then measuring the change in the angle, and setting
  41 KB (7,169 words) - 14:00, 1 December 2015
• Linear algebra: final
  ...(b) State the Cauchy-Schwarz inequality. (c) Define the angle between two vectors in an inner product space. Prove that it's well-defined. ...an element of an inner product space $V$ and suppose $S$ is the set of all vectors orthogonal to $a$, plus $0$. Prove that $S$ is a subspace of $V$.
  2 KB (376 words) - 20:27, 13 June 2011
• Elementary Linear Algebra -- Spring 2018 -- midterm
  $\bullet$ '''3.''' Vectors $A$ and $B$ are given below. Copy the picture and illustrate graphically: ( $\bullet$ '''4.''' Find the angle between the vectors $<1,1,1>$ and $<1,0,0>$. Don't simplify.
  2 KB (308 words) - 23:06, 14 March 2018
• Geometry in calculus
  ...know the sign of the derivative or the exterior derivative to distinguish between increasing and decreasing behavior. But the latter only uses the topologica ...ch location in some region $D$ (the domain) in this space and each pair of vectors (directions) at that location:
  9 KB (1,604 words) - 18:08, 27 August 2015
• Calculus 3: test 1
  1. Vectors $a$ and $b$ are given below. Copy the picture and illustrate graphically (a 2. Find the angle between the vectors $<1,1,1>$ and $\mathbf{i}$. Don't simplify.
  799 bytes (132 words) - 15:21, 9 March 2014
• Products
  The question is: what is the relation between the topology of the $xy$-plane and the topologies of the $x$- and $y$-axes? where $\phi$ is the angle of the rod and $R$ is the (fixed) length of the rod.
  44 KB (7,951 words) - 02:21, 30 November 2015

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