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Topic: Tangent vector

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	PlanetMath: vector field (Site not responding. Last check: 2007-10-08)
		The problem is that the tangent spaces form a fiber bundle, and this may not be trivial.
		Vector fields on a smooth manifold support an operation called the Lie bracket, making them into a Lie algebra; this construction produces an intimate link between Lie algebras and Lie groups, which are of great interest to physicists and mathematicians alike.
		This viewpoint on vector fields emphasizes the machinery of modern geometry, namely sheaves, local rings, and bundles; this machinery is useful in differential geometry, important in complex analtyic geometry, and foundational in algebraic geometry -- schemes cannot be described without it.
	planetmath.org /encyclopedia/VectorField.html (770 words)

	PlanetMath: tangent bundle (Site not responding. Last check: 2007-10-08)
		forgetting the tangent vector and remembering the point, is a vector bundle.
		Cross-references: fibers, base, obvious, section, vector field, vector bundle, tangent vector, projection, differentiable, bijective, derivative, map, diffeomorphism, neighborhood, isomorphic, structure, tangent spaces, disjoint union, differentiable manifold
		This is version 2 of tangent bundle, born on 2003-10-06, modified 2003-10-06.
	planetmath.org /encyclopedia/TangentBundle.html (137 words)

	Normal Vector and Curvature
		The tangent line, binormal line and normal line are the three coordinate axes with positive directions given by the tangent vector, binormal vector and normal vector, respectively.
		Thus, the unit-length tangent vector is (-sin(u), cos(u), 0), the binormal vector is (0, 0, 1), and the normal vector is (-cos(u), sin(u), 0).
		Therefore, you are moving in the direction of the tangent vector, your "up" vector is in the direction of the binormal vector and the rate of turning and turning direction are given by the curvature and the direction of the normal vector, respectively.
	www.cs.mtu.edu /~shene/COURSES/cs3621/NOTES/curves/normal.html (890 words)

	No Title
		A vector from the tangent space at point 1 is parallel-transported along the curve to point 2, and then it is embedded in the bigger space.
		We require that the resulting vector must be the same as the one obtained in the opposite way: The vector at point 1 is first embedded into a bigger euclidean space and then parallel-transported along the image of the curve connecting points 1 and 2.
		on a constant vector in (8) is irrelevant for the embedding of the tangent space.
	www.physik.fu-berlin.de /~kleinert/kleiner_re259/embed.html (2902 words)

	Tangent space -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-08)
		All the tangent spaces have the same (The magnitude of something in a particular direction (especially length or width or height)) dimension, equal to the dimension of the manifold.
		Once tangent spaces have been introduced, one can define (Click link for more info and facts about vector field) vector fields, which are abstractions of the velocity field of particles moving on a manifold.
		All the tangent spaces can be "glued together" to form a new differentiable manifold of twice the dimension, the (Click link for more info and facts about tangent bundle) tangent bundle of the manifold.
	www.absoluteastronomy.com /encyclopedia/t/ta/tangent_space.htm (1389 words)

	cotangent.vector (Site not responding. Last check: 2007-10-08)
		Two such curves yield the same tangent vector if they are tangent to each other, and if increments in their respective parameters move them along at the same rate at the point in question.
		You can associate to the cotangent vector the tangent vector which suggests moving in the direction of fastest increase of the function, and whose length is the rate of increase.
		If you multiply the coordinates of all the points by 10, then the coordinates of a tangent vector also get multiplied by 10, but the coordinates of a cotangent vector are reduced by a factor of 10: the amount by which the function increases per "unit" change in a coordinate is less, not greater.
	math.ucr.edu /home/baez/gr/cotangent.vector.html (695 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		Two curve tangents define a plane.The first curve selected is modified to have its tangent vector lie in the plane defined by the tangent vectors of the other two curves.
		Adjusting the rotation is rotating the tangent vector on the tangent plane defined by the surface intersection.
		Tangent the curve is modified by projecting its tangent vector where it intersects the surface onto the tangent plane of the surface.
	www.cclabs.missouri.edu /things/instruction/aw/Projecttangent.html (2415 words)

	ENGR 1405 Chapter 4: Parametric Vector Functions
		These two vectors can then be used to rewrite (decompose) the velocity and acceleration vectors studied in the mechanics course in terms of their tangential and normal components (NOTE: they also span a subspace).
		In this expression for the unit tangent vector the quantity “ds” is the element of arc length defined previously.
		However we determined above that the derivative of the tangent vector is proportional to the unit principal normal vector.
	www.engr.mun.ca /~ggeorge/1405/notes/ch4/c4.html (2723 words)

	2511.html
		A derivative of a vector function, is also a vector function, and it is obtained from the original function by simply differentiating the component functions.
		In order to find a perpendicular vector to the tangent line, we use the fact that if a differentiable vector function, g(t), has a constant magnitude, that is g(t)=c where c is a constant, then g'(t) and g(t) are perpendicular to each other.
		The unit vector of T'(t) is called the unit normal vector, and it is denoted by n(t).
	www.math.csusb.edu /faculty/fejzic/images/25111.html (806 words)

	Lecture(Week 3) (Site not responding. Last check: 2007-10-08)
		A vector function (or vector-valued function) is a function r(t) of a real variable t whose values are vectors (usually in 3 dimensional space).
		A good way to think of a vector function is as a function describing the motion of a particle in space, with t being a time parameter and the value r(t) denoting the position vector of the particle at time t.
		The derivative r'(t) is also a vector function, which can be computed by differentiating each component of r(t), We have seen that the direction of r'(t) is a tangent vector to the curve at time t.
	www.math.uiuc.edu /~dikim/m242/lec3.html (918 words)

	Tangent Vector and Tangent Line (Site not responding. Last check: 2007-10-08)
		As point P moves toward X, the vector from X to P approaches the tangent vector at X.
		Computing the tangent vector at a point is very simple.
		In general, the length of the tangent vector f'(u) is not one, and normalization is required.
	www.cs.mtu.edu /~shene/COURSES/cs3621/NOTES/curves/tangent.html (202 words)

	II . HYPOTHETICAL APPLICATION 1 (Site not responding. Last check: 2007-10-08)
		However, the method of determining these tangent vector is a major undertaking requiring matrix computations, inverse matrix computations, etc., and it is impossible for an ordinary curved surface generating apparatus on the personal computer level to determine the tangent vectors.
		In accordance with the present invention, the arrangement is such that a tangent vector at each point of a point sequence is obtained by using three or five consecutive points.
		With the method of obtaining tangent vectors in accordance with Eq.(2) using five points, a curve can be minutely altered by changing the manner in which each weighting coefficient is decided, thereby making it possible to generate the desired curve.
	www.jpo.go.jp /saikine/tws/app11.htm (2132 words)

	Tangent vector (Site not responding. Last check: 2007-10-08)
		Once tangent spaces have been introduced, one can define vector fields,which are abstractions of the velocity field of particles moving on a manifold.
		Such a vector field serves to define a generalized ordinary differential equation on amanifold: a solution to such a differential equation is a differentiable curve on themanifold whose derivative at any point is equal to the tangent vector attached to that point by the vector field.
		This is an equivalencerelation, and the equivalence classes are known as thetangent vectors of M at p.
	www.therfcc.org /tangent-vector-35871.html (1078 words)

	Booyah.com - 3D Game and Shader Development (Site not responding. Last check: 2007-10-08)
		Tangent space, or vertex space, is a coordinate system based on each vertex in which you create 3 basis vectors (creating an orthonormal basis).
		One vector is perpendicular to the surface at that vertex (the normal) and the other two vectors create the tangent plane at that point.
		These two vectors, tangent to the surface, are called the binormal and tangent vectors.
	www.booyah.com /article06-dx9.html (1745 words)

	What is a tangent vector and tangent space. - GameDev.Net Discussion Forums
		The tangent vectors are used to rotate the light the opposite of the texture.
		Tangent and bitangent are attributes of a point on a 3D SURFACE.
		These vectors sometimes are the directions of maximum and minimum curvature of a quadric approximation of the surface at that point, but don't need to be.
	www.gamedev.net /community/forums/ViewReply.asp?id=1878832 (2388 words)

	tangent.vector
		After all, if we had an ambient 3d space as before, we could ignore the difference between a vector tangent to the pumpkin, and a vector actually drawn on the pumpkin, in the limit where the arrow became very small.
		Think of a tangent vector at a point of spacetime, if you like, as a wee arrow whose tail is pinned to that point.
		Thus we may unabashedly imagine a tangent vector to a pumpkin as an vector tangent to the pumpkin, but infinitesimal, so that it doesn't cruise off into the 3d space which is, after, quite nonexistent to the Pumpkin People, the 2-dimensional beings who inhabit the surface of the pumpkin.
	math.ucr.edu /home/baez/gr/tangent.vector.html (694 words)

	11
		Since the velocity vector at time t points in the direction of motion, it must be tangent the the curve C traced by the vector valued function at time t.
		Note that the unit tangent vector points in the direction of motion at time t, and N(t) is orthogonal to T(t) and points in the direction that the object is turning.
		As an example, the acceleration vector (normal) of a projectile always points down, and would only be orthogonal to the tangent vector when it is at its max imum height.
	www.ac.cc.md.us /~donr/CalcIII/unit2/lesson4/u2l4.html (677 words)

	Visualizing a Tangent Vector Field (Site not responding. Last check: 2007-10-08)
		A common approach is to draw integral curves through the vector field, either as curvilinear segments or a s a texture applied to the surface.
		An alternative approach is to reveal the vector field indirectly, using it only to govern the reflective properties of the surface.
		A new illumination model uses the vector field to produce anisotropic reflections from the surface.
	www.icase.edu /docs/hilites/banks/tanVector.html (230 words)

	covariant vs contravariant - Physics Help and Math Help - Physics Forums
		A covariant vector is specifically a vector which transforms with the basis vectors, a contravariant vector on the other hand is a vector that transforms against the basis vectors.
		There are displacement vectors, used mostly in flat spaces such as the flat spacetime of SR and then there are tangent vectors which one uses in curved spaces such as the curved spaces one often finds in GR.
		In this sense a a covariant vector is a vector in the abstract mathematical sense of the term.
	www.physicsforums.com /showthread.php?t=58257&page=2 (2663 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		It may seem more reasonable to call a tangent to a curve as a tangent vector, but that idea is equivalent to this one, and less intrinsic.
		Even thinking about a tangent vector as a tangent to a curve in Euclidean space, one thing you can do is find a direction al derivative in that direction.
		However, now that we have a definition of tangent vectors to work with, we should define what we mean by tangents to curves.
	www.lehigh.edu /dlj0/public/www-data/courses/423f02-lect4.lyx (1504 words)

	covariant vs contravariant - Page 2 - Physics Help and Math Help - Physics Forums
		then a non zero cotangent vector, being by definition a non zero linear function on this space with real values, is determined up to a constant multiple by the subspace of tangent vectors which are mapped to zero, hence by a line through the origin.
		A tangent vector at p is represented by a curve passing through p.
		in differential geometry, "contravariant vectors" are the tangent vectors that transform in the SAME direction as the mapping on points, while "covariant vectors" or "covectors" are the ones that transform in the opposite direction.
	www.physicsforums.com /showthread.php?t=58257&page=2 (3806 words)

	Tangent Vectors (Site not responding. Last check: 2007-10-08)
		The tangent is defined as F'(x) of a parameterized...
		Tangent, Normal and Binormal Vectors for Physics of Walking/Marching and Tossing...
		Tangent Vectors and Arc Length of Curves in 3-D...
	www.scienceoxygen.com /math/704.html (100 words)

	untitled
		Draw the vector v in the plane, and a filament surrounding him with a given twist and in a certain interval.
		Draw the vector v in the plane, and a filament surrounding him with a width which decrements as a negative exponential, with a given twist and in a certain interval.
		in one time, from the equation of a vector, the twist function and an interval with the total number of points to be discretisized.
	cso.ulb.ac.be /cso/povweb/reference.html (4159 words)

	8.5.1 Tangent Space (Site not responding. Last check: 2007-10-08)
		, the tangent vector, is parallel to the direction of increasing s or t on a parametric surface.
		vectors are defined in eye space, then it converts from eye space to tangent space.
		Use the transformed x and y components of the light vector to shift the texture coordinates at the vertex.
	opengl.org /resources/tutorials/advanced/advanced98/notes/node108.html (850 words)

	Physics Help and Math Help - Physics Forums - tagent vector and vector field difference
		In the physics literature, a vector field is sometimes defined as a function that takes each point p in a subset U of M to a vector in the tangent space at p.
		In a certain sense, while a tangent vector IS a derivative (the gradient of a function), a vector field is a differential equation.
		A tangent vector is always a member of the tangent space of the manifold at a particular point.
	www.physicsforums.com /printthread.php?t=47610 (1141 words)

	Kinematics with Vector Calculus
		The velocity of a curve is the derivative of the vector function defining the curve:
		The direction of the unit tangent vector found in the previous equation depends on the direction in which t is increasing.
		The unit tangent vector found will always point in the direction of increasing t.
	omega.albany.edu:8008 /calc3/kinematics-dir/lecture.html (297 words)

	Math B6C
		Thus a tangent vector to the curve is given by
		is perpendicular to the tangent vector (since their dot products is zero), it must be a normal vector to the curve at the given point.
		Let vector W be in the direction of the line, which looking at the parametrization we see can be chosen to be the vector
	www2.bc.cc.ca.us /resperic/6C/calciii/Chapter10quizsols/ch10sols.htm (679 words)

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