What in mathematics is Poincare Conjucture? Has it any practical applications?
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http://mathworld.wolfram.com/PoincareConjecture.html
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Most introductory books on differential geometry discuss this. Looks at 'From Calculus to Cohomology' and 'Introduction to Smooth Manifolds' as references.
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you can see the text book ENGINEERING MATHEMATICS by SINGARAVELU. All the best....
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Counterexample.  Let u(x), v(x) be members of {sin x, cos x}.  Both of those are solutions, but you do not get zero by adding one to the other, or by doubling one of them.
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To solve this it's a pretty straightforward application of Laplace transforms. You can do this question using convolutions as follows:\n\nFirst take the laplace transfom of the ODE:\n\nlet L{y(t)}(s) = Y(s) be the laplace transform of y(t).  Then we have:\ns^2(Y(s))  - 2s + 3Y(s) = 1/s^2 - (e^-s)/s\n\nrearranging we have Y(s) = (1/s^2 - e^-s/s + 2s) (1/(s^2+3)) = G(s) H(s) where G(s) = (1/s^2 - (e^-s)/s)  and H(s) =  (1/(s^2+3)) \n\nnow using the inverse laplace formula for the convolution of two functions:\n\ny(t) = L(^-1){Y(s)} = integral from 0 to t of (h(t-x)g(x)dx) + 2cos(sqrt(3)t)\n\n where h(t) = 1/sqrt(3) sin(sqrt(3)t) and g(t) = (t - u1(t))\n\nand the integral can be evaluated. \n\nSorry for the readability but I don't think yahoo answers can render math text properly. Also, as a note, this question would be far easier to solve using a partial fraction method rather than unecesarily using convolutions.
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I know of one application in an abstruse area of physics: The sinh( ) of a scalar field occurs in gravitational theories where, for whatever reason, one emplys a metric disformally related to the 'physical' metric tensor.\n\nAlso, the shape of a freely hanging level chain is cosh( ) of displacement along the line joining ends
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Yes.\nThe closure relation on a function space produces a dirac delta function and the dirac delta function can be expanded in any basis set, in particular in a complete orthonormal family. This can be used to solve PDEs by using eigenfunction expansions.\n\nFor example, the dirac delta function has a representation in the sine basis functions, a representation in the cosine basis functions and also a representation in the fourier series basis functions.
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DO YOU REFER TO CLASSICAL MECHANICS?\n\nIn classical Mechanics it may be useful to study changes from some generalized coordinates q_i and associated gen. momenta p_i to some other general coordinates Q_i and associated gen. momenta P_i. Most textbooks define a Transformation \n\n(q,p)->(Q(q,p,t),P(q,p,t)) (1) (where q is the "vector" of all q_i, etc.) as\n\nCanonical in the extended sense if and only if:\n\nFor any function H(p,q,t) fulfilling\n\ndq_i/dt=dH/dp_i and dp_i/dt= -dH/dq_i (partial derivatives on the right side)\n\nthere exists a Hamilton-function K(Q,P,t) full filling:\n\ndQ_i/dt=dK/dP_i and dP_i/dt= -dK/dQ_i (partial derivatives on the right side)\n\nIn other words: For any Hamiltonfct. of the p,q - Space there must exist a Hamiltonfct. of the P,Q- Space, (because Hamiltonfuncions of Systems  fulfill the relations above). Then and only then the transformation (1) is called canonical in ext. sense, because the form of Hamilton's equations is conserved.\n\nABOUT THE OTHER TRANSF. I DON'T KNOW ANYTHING. SEE THE LINK BELOW.
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Imaginary numbers, actually complex numbers, are enormously useful in many areas of engineering.  One example is in trigonometry where difficult trigonometric identities become almost trivial to derive.  Another is in analyzing circuits.  Using imaginary numbers allows Ohm's law to be applied to passive (resistors, capacitors and inductors) AC circuits.  Euler's identity allows cubic equations to be solved analytically.  Applying the concept of graphical representation of 2-D vectors provides very powerful tools for the solution of 2-D kinematics problems and difficult geometry problems.\n\nI am a research engineer and I use complex numbers all the time in my work.  It is by far one of the most useful tools that I have to solve difficult problems.
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Example:\nIf z=f(u,v),\nu=g(x,y) and\nv=h(x,y), then\n(all derivatives are supposed to be partial derivatives)\ndz/dx=dz/du*du/dx+dz/dv*dv/dx\ndz/dy=dz/du*du/dy+dz/dv*dv/dy\n\nAlternatively, you can set of the matrices\n[[dz/du dz/dv]]\nand\n[[ du/dx du/dy]\n[dv/dx dv/dy]]\nand get that\n[[dz/dx dz/dy]]\nis the product of the two matrices.
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A conformal mapping, also called a conformal map, conformal transformation, angle-preserving transformation, or biholomorphic map, is a transformation  that preserves local angles. An analytic function is conformal at any point where it has a nonzero derivative. Conversely, any conformal mapping of a complex variable which has continuous partial derivatives is analytic. Conformal mapping is extremely important in complex analysis, as well as in many areas of physics and engineering. \n\nA mapping that preserves the magnitude of angles, but not their orientation is called an isogonal mapping (Churchill and Brown 1990, p. 241). \n\n \n \nSeveral conformal transformations of regular grids are illustrated in the first figure above, and are implemented as ComplexMap in the Mathematica add-on package Graphics`ComplexMap` (which can be loaded with the command <<Graphics`) . In the second figure above, contours of constant  are shown together with their corresponding contours after the transformation. Moon and Spencer (1988) and Krantz (1999, pp. 183-194) give tables of conformal mappings. \n\nA method due to Szegö gives an iterative approximation to the conformal mapping of a square to a disk, and an exact mapping can be done using elliptic functions (Oberhettinger and Magnus 1948; Trott 2004, pp. 71-77). \n\nLet  and  be the tangents to the curves  and  at  and  in the complex plane, \n\n (1) \n\n (2) \n\nThen as  and , \n\n (3) \n\n (4) \n\nA function  is conformal iff there are complex numbers  and  such that \n\n (5) \n\nfor  (Krantz 1999, p. 80). Furthermore, if  is an analytic function such that \n\n (6) \n\nthen  is a polynomial in  (Greene and Krantz 1997; Krantz 1999, p. 80). \n\nConformal transformations can prove extremely useful in solving physical problems. By letting , the real and imaginary parts of  must satisfy the Cauchy-Riemann equations and Laplace's equation, so they automatically provide a scalar potential and a so-called stream function . If a physical problem can be found for which the solution is valid, we obtain a solution--which may have been very difficult to obtain directly--by working backwards. \n\nFor example, let \n\n (7) \n\nthe real and imaginary parts then give \n\n   (8) \n   (9) \n\n \nFor , \n\n   (10) \n   (11) \n\nwhich is a double system of lemniscates (Lamb 1945, p. 69). \n\n \nFor , \n\n   (12) \n   (13) \n\nThis solution consists of two systems of circles, and  is the potential function for two parallel opposite charged line charges (Feynman et al. 1989, §7-5; Lamb 1945, p. 69). \n\n \nFor , \n\n   (14) \n   (15) \n\n gives the field near the edge of a thin plate (Feynman et al. 1989, §7-5). \n\n \nFor , \n\n   (16) \n   (17) \n\ngiving two straight lines (Lamb 1945, p. 68). \n\n \nFor , \n\n (18) \n\n gives the field near the outside of a rectangular corner (Feynman et al. 1989, §7-5). \n\n \nFor , \n\n   (19) \n   (20) \n   (21) \n\nThese are two perpendicular hyperbolas, and  is the potential function near the middle of two point charges or the field on the opening side of a charged right angle conductor  (Feynman 1989, §7-3).
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I'm a fan of MathWorld: http://mathworld.wolfram.com/\nIt's affiliated with the software package Mathematica.  Also at the same site is ScienceWorld: http://scienceworld.wolfram.com/
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Try here (the content is very techinical).  http://scienceworld.wolfram.com/physics/ContinuityEquation.html\nhttp://www.efm.leeds.ac.uk/CIVE/CIVE1400/Section3/continuity.htm
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Good Question, Mandelbrot's book and Barnsley's book will provide you with many examples.
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Why is this in Mathematics?
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Differential geometry is the study of smooth manifolds and differential equations on such manifolds. As such, it mostly gives examples for these other areas of study. For example, the DeRham complex is a relatively easy to understand example of a chain complex. The homology of this chain complex turns out to be the same as for singular homology. The collection of continuous functions (or various levels of differentiable functions) gives an example of a Banach space, which is the fundamental concept in functional analysis. Also, a good part of the Hodge theory uses function analytic techniques. Finally, the differential equations that come up in differential geometry are often of practical use and need to be solved, at least approxiamtely, so numerical analysis comes into play.
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just investigate for example ODE y"=-y and then add to the left part increasingly small non-linearity +c*y^2 and find solution of the first order of c - you will see how different modes interact with each other.