What is integral-invariant (or Poincare integral) in analitical dynamics?
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1. Zero is a natural number.\n2. Zero is not the successor of any natural number.\n3. Every natural number has a successor, which is a natural number.\nIf the successor of natural number a is equal to the successor of natural number b, then a and b are equal. 4.\nSuppose:\n(i) zero has a property P;\n(ii) if every natural number less than a has the property P then a also has the property P.\nThen every natural number has the property P. (This is the principle of complete induction.)\nFor any x and any y, x + y = y + x\nis not directly verifiable. We can choose an arbitrary pair of natural numbers a and b, and we can verify that a+b = b+a; but there is an infinite number of admissible choices of pairs, so the verification is always incomplete. In other words, the verification of the commutative law is an analytical method by means of which we can verify every particular instance of a general theorem, while the proof of the theorem itself is synthetic reasoning which really extends our knowledge, Poincaré believed.
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Differential equations.\nLinear algebra.\nNumerical analysis.
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I am not sure of the final answer you are looking for.  If you are looking for the number of molecules transferred after a certain time t, the answer is simply n=-DAt(dc/dz).  This of course assumes that dc/dz remains relatively constant.  If not you will need a dc/dz as a function of t.
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It's called the integration by parts formula.\nInt u dv=uv- int v du\n\nBoth u and v need to be absolutely continuous for this to work. A continous derivative for both is plenty.
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yes, it's a sine wave, and velocity is integral of that, ie a cosine\n(this is considering only fist order effects)
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Try this:\n\nStructure and Interpretation of Classical Mechanics\nhttp://mitpress.mit.edu/SICM/\nbook-Z-H-20.html#%_chap_2
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When V(t)=0 the particle stops.\nIt can have a(t)=0, but still moving (constant speed)
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The result is still under debate because all transformations have still not been verified. The Sobolev space can be bi variant if the gauge is not satisfied.\n\nI would suggest looking up the bulletins of the London Mathematical Society, their late 2004 editions came out with one such treatment.
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Stationary in reference to what fixed point?
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You are given P(t=0), Q(t=0); P'(t=0), Q'(t=0); and P''(t=0), Q''(t=0). Given that P'' and Q'' remain the same (that is, the motion of P and Q do not change), you need to show that there is a time T such that P(t=T) = Q(t=T).\n\nIt's late and I'm drunk, so you can do the actual working for yourself :D
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Here are a couple of web sites that approach this in different ways. \n\nhttp://www.circletrack.com/techarticles/84199/\n\nhttp://www.diracdelta.co.uk/science/source/e/n/engine%20excitation%20mechanisms/source.html\n\nHope these help.
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i made a little comment for you on your integral answer:)
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Use the ideal gas law \n\nPV/T = constant for the final and initial states.
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You might try reading Whitehead and Russell's Principia Mathematica.  I can't promise it will help, as I haven't read it, but I've been told that this issue is addressed.
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Anytime you have a problem that is continuous, as opposed to discrete, you will need to use either integration or differentiation to characterize the solution.  That means to say that every problem that involves time in one way or another will necessitate the use of calculus to find an answer at some instant in time.  If the problem is such that you can approximate the answer, then you can discretize the parameters as small slices of time and come up with an approximate answer - without using calculus.
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the integration for any intgral from (a) to (-a) is always equal zero.\nso the intgral for this is zero\nbut if there is no interval the integration is very complicated
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I have never used them, but I have heard they have uses in certain kinds of electrical engineering.  I think it involves phase shifts of period functions like sine waves.  Also, there are occasional problems where you start in the real numbers and end in the real numbers, but by doing the problem in the complex domain, certain difficult problems become solvable.  I think Cauchy integrals are used for this purpose.