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Q: What is the derivation of Navier-Stokes equation in cylindrical coordinates?

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it is easy you can see any textbook........

Integration results in an equation which gives the area under the original equation between the bounds. Derivation results in an equation which gives the slope of the original line at any point.

See wikipedia article on polytropic processes.

The graph (on Cartesian coordinates) of a quadratic equation is a parabola.

in helmholtz vector equation why F=-∆ф+∆xA?

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it is easy you can see any textbook........

http://en.wikipedia.org/wiki/Navier-Stokes_equations Please go to this page.

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Integration results in an equation which gives the area under the original equation between the bounds. Derivation results in an equation which gives the slope of the original line at any point.

See wikipedia article on polytropic processes.

There are basically SEVERAL continuity equations, one for each conserved quantity. The equations themselves are simply statements that matter (in the example of conservation of mass) will not appear out of nothing, or suddenly teleport to a far-away place.

polar

The graph (on Cartesian coordinates) of a quadratic equation is a parabola.

in helmholtz vector equation why F=-∆ф+∆xA?

Assume the equation is y = kx + c Put in the x and y values of your known coordinates and sove the simultaneous equations.