By Randima Fernando
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Additional resources for The Cg Tutorial: The Definitive Guide to Programmable Real-Time Graphics
These five numbers show up all over the place in mathematics and Euler combined them all in one equation. This is pure genius. What it really says in plain language is: “if you rotate a vector counterclockwise in the plane by 180° it will point in the opposite direction. ” That seems pretty obvious. Euler’s genius was to concoct an equation from this obvious fact that involves very different mathematical constants. They show up all over the place in different mathematical fields but here they are all united by Euler’s formula.
It involves both the entire shape and the entire fluid. Euler on the other hand came up with a local property which guarantees that the area is preserved for small pieces of the fluid. Piecing together these local conditions results in a global condition in practice. More details about this connection will be given later. 5 The area of two shapes is preserved exactly over time in an ideal continuous fluid. * I tried my best in Adobe Illustrator to make the two shapes have an equal area. But hopefully you get the idea.
Then he climbed the ladder and poured water into the pipe. This raises the overall pressure in the barrel. And since the barrel was poorly made, it burst: crevé. The burst factor depends both on Observations, Equations, and Numbers ◾ 19 the strength of the barrel and on the height of the pipe. More concretely, the necessary height of the pipe to burst the barrel is proportional to the strength of the barrel. No wonder the unit of pressure is named after Pascal. It is equal to the amount of force exerted per area.