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A detailed tutorial on mass-energy equivalence. Step by step tutorial including several examples of mass-energy equivalence for reference.

Mass-energy equivalence is the concept that the mass of a body is the measure of its energy content. This is often expressed by a formula written by Einstein, who is also the one that proposed the idea of mass-energy equivalence. This formula is , where E is energy, m is the mass, and c is the speed of light in a vacuum.

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A detailed tutorial on Lorentz transformations. Step by step tutorial including several examples of Lorentz transformations for reference.

A Lorentz transformation is a way of describing how two different measurements of space and time can be converted into one frame of reference. This is because it was discovered that people who are moving at different velocities will report different times of certain events, or even a different order of events. The speed or velocity at which they are moving will throw things off. So by using a Lorentz transformation, you can get two different accounts to match up. Typically, a Lorentz transformation is a linear transformation.

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A detailed tutorial on how to solve for Mach numbers. Step by step tutorial including several examples of how to solve for Mach numbers for reference.

A Mach number is the speed of an object moving through the air, or any fluid substance, divided by the speed of sound as it is in that substance. It is often used to represent an object such as an aircraft or a missile’s speed, when it is travelling at the speed of sound or multiples of the speed of sound. This can be portrayed mathematically in the equation M = vs / u, where M is the Mach number, vs is the speed of the source (the object relative to the medium), and u is the speed of sound in the medium.

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A detailed tutorial on superelevation. Step by step tutorial including a visual example of superelevation of a road for reference.

The superelevation of a road or of a railway is the difference in elevation between the two edges. A non-zero superelevation – meaning that the edges of the road or railway are at different heights – allows for a bank turn, letting vehicles traverse the turns at higher speeds than would otherwise be possible. Superelevation is sometimes referred to as the cant of a road or railway. An important calculation in superelevation is the maximum speed of a vehicle on a curved road. It is determined by the formula .

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A detailed tutorial of uniform convergence. Step by step tutorial including several example problems of uniform convergence for reference.

Uniform convergence is a very strong type of convergence, even stronger than pointwise convergence. A sequence {fn} of functions converges uniformly to a limiting function f if the speed of convergence of fn(x) to f(x) does not depend on x. This concept is important because several properties of these functions are transferred to the limit f if the convergence is uniform.