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Thursday, July 16, 2009

ONE EQUALS ZERO

Consider the following integral:
INTEGRAL (1/x) dx
Perform integration by parts:
let u = 1/x , dv = dx du = -1/x2 dx , v = x
Then obtain:
INTEGRAL (1/x) dx = (1/x)*x - INTEGRAL x (-1/x2) dx
= 1 + INTEGRAL (1/x) dx
which implies that 0 = 1.

USES OF LAPLACE TRANSFORMS



 PIERRE SIMON LAPLACE


  • The Laplace transform is a simple way of converting functions in one domain to functions of another domain.

Here's an example
:
Suppose we have a function of time, such as cos(w*t). With the Laplace transform, we can convert this to a function of frequency, which is

cos(w*t) ----L{}-----> w / (s^2 + w^2)
    • This is useful for a very simple reason: it makes solving differential equations much easier.
      • The development of the logarithm was considered the most important development in studying astronomy. In much the same way, the Laplace transform makes it much easier to solve differential equations. 
        • Since the Laplace transform of a derivative becomes a multiple of the domain variable, the Laplace transform turns a complicated n-th order differential equation to a corresponding nth degree polynomial. Since polynomials are much easier to solve, we would rather deal with them. This occurs all the time.
          • In brief, the Laplace transform is really just a shortcut for complex calculations. It may seem troublesome, but it bypasses some of the most difficult mathematics.  
          • Laplace transform is a technique mainly utilized in engineering purposes for system modeling in which a large differential equation must be solved.
          • The Laplace transform can also be used to solve differential equations and is used extensively in electrical engineering.
          • Laplace Transform is used in electrical circuits for the analysis of linear time-invariant systems

            THE PROOF 1 + 1 = 0

            1+1
            = 1+ sqrt (1)
            = 1+ sqrt [(-1) (-1)]
            = 1+ [sqrt (-1) * sqrt (-1)]
            = 1+ [i * i]
            = 1+ (-1 )
            = 1 -1
            = 0

            MATH INVOLVED IN ELECTRICAL AND ELECTRONIC ENGINEERING



            • There's really a lot of math involved in electrical and electronicengineering. How much you do depends on what area of EE (shorthand for electrical and electronic engineer) you do.





            • For example, there's a lot more abstract math in communication theory and signal processing, and many more very direct calculation differential equations in circuit theory and systems design.





            • Circuit theory at its simplest form is really differential equations, which is basically solving equations involving derivatives, so you need some CALCULUS and ALGEBRA and TRIGONOMETRY are fundamental to understanding it. Every basic circuit element (resistor, capacitor, inductor) has arelated current-voltage relation determined by its impedance. This iswhere COMPLEX NUMBERS come in.




            • If we move on to the theory of "how" electromagnetism works, we haveMaxwell's equations. These pretty much form the basis for EE. They are written in both integral and derivative forms and involve vectors. So, suddenly, we also have VECTOR CALCULUS.




            • If we move to Communication Theory/Information Theory, a mathematician named Claude Shannon developed a mathematical theory to explain various quantities related to how to communicate between devices.Communication Theory is used everywhere, from RADAR, to telephones, to devices within computers. The underlying theory requires at least CALCULUS , some LINEAR ALGEBRA , some MEASURE THEORY, etc.








            • Even wavelets, which have revolutionized signal processing, were discovered by mathematicians early in the 20th century, but not used by engineers until 20 years ago.






            In general, it is not possible to do EE without math.


            Each abstract mathematical theorem somehow finds its use in EE.
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