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  • mbenkerumass 8:49 am on December 22, 2019 Permalink | Reply
    Tags: Calculus,   

    Review of Fourier Series 

    The French mathematician Fourier discovered that any periodic waveform can be expressed as a series of harmonically related sinusoids.

    Any periodic waveform can be expressed as the following:

    series

    The first term a0/2 is the constant DC or average component of f(t). The terms with coefficients a1 and b1 represent the fundamental frequency components of f(t). Coefficients a2 and b2 are the second harmonic components at frequency 2w. The frequency doubling on the second order harmonic is computed as a result of the multiplication of sinusoids.

    harmonics

    In order to determine the coefficients of a harmonic series ai and bi, multiply both sides of the above formula by 2sin(2wt). In this case for simplicity, let w = 1.

    calcaibi

    Next, integrate from zero to 2*pi.

    sereis2

    The following relations are then found:

    seriesrelations

    The Fourier series are often expressed in exponential form:

    series3

    The MATLAB function int(f,t,a,b) is often a useful tool, where f is the function, t is the symbolic variable, and a and b are the bounds of integration.

    matlabint

     

     

     
  • mbenkerumass 9:00 am on December 21, 2019 Permalink | Reply
    Tags: Calculus   

    Del Operator, Curl, Divergence, Gradient, Laplacian 

    • Del Operator:del1delop

     

    • Curl:curl1curl2

     

     


    • Divergence: divergencediv2

     

     

     


    • Gradient: gradientofscalargradientofscalar2

     

     

     


    • Laplacian: laplacelaplace1laplace3
     
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