Understanding Fourier Series

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Comparing Functions and vectors.

\vec{v} function f(x)

Finite dimensional Infinite dimensional

A vector can be written in the following different ways,
\vec{V} = V_x \hat{x} + V_y \hat{y} + V_z \hat{z}
\hskip .5cm = (V \cdot \hat{x}) \hat{x} + (V \cdot \hat{y}) \hat{y} + (V \cdot \hat{z}) \hat{z}
If the decomposition is along an orthogonal frame along the vectors \vec{a},\, \vec{b} and \vec{c} then the expression would be,
\vec{V} = (\vec{V} \cdot \hat{a}) \hat{a} + (\vec{V} \cdot \hat{b}) \hat{b} + (\vec{V} \cdot \hat{c}) \hat{c}
\hskip .5cm = \frac{\vec{V} \cdot \vec{a}}{\vec{a}\cdot\vec{a}} \vec{a} + \frac{\vec{V} \cdot \vec{b}}{\vec{b}\cdot\vec{b}} \vec{b} + \frac{\vec{V} \cdot \vec{c}}{\vec{c}\cdot\vec{c}} \vec{c}

In general the dot product of two $n-$dimensional vectors \vec{V} = (V_1, V_2,...,V_n) and \vec{W} = (W_1,W_2,...,W_n), can be written as,
\vec{V} \cdot \vec{W} = \sum_{i=1}^n V_i W_i.

It is useful to think of a real function f(x) over an interval
[a,b] as a vector with infinite components. Here the argument serves
as an index and the function value as the vector component. Analogous to vector dot product, the dot product between two functions $f$ and $g$ defined over the same interval can be written as,
(f,g) = \int_a^b f(x) g(x) dx.

Using this definition of the dot product, one can show that the following functions
are orthogonal to each-other (mutual dot products are zero) on the interval
[0, 2\pi].
f_1(x) = 1, \sin{x}, \sin{2x}, \sin{3x}, ...,\cos{x}, \cos{2x}, \cos{3x},...

Thus in parallel with writing a vector in terms of it’s components, one can write any (finite, smooth and continuous on [0, 2\pi] (I am not trying to be mathematically precise, the aim is to give an intuitive feel)) function in terms of
the above basis functions in the same manner,
f(x) = \frac{(f(x),1)}{(1,1)} 1 + \frac{(f(x),\sin(x)}{(\sin(x), \sin(x))} \sin(x) + \frac{(f(x),\sin(2x)}{(\sin(2x), \sin(2x))} \sin(2x) + ...
\hskip.2cm + \frac{(f(x),\cos(x)}{(\cos(x), \cos(x))} \cos(x) + \frac{(f(x),\cos(2x)}{(\cos(2x), \cos(2x))} \cos(2x) +....
Notice the similarity of the expression of a function in terms of it’s components and a vector in terms of it’s components. Hence decomposition of a function in its Fourier components is quite akin to decomposition of a vector in its Cartesian components.

Encyclopedia of Ignorance

Since a long time I have been thinking of creating an encyclopedia of ignorance for the courses that I teach. When I correct papers, I see some standard mistakes students make.

Let me start randomly collecting them. May be some day I shall put them in order. I think I shall keep updating this page.

Physics:

    1. The \vec{E} and \vec{B} field in an electromagnetic wave are point quantities. Text book diagram often makes people think about them as quantities extended in space.
    2. Density is also a point quantity which is defined as the ratio of mass ina volume to the volume, in the limit volume going to zero. Though theoretical such definitions are very useful practically.
    3. It is possible to have the function value zero at a point while its gradient is nonzero.Which also implies that the magnetic field at a point may be zero but the force on a dipole at that point may be nonzero.
    4. Keeping a body magnetically levitated does not require work.

Mathematics:

  1. d^2y/dx^2 is not the same as (dy/dx)^2, many student wrote this in their answers in the last exam. I tell them this would be like saying a = v^2 that usually makes sense to them.
  2. Over and over again I have to remind the students to try to make sense of the equations they write. For example while applying Lagrange’s mean value theorem to x \ln x on [1, x] students wrote,¬† \frac{x \ln x}{x-1} = 1 + \ln x, replacing \ln c on the RHS by \ln x. The formula when simplified gives, \ln x = x-1 which should raise a serious doubt, provided they stop doing mathematics mechanically and start thinking about meaning of the formulas.
  3. Exponential processes in a variable x can also be of the form a^x, somehow many students think that only e^x would be an exponential process but any given number a can be written as a= e^b hence  a^x can be expressed as e^{bx}.