Mathematical Review

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Mathematical Review
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Partial Derivatives
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• The partial derivative w.r.to x assumes that we
  hold y constant

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Total variation of a function F(x,y)
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• The traditional definition of a ?(x) function is
  a function that is zero everywhere except at x0
  where it is infinite in such away that
• 
  
• 
  (1)

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• Now this is mathematical nonsense
• It is impossible to introduce a rigorous
  mathematical theory of integration where the
  value of the integrand at one point can affect
  the integral.
• It is possible to put the ? function on a
  rigorous mathematical basis by treating it as a
  distribution or generalized function i.e.

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Square integrable functions
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Square integrable

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Square integrable
exists

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Square integrable

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• However the best way to think of the delta
  function is that it is an integral waiting to
  happen.

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• For example it is straightforward to establish
  quite rigourously the Fourier integral theorem

(2)
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• And one may define

(3)
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• With the understanding that it will eventually be
  integrated over x

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Another representation
(4)
Where it is understood that the limit
will be taken after the integration
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(5)
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• Consider

(6)
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• Consider

Break integral in 3
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(7)
Cauchy Principal value
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Consequently
(8)
Feynman integral relation
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Another Example

• Now integrate by parts assuming that

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Consequently we can define a new generalized
function
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Excercise

• In a distribution(generalized function sense)

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? Function in higher dimensions
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Spherical Coordinates(r,?,F)
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Spherical Coordinates(r,?,F)
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Recall
(3)
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hence

• And one may define

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• vector space, consists of a set of vectors
• agt,bgt,cgt, s.t.

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• vector space, consists of a set of vectors
• agt,bgt,cgt, s.t.

For All
Complex numbers
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Where
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Consider the set of vector

• This set is linearly independent if

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The set is said to span the space
or form a basis

• And the set is linearly
  independent

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• The number of vectors in a basis gives us the
  dimension of the vector space
• Clearly because of the linear independence
  condition
• If both span
  the space
• Then

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• For the moment assume space to be finite
  dimensional,
• Typically, as we will, see the Vector
  Space(Hilbert Space) in quantum problems will be
  infinite dimensional