I3. Partial Differential Equations and Integration, Pages 34 – 36.

The Laplacian is defined as:  Df  = 2f/x2 + 2f/y2 + 2f/z2  The Laplacian may be expanded to more or less than three variables.  The heat equation is written:   T/t = kDT.  This equation describes the temperature of a 3-dimensional point as a function of time and the temperature near the point.  The Laplace equation is  Df  = 0.   The Laplace equation says that that f is equal to the average of the points surrounding the point in question.  The one dimensional wave equation is written: (1/v2)2h/t2 = 2h/x2.  This equation related the displacement of a vibrating string as a function of time. 

The three dimensional wave equation may be written: (1/v2)2h/t2 = Dh.  This equation can be rewritten with the operation p2 where p2h = Dh – (1/v2)2h/t2.  This operation is called the d’Alembertian after d’Alembert who was the first to formulate the wave equation.  With this notation the wave equation becomes: p2h = 0.

Reimann integration examples are provided.  Here a function, f, is approximated or bracketed  by a mesh in a set of points S of step functions.  The mesh is then made finer and finer as the approximation tends to a limit.   If f is the function and S is the set then the total amount of f in S is the integral of f over S written as: òS f(x) dx.  Here x is an element of S.

The fundamental theorem of calculus states that integration and differentiation are inverse functions (up to an additive constant) for functions with reasonable continuity properties.

The Heaviside step function, H(x), does not have the required continuity properties.  H(x) = 0 when x < 0 and H(x) = 1 when x s 0.  The problem occurs at the point zero where the derivative approaches infinity from below.


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