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      1.3.2 Big O and Little o Notation

      For many applications we need a definition of the asymptotic behaviour of quantities such as functions and series; in particular we wish to find bounds on mathematical expressions and applications in computer science. To this end, we introduce the Landau symbols O and o.

      Definition 1.2 (O-Notation).

An example is:

      Definition 1.3 (O-Notation).

      An example is:

      We note that complexity analysis applies to both continuous and discrete functions.

1.4 PARTIAL DERIVATIVES

      In general, we are interested in functions of two (or more) variables. We consider a function of the form:

      The variables x and y can take values in a given bounded or unbounded interval. First, we say that f (x, y) is continuous at (a, b) if the limit:

      exists and is equal to f (a, b). We now need definitions for the derivatives of f in the x and y directions.

      In general, we calculate the partial derivatives by keeping one variable fixed and differentiating with respect to the other variable; for example:

We now discuss the situation when we introduce a change of variables into some problem and then wish to calculate the new partial derivatives. To this end, we start with the variables (x, y), and we define new variables (u,

). We can think of these as ‘original’ and ‘transformed’ coordinate axes, respectively. Now define the function z(u,

) as follows:

      This can be seen as a function of a function. The result that we are interested in is the following: if z is a differentiable function of (u,

) and u,

are themselves continuous functions of x, y, with partial derivatives, then the following rule holds:

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