4-PrincipCalculus Page 139 Friday, March 12, 2004 12:39 PM
Principles of Calculus 139∂^2 f –(x^2 + σxy + y^2 )^2
+ ( 2 y + σx)^2 e –(x(^2) + σxy + y )
---------= – 2 e
∂y^2
∂^2 f^2 – (x^2 + σxy + y^2 )
-------------= ( 2 x + σy)( 2 y + σx)e –(x
(^2) + σxy + y )
- σe
∂x∂y
In bond analysis, we can also compute partial derivatives in the case
where each interest rate is not the same for each time period in the bond
valuation formula. In that case, derivatives can be computed for each
time period’s interest rate. When the percentage price sensitivity of a
bond to a change in the interest rate for a particular time period is com-
puted, the resulting measure is called rate duration or partial duration.^12
The definition of the integral can be obtained in the same way as in
the one variable case. The integral is defined as the limit of sums of
multidimensional rectangles. Multidimensional integrals represent the
ordinary concept of volume in three dimensions and n-dimensional
hypervolume in more that three dimensions. A more general definition
of integral that includes both the Riemann and the Riemann-Stieltjes as
special cases, will be considered in the chapter on probability.SUMMARY
We can now summarize our discussion of calculus as follows:■ The infinitesimally small and infinitely large. Through the concept of
the limit, calculus has rendered precise the notion of infinitesimally
small and infinitely large.
■ Rules for computing limits. A sequence or a function tends to a finite
limit if there is a number to which the sequence or the function can get
arbitrarily close; a sequence or a function tends to infinity if it can
exceed any given quantity. Starting from these simple concepts, rules
for computing limits can be established and limits computed.
■ Derivatives. A derivative of a function is the limit of its incremental
ratio when the interval tends to zero. Derivatives represent the rate of
change of quantities.
■ Integrals. Integrals represent the area below a curve; they are the limit
of sums of rectangles that approximate the area below the curve. More(^12) There is a technical difference between rate duration and partial duration but the
difference is not important here.