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If the value of the function and its derivatives are known at a single point, a, the function can often be represented by a power series. This, however, is usually an infinite series that must be truncated for practical application. The trick is to truncate it in such a way that it accurately approximates the given function, at least in some neighborhood of a. The following series, known as a Taylor series, gives a representation of an analytic1 function, f(x). If a = 0, the series is known as a Maclaurin series. f (x) =

k =0

f ( k ) (a) ( x a) k k!

n dS 0, where n is the outward drawn normal to any closed surface S. [Hint: Let A c, Express the divergence theorem in this special case. Use the

f ( k ) (a) represents the kth derivative of f evaluated at a. If k = 0, it represents f(a).

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If we truncate this infinite series by omitting all terms of degree greater than n, we obtain the nth degree Taylor polynomial of f about a. We shall represent this polynomial as pn ( x ). If a = 0, the polynomial is called a Maclaurin polynomial. Sum command or the symbol from the Basic Math Input palette. Here are three different ways the polynomial can be generated:

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10.67. If n is the unit outward drawn normal to any closed surface S bounding the region V, prove that div n dV S

f[x_] = Exp[x]; (a) Sum[(D[f[x], {x, k}]/.x 0)/k! * xk, {k, 0, 5}]

Abs*

Fi*

On the server side, DNS is implemented under UNIX by the named daemon. Usually only a pair of DNS servers, primary and secondary, are configured for a given organization. It is not necessary to run named on each machine in a network. The named daemon obtains configuration information from the set of configuration files listed in Table 2.5. The full scope of the format and syntax of these tables is often complex and beyond the purpose of this text. The reader is directed to documentation regarding DNS setup and configuration for the given UNIX platform.

STOKES THEOREM 10.68. Verify Stokes theorem for A 2yi 3xj z2 k, where S is the upper half surface of the sphere x2 y2 z2 9 and C is its boundary. Ans. common value 9 10.69. Verify Stokes theorem for A y z i xzj y2 k, where S is the surface of the region in the rst octant bounded by 2x z 6 and y 2 which is not included in the (a) xy plane, (b) plane y 2, (c) plane 2x z 6 and C is the corresponding boundary. Ans. The common value is (a) 6; b 9; c 18 10.70. Evaluate z 2 r A n dS, where A x z i x3 yz j 3xy2 k and S is the surface of the cone Ans: 12 10.71. If V is a region bounded by a closed surface S and B r A, prove that

There are many specialized functions and procedures that are not loaded when Mathematica is initially invoked. Rather, they must be loaded separately from files in the Mathematica directory on the hard drive. These files are of the form filename.m.

EXAMPLE 26 A map of the world can be obtained from the command WorldPlot which is located in the package WorldPlot`. To load this command, simply type (note the ` at the end)

or Needs["WorldPlot`"]

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