1000 Solved Problems in Modern Physics

(Romina) #1
12 1 Mathematical Physics

which are to be solved as ordinary algebraic equations to determine the best
values ofmandC.
(b) Parabola:y=a+bx+cx^2


Residue:S=

∑n
i= 1 (yi−a−bxi−cx

2
i)
2
Minimize the residue:∂∂as=0;∂∂sb=0;∂∂cs= 0
The normal equations are:

yi−na−b


xi−c


xi^2 = 0

xiyi−a


xi−b


x^2 i−c


xi^3 = 0

x^2 iyi−a


xi^2 −b


x^3 i−c


x^4 i= 0

which are to be solved as ordinary algebraic equations to determine the best
value ofa,bandc.

Numerical integration
Since the value of a definite integral is a measure of the area under a curve, it follows
that the accurate measurement of such an area will give the exact value of a definite
integral;I=

∫x 2
x 1 y(x)dx. The greater the number of intervals (i.e. the smallerΔxis),
the closer will be the sum of the areas under consideration.

Trapezoidal rule

area=

(

1

2

y 0 +y 1 +y 2 +···yn− 1 +

1

2

yn

)

Δx (1.76)

Simpson’s rule

area=

Δx
3

(y 0 + 4 y 1 + 2 y 2 + 4 y 3 + 2 y 4 +···yn),nbeingeven. (1.77)

Fig. 1.1Integration by
Simpson’s rule and
Trapezoidal rule
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