1000 Solved Problems in Modern Physics

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