12
1 Mathematical Physics
which are to be solved as ordinary algebraic equations to determine the best
values of m and C.
(b) Parabola: y = a + bx + cx
2
Residue: S =
n
i=1 (y i − a − bx i − cx
2
i )
2
Minimize the residue:
∂s
∂a
= 0;
∂s
∂b
= 0;
∂s
∂c
= 0
The normal equations are:
y i − na − b
x i − c
x
2
i = 0
x i y i − a
x i − b
x
2
i − c
x
3
i = 0
x
2
i y i − a
x
2
i − b
x
3
i − c
x
4
i = 0
which are to be solved as ordinary algebraic equations to determine the best
value of a, b and c.
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 =
x2
x 1
y(x)dx. The greater the number of intervals (i.e. the smaller Δx is),
the closer will be the sum of the areas under consideration.
Trapezoidal rule
area =
1
2
y 0 + y 1 + y 2 + · · · y n−1 +
1
2
y n
Δx
(1.76)
Simpson’s rule
area =
Δx
3
(y 0 + 4y 1 + 2y 2 + 4y 3 + 2y 4 + · · · y n ), n being even.
(1.77)
Fig. 1.1 Integration by
Simpson’s rule and
Trapezoidal rule