- #1
EmilyRuck
- 136
- 6
Homework Statement
Hello!
My problem is with a variational expression. I have a quantity, say L, which could be determined by the ratio:
[itex]K = \displaystyle \frac{\left[\int_a^b E(x)e_n(x)dx\right]^2}{\left[\int_a^b E(x)e_1(x)dx\right]^2}[/itex]
Where [itex]e_1(x), e_n(x)[/itex] are known functions and [itex]n[/itex] is an integer, with [itex]n > 1[/itex].
If I couldn't know exactly [itex]E(x)[/itex] and I substitute it with [itex]E_0(x) + \delta E(x)[/itex], I have to demonstrate that the corresponding [itex]\delta K[/itex] is proportional to [itex][\delta E(x)]^2[/itex].
So an error of 10 % made during the estimation of [itex]E(x)[/itex] is an error of only 1 % for the corresponding estimation of [itex]K[/itex].
Homework Equations
The Attempt at a Solution
I tried to write the square of numerator and denominator:
[itex]K + \delta K = \displaystyle \frac{\left[\int_a^b [E(x) + \delta E(x)] e_n(x)dx\right]^2}{\left[\int_a^b [E(x) + \delta E(x)] e_1(x)dx\right]^2} = [/itex]
[itex] \displaystyle = \frac{\left[\int_a^b E(x)e_n(x)dx\right]^2 + 2\int_a^b E(x)e_n(x)dx \int_a^b \delta E(x)e_n(x)dx + \left[\int_a^b \delta E(x)e_n(x)dx\right]^2}{\left[\int_a^b E(x)e_1(x)dx\right]^2 + 2\int_a^b E(x)e_1(x)dx \int_a^b \delta E(x)e_1(x)dx + \left[\int_a^b \delta E(x)e_1(x)dx\right]^2}[/itex]
We can easily substitute the integral expression with letters and write:
[itex]K + \delta K = \displaystyle \frac{a^2 + 2ab + b^2}{c^2 + 2cd + d^2}[/itex]
How could I do now?
Could I neglect the [itex]b^2[/itex] and [itex]d^2[/itex] terms because they are small? If I do this, then I could divide both numerator and denominator by [itex]c^2[/itex] and take the Taylor series expansion of the denominator truncated to the first order, so
[itex]K + \delta K \simeq \frac{\displaystyle \frac{a^2}{c^2} + \displaystyle \frac{2ab}{c^2}}{1 + \displaystyle \frac{2d}{c}} \simeq \left(\displaystyle \frac{a^2}{c^2} + \frac{2ab}{c^2}\right)\left(1 + \displaystyle \frac{2d}{c}\right)[/itex]
But in this way I can't separate yet [itex]a[/itex] and [itex]c[/itex] from [itex]b[/itex] and [itex]d[/itex], and I can't write a term with only [itex]c^2[/itex] or [itex]d^2[/itex]. How can I proceed?
Thank you if you read this post,
Emily