91Ó°ÊÓ

Write and solve the Euler equations to make the following integrals stationary. In solving the Euler equations, the integrals in Chapter 5, Section 1, may be useful.

${\mathbf{∫}}_{{\mathbf{x}}_{\mathbf{1}}}^{{\mathbf{x}}_{\mathbf{2}}}\left(y{\text{'}}^2+\sqrt{y}\right)\mathit{d}\mathit{x}$

A particle moves on the surface of a sphere of radius ‘$\mathbf{a}$’ under the action of the earth’s gravitational field. Find the $\mathbf{θ}\mathbf{,}\mathbf{φ}$equations of motion. (Comment: This is called a spherical pendulum. It is like a simple pendulum suspended from the center of the sphere, except that the motion is not restricted to a plane.)

Test the following series for convergence

$\underset{\mathbf{n}\mathbf{-}\mathbf{1}}{\overset{\mathbf{∞}}{\mathbf{∑}}}\frac{{\mathbf{(}\mathbf{-}\mathbf{1}\mathbf{)}}^{\mathbf{n}}\mathbf{n}}{\mathbf{n}\mathbf{+}\mathbf{5}}$

Show that if $\mathit{C}$is a matrix whose columns are the components $\left(x_1,y_1\right)$and $\left(x_2,y_2\right)$of two perpendicular vectors each of unit length, then $\mathit{C}$is an orthogonal matrix. Hint: Find${\mathit{C}}^{\mathbf{T}}\mathit{C}$

Use the preliminary test to decide whether the following series are divergent or require further testing. Careful:Do notsay that a series is convergent; the preliminary test cannot decide this.

6.$\underset{\mathbf{n}\mathbf{=}\mathbf{1}}{\overset{\mathbf{∞}}{\mathbf{∑}}}\frac{\mathbf{n}\mathbf{!}}{\mathbf{(}\mathbf{n}\mathbf{+}\mathbf{1}\mathbf{)}\mathbf{!}}$

Given ,$\mathbf{z}\mathbf{=}{\left(x+y\right)}^{\mathbf{5}}\mathbf{}\mathbf{,}\mathbf{}\mathbf{y}\mathbf{=}\mathbf{sin}\mathbf{10}\mathbf{x}\mathbf{}\mathbf{,}\mathbf{}\mathbf{find}\mathbf{}\frac{\mathbf{dz}}{\mathbf{dx}}\mathbf{.}$.

There are 9 one-digit numbers (1 to 9), 90 two-digit numbers (10 to 99). How many three-digit, four-digit, etc., numbers are there? The first 9 terms of the harmonic series $\mathbf{1}\mathbf{+}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{+}\frac{\mathbf{1}}{\mathbf{3}}\mathbf{+}\mathbf{ }\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{+}\frac{\mathbf{1}}{\mathbf{9}}$are all greater than $\frac{\mathbf{1}}{\mathbf{10}}$; similarly consider the next 90 terms, and so on. Thus prove the divergence of the harmonic series by comparison with the series

$$\begin{gathered} \left[\frac{1}{10}+\frac{1}{10}+... \left(9  terms  each=\frac{1}{10}\right)+\left[90 terms each=\frac{1}{100}\right]\right]\mathbf{+}\mathbf{.}\mathbf{.}\mathbf{.} \\ \mathbf{=}\frac{\mathbf{9}}{\mathbf{10}}\mathbf{+}\frac{\mathbf{90}}{\mathbf{100}}\mathbf{+}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{=}\frac{\mathbf{9}}{\mathbf{10}}\mathbf{+}\frac{\mathbf{9}}{\mathbf{10}}\mathbf{+}\mathbf{.}\mathbf{.}\mathbf{.} \end{gathered}$$

(a) Find the Fourier series of period 2$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}{\left(x-1\right)}^{\mathbf{2}}$on (0,2)$$" width="9" height="19" role="math">$$" width="9" height="19" role="math">" width="9" height="19" role="math">

(b) Use your result in (a) to evaluate$\underset{}{\overset{}{\mathbf{∑}}}\mathbf{1}\mathbf{/}{\mathit{n}}^{\mathbf{4}}$.

Given

$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\left\{\begin{array}{c}1,\\ -1,\end{array}\begin{array}{ccc}-2& <x& <0\\ 0& <x& <2\end{array}\right\}$

find the exponential Fourier transform $\mathit{g}\left(\mathrm{Î\pm }\right)$and the sine transform${\mathit{g}}_{\mathbf{s}}\left(\mathrm{Î\pm }\right)$ .Write f(x) as an integral and use your result to evaluate

${\mathbf{∫}}_{\mathbf{0}}^{\mathbf{∞}}\frac{\left(cos2\mathrm{Î\pm }-1\right)\mathbf{}\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{2}\mathbf{Î\pm }}{\mathbf{Î\pm }}\mathit{d}\mathit{Î\pm }$

Given

$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\left(\begin{array}{c}x,\\ 2-x,\\ 0\end{array}\begin{array}{ccccc}0& ≤& x& ≤& 1\\ 1& ≤& x& ≤& 2\\ & x& â‰\yen & 2& \end{array}\right)$

find the cosine transform of f(x) and f(x) use it to write as an integral. Use your result to evaluate

${\mathbf{∫}}_{\mathbf{0}}^{\mathbf{∞}}\frac{\mathbf{c}\mathbf{o}{\mathbf{s}}^{\mathbf{2}}\mathbf{Î\pm }\mathbf{}\mathbf{s}\mathbf{i}{\mathbf{n}}^{\mathbf{2}}\mathbf{Î\pm }\mathbf{/}\mathbf{2}}{{\mathbf{Î\pm }}^{\mathbf{2}}}\mathit{d}\mathit{Î\pm }$