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For the scalar surface integral over an obliquely cut cylinder, split the scalar surface integral into the curved cylinder and its two planar caps. Write $s=z-x$, so the solid has $x^2+y^2\le 4$ and $-3\le s\le 2$. On the curved cylinder, use the parametrizationThe surface area element follows fromConsequently the curved contribution isOn either cap $z=x+s$, the surface area element is $\sqrt{1+z_x^2+z_y^2}dxdy=\sqrt{2}dxdy$ over a radius-two disk. The lower cap has $x-z=3$ and the upper cap has $x-z=-2$. Thusand the complete scalar surface integral isThe surface area element is unsigned: opposite outward normals on the caps do not introduce extra minus signs.

The gradient of the scalar field isFor a unit vector $\hat{\mathbf{n}}$, the directional derivative is $D_{\hat{\mathbf{n}}}f=\nabla f\cdot \hat{\mathbf{n}}$. The Cauchy-Schwarz inequality bounds its magnitude by $\Vert \nabla f\Vert =2\sqrt{3041}$, with equality precisely in the two directions parallel to the gradient. ThereforeThe plus choice gives the greatest positive directional derivative, and the minus choice gives the most negative directional derivative. Both maximize its magnitude.

Differentiate the parametric curve:The arc length element is $ds=\sqrt{{\dot{x}}^2+{\dot{y}}^2}dt=|a|tdt$ on $0\le t\le 2$. ThusFor a positive geometric scale $a$, this is $2a$. The absolute value is necessary when the unspecified real constant $a$ is negative; $a=0$ gives zero arc length.

Apply the product-to-sum formula $2\cos u\cos v=\cos (u+v)+\cos (u-v)$ to the amplitude modulation:The amplitude modulation sidebands have angular frequencies $\omega +{\omega }_m$ and $\omega -{\omega }_m$, each with amplitude $b/2$ in this real cosine convention. The carrier wave has angular frequency $\omega$ and amplitude $a$.

This identity is exact for the supplied space-time expression. A physical qualification is needed if every term is meant to be a freely propagating electromagnetic wave with speed of light $c$: the dispersion relation requires $k_{\pm }=(\omega \pm {\omega }_m)/c$, whereas the prescribed modulation at every position leaves all three wavenumbers equal to $k=\omega /c$. A freely propagating modulated wave instead has envelope $a+b\cos ({\omega }_m(t-x/c))$; its analogous expansion has $k_{\pm }$. For example, at $x=t=0$ the prescribed expression has $A_{tt}-c^2{A}_{xx}=-b{\omega }_m^2$, which is nonzero for nontrivial modulation. The given formula can describe modulation imposed throughout space or the temporal signal at one position.