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Let $D$ be the disk $x^2+y^2\le 3$. The upper boundary is the graph $z=3-x^2-y^2$, with outward vector vector surface element $(2x,2y,1)dxdy$. Thus its flux isThe lower disk has outward normal vector $-{\mathbf{e}}_z$ and zero flux, since $F_z=0$. The total outward surface integral is therefore $9\pi$.

Independently, $\nabla \cdot \mathbf{F}=2$ and the enclosed volume isConsequently ${\iiint }_V\nabla \cdot \mathbf{F}dV=2V=\boxed{9\pi }$, agreeing with the divergence theorem. Both the orientation of the cap and inclusion of the closing disk have been accounted for.

Subtract the Maclaurin series of $\log (1-x)$ from that of $\log (1+x)$. Their even powers cancel, givingThe full power series is $2{\sum }_{n\ge 0}{x}^{2n+1}/(2n+1)$, with radius of convergence one. For the real logarithm, the stated series applies on $|x|<1$.

The field is the gradient of a globally defined polynomial:It is therefore a conservative vector field on all of ${\mathbb{R}}^3$. The fundamental theorem for line integrals givesfor every closed piecewise smooth loop. Constructing the global vector-field potential avoids relying on a zero-curl test without checking the topology of the domain.

The total derivative follows the curve $x\mapsto (x,y(x))$, so the chain rule givesFor the stated functions, $u_x=6x$, $u_y=2y$ and $y^{\prime }=\cos x$, henceDirect differentiation of $u(x,\sin x)=3x^2+{\sin }^{2}x$ gives the same result.

For a dispersion relation $\omega (k)$, phase velocity is $v_p=\omega /k$ and group velocity is $v_g=d\omega /dk=v_p+kd{v}_p/dk$. Their local equality of phase and group velocity holds throughout an interval for a nondispersive branch $\omega =ck$. Equality at one particular $k$ requires only that $v_p$ have zero derivative there, so it does not imply a globally nondispersive medium.

Use $k=2\pi /\lambda$ in the prescribed gravity-capillary wave relation. Then ${\omega }^2=gk+(T/\rho )k^3$, and differentiation givesIn the gravity-dominated limit, $v_g=v_p/2$; in the surface-tension-dominated limit, $v_g=3v_p/2$. For $g,T,\rho >0$, the two are equal at $\lambda =2\pi \sqrt{T/(\rho g)}$, the minimum of the phase velocity for this dispersion relation.