91Ó°ÊÓ

The spherical coordinates.

The coordinate system primarily utilized in three-dimensional systems is the spherical coordinates of the system. The spherical coordinate system is used to find the surface area in three-dimensional space.

Find the value of ${∇}^{2}r$.

role="math" localid="1659264897805" $$\begin{gathered} {∇}^{2}r=\frac{{∂}^{2}r}{∂r^2}+\frac{2}{r}\frac{∂r}{∂r} \\ {∇}^{2}r=\frac{2}{r} \end{gathered}$$

Find the value of ${∇}^2\left(r^2\right)$.

role="math" localid="1659264942317" $$\begin{gathered} {∇}^2\left(r^2\right)=\frac{{∂}^2}{∂r^2}\left(r^2\right)+\frac{2}{r}\frac{∂r}{∂r}\left(r^2\right) \\ {∇}^2\left(r^2\right)=2+4 \\ {∇}^2\left(r^2\right)=6 \end{gathered}$$

Find the value of ${∇}^2\left(\frac{1}{r^2}\right)$.

role="math" localid="1659264955541" $$\begin{gathered} {∇}^2\left(\frac{1}{r^2}\right)=\frac{{∂}^2}{∂r^2}\left(\frac{1}{r^2}\right)+\frac{1}{r}\frac{∂}{∂r}\left(\frac{1}{r^2}\right) \\ {∇}^2\left(\frac{1}{r^2}\right)=\frac{6}{r^4}-\frac{2}{r^4} \\ {∇}^2\left(\frac{1}{r^2}\right)=\frac{4}{r^4} \end{gathered}$$

Findthe value of ${∇}^2\left(e^{ikrcos\mathrm{θ}}\right)$.

$$\begin{gathered} {∇}^2\left(e^{ikrcos\mathrm{θ}}\right)=\left({\left(ik\cos \mathrm{θ}\right)}^2+\frac{2ik}{r}\cos \mathrm{θ}-\frac{2ik}{r}\cos \mathrm{θ}+{\left(ik\sin \mathrm{θ}\right)}^2\right)e^{ikrcos\mathrm{θ}} \\ {∇}^2\left(e^{ikrcos\mathrm{θ}}\right)=-k^2{e}^{ikrcos\mathrm{θ}} \end{gathered}$$

The required values are mentioned below.

role="math" localid="1659264964168" $$\begin{gathered} {∇}^{2}r=\frac{2}{r} \\ {∇}^2\left(r^2\right)=6 \\ {∇}^2\left(\frac{1}{r^2}\right)=\frac{4}{r^4} \\ {∇}^2\left(e^{ikrcos\mathrm{θ}}\right)=-k^2{e}^{ikrcos\mathrm{θ}} \end{gathered}$$