91Ó°ÊÓ

The cylindrical coordinates.

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

Find the value of ${∇}^{2}u.$.

role="math" localid="1659263309926" ${∇}^{2}u=\frac{{∂}^{2}u}{∂r^2}+\frac{1}{r}\frac{∂u}{∂r}+\frac{1}{r^2}\frac{{∂}^{2}u}{∂{\mathrm{θ}}^2}+\frac{{∂}^{2}u}{∂z^2}$

Find the value of ${∇}^{2}r.$

role="math" localid="1659263331235" $$\begin{gathered} {∇}^{2}r=\frac{{∂}^{2}r}{∂r^2}+\frac{1}{r}\frac{∂r}{∂r} \\ {∇}^{2}r=\frac{1}{r} \end{gathered}$$

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

role="math" localid="1659263371924" $$\begin{gathered} {∇}^2\left(\frac{1}{r}\right)=\frac{{∂}^2}{∂r^2}\left(\frac{1}{r}\right)+\frac{1}{r}\frac{∂}{∂r}\left(\frac{1}{r}\right) \\ {∇}^2\left(\frac{1}{r}\right)=\frac{1}{r^3} \end{gathered}$$

Findthe value of ${∇}^{2}lnr.$.

role="math" localid="1659263388437" $$\begin{gathered} {∇}^{2}lnr=\frac{{∂}^2}{∂r^2}(lnr)+\frac{1}{r}\frac{∂}{∂r}(lnr) \\ {∇}^{2}lnr=0 \end{gathered}$$

The required values are mentioned below.

role="math" localid="1659263401664" $$\begin{gathered} {∇}^{2}u=\frac{{∂}^{2}u}{∂r^2}+\frac{1}{r}\frac{∂u}{∂r}+\frac{1}{r^2}\frac{{∂}^{2}u}{∂{\mathrm{θ}}^2}+\frac{{∂}^{2}u}{∂z^2} \\ {∇}^{2}r=\frac{1}{r} \\ {∇}^2\left(\frac{1}{r}\right)=\frac{1}{r^3} \\ {∇}^{2}lnr=0 \end{gathered}$$