Question

In cylindrical coordinates${\mathbf{\nabla }}^{\mathbf{2}}\mathit{r}\mathbf{,}{\mathbf{\nabla }}^{\mathbf{2}}\left(\frac{1}{R}\right)\mathbf{,}{\mathbf{\nabla }}^{\mathbf{2}}\mathit{l}{\mathit{n}}_{\mathbf{r}}\mathbf{.}$

Short answer

The required values are mentioned below.

$$\begin{gathered} {\nabla }^{2}u=\frac{{\partial }^{2}u}{\partial r^2}+\frac{1}{r}\frac{\partial u}{\partial r}+\frac{1}{r^2}\frac{{\partial }^{2}u}{\partial {\theta }^2}+\frac{{\partial }^{2}u}{\partial z^2} \\ {\nabla }^{2}r=\frac{1}{r} \\ {\nabla }^2\left(\frac{1}{r}\right)=\frac{1}{r^3} \\ {\nabla }^{2}lnr=0 \end{gathered}$$

Step-by-step solution

Given Information

The cylindrical coordinates.

Definition of 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.

Find the value of ${\nabla }^{2}u.$.

role="math" localid="1659263309926" ${\nabla }^{2}u=\frac{{\partial }^{2}u}{\partial r^2}+\frac{1}{r}\frac{\partial u}{\partial r}+\frac{1}{r^2}\frac{{\partial }^{2}u}{\partial {\theta }^2}+\frac{{\partial }^{2}u}{\partial z^2}$

Find the value of ${\nabla }^{2}r.$

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

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

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

Findthe value of ${\nabla }^{2}lnr.$.

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

The required values are mentioned below.

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