Question

In spherical coordinates${\mathbf{\nabla }}^{\mathbf{2}}\mathit{r}\mathbf{,}{\mathbf{\nabla }}^{\mathbf{2}}\mathbf{\left(}{\mathit{r}}^{\mathbf{2}}\mathbf{\right)}\mathbf{,}{\mathbf{\nabla }}^{\mathbf{2}}\left(\frac{1}{r^2}\right)\mathbf{,}{\mathbf{\nabla }}^{\mathbf{2}}{\mathit{e}}^{\mathbf{i}\mathbf{k}\mathbf{r}\mathbf{}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{\theta }}$.

Short answer

The required values are mentioned below.

$$\begin{gathered} {\nabla }^{2}r=\frac{2}{r} \\ {\nabla }^2\left(r^2\right)=6 \\ {\nabla }^2\frac{1}{r^2}=\frac{4}{r^4} \\ {\nabla }^2\left(e^{ikrcos\theta }\right)=-k^2{e}^{ikrcos\theta } \end{gathered}$$

Step-by-step solution

Given Information

The spherical coordinates.

Definition of 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.

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

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

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

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

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

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

Findthe value of ${\nabla }^2\left(e^{ikrcos\theta }\right)$.

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

The required values are mentioned below.

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