Laplacian and gradient in curvilinear coordinates

Given $x=r\cos \varphi$, $y=r\sin \varphi$, transform the following expression into polar coordinates $r$ and $\varphi$: $w=\left(\dfrac {\partial u}{\partial x}\right)^{2}+\left(\dfrac {\partial u}{\partial y}\right)^{2}$.

Given $x=r\cos \varphi$, $y=r\sin \varphi$, transform the following expression into polar coordinates $r$ and $\varphi$: $w=\dfrac {\partial^{2}u}{\partial x^{2}}+\dfrac {\partial^{2}u}{\partial y^{2}}$.

Take $u$ and $v$ as new independent variables, and transform the following equation: $\dfrac {\partial^{2}z}{\partial x^{2}}+\dfrac {\partial^{2}z}{\partial y^{2}}+m^{2}z=0$, where $x=e^{u}\cos v$, $y=e^{u}\sin v$.

Suppose $u=f(r)$, where $r=\sqrt{x^{2}+y^{2}}$, transform the equations

Given

express the quantities $\Delta_{1}u=\left(\dfrac {\partial u}{\partial x}\right)^{2}+\left(\dfrac {\partial u}{\partial y}\right)^{2}+\left(\dfrac {\partial u}{\partial z}\right)^{2}$ and $\Delta_{2}u=\dfrac {\partial^{2}u}{\partial x^{2}}+\dfrac {\partial^{2}u}{\partial y^{2}}+\dfrac {\partial^{2}u}{\partial z^{2}}$ in terms of spherical coordinates.

Find the expression for the divergence of a plane vector $\overrightarrow {a}=\overrightarrow {a}(r,\varphi )$ represented in polar coordinates $r$ and $\varphi$.

Let

express $\operatorname {div}\overrightarrow {a}(x,y,z)$ in orthogonal curvilinear coordinates $u$, $v$, $w$.

As special cases, find the expressions for $\operatorname {div}\overrightarrow {a}$ in cylindrical coordinates and in spherical coordinates.

Transform to polar coordinates, setting $x = r\cos \varphi$, $y = r\sin \varphi$, the expression $w = \left(\frac{\partial u}{\partial x}\right)^{2}+\left(\frac{\partial u}{\partial y}\right)^{2}$.

Transform to polar coordinates, setting $x = r\cos \varphi$, $y = r\sin \varphi$, the expression $w = \frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}$.

What form does the equation $\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}+u = 0$ take, if $u = \varphi (r)$, where $r = \sqrt{x^{2}+y^{2}}$?

The same question for the equation

What form does the equation $\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}+\frac{\partial^{2} u}{\partial z^{2}}+u = 0$ take, if $u = \varphi (r)$, $r = \sqrt{x^{2}+y^{2}+z^{2}}$?

Transform the expressions

and

to spherical coordinates, setting

Express $\operatorname {grad} u$ :

Find $\operatorname {div}$ a of the plane field a in polar coordinates.

Find div a of a three-dimensional field:

Find the components of $\operatorname {rot} \mathbf{a}$ of the plane field a in polar coordinates.

Find the components of $\operatorname {rot} \mathbf{a}$ of a three-dimensional field a:

Write $\Delta u = \operatorname {div} \operatorname {grad} u$ :

Let an orthogonal system of curvilinear coordinates $(\xi ; \eta ; \zeta )$ be introduced in the region $\Omega$ :

where the right-hand sides are continuously differentiable functions. Let $\mathbf{e}_{\xi }, \mathbf{e}_{\eta }, \mathbf{e}_{\zeta }$ be the unit vectors of this system (vectors tangent to the coordinate lines and directed toward increasing coordinates, $\mathbf{e}_{\xi } \perp \mathbf{e}_{\eta }, \mathbf{e}_{\eta } \perp \mathbf{e}_{\zeta }, \mathbf{e}_{\zeta } \perp \mathbf{e}_{\xi }$)$^{*}$. Let $H_{\xi }, H_{\eta }, H_{\zeta }$ be the Lame coefficients, i.e. $H_{\xi } = \sqrt{\left(\frac{\partial x}{\partial \xi }\right)^{2}+\left(\frac{\partial y}{\partial \xi }\right)^{2}+\left(\frac{\partial z}{\partial \xi }\right)^{2}}$ etc. Prove that:

Here, for orthogonal curvilinear coordinates $(\xi ; \eta ; \zeta )$ with unit vectors $\mathbf{e}_{\xi }, \mathbf{e}_{\eta }, \mathbf{e}_{\zeta }$ and Lame coefficients $H_{\xi }, H_{\eta }, H_{\zeta }$ (see problem 66), formulas (39)-(41) are

Using formulas (39)-(41), derive expressions for $\operatorname {grad} u, \operatorname {div} \mathbf{a}, \operatorname {rot} \mathbf{a}$ :