Continuity Equation in Cartesian Coordinates Using Time Also as a Parameter

Water and Fine Sediment Circulation

R. Torres , R.J. Uncles , in Treatise on Estuarine and Coastal Science, 2011

2.17.2.3 The 2D Width-Averaged Equations

In Cartesian coordinates, the width-averaged governing equations for longitudinal and vertical flow in an estuary of width B are (e.g., Ford et al., 1990)

${\partial }_{t}B\widehat{u}+{\partial }_{x}B{\widehat{u}}^2+{\partial }_{z}B\widehat{u}\widehat{w}+gB{\partial }_x\varsigma +C_{\text{d}}\widehat{u}\left|\widehat{u}\right|\cdot \left|{\partial }_{z}B\right|+g{\rho }_0^{-1}B{\displaystyle {\int }_z^{\varsigma }{\partial }_x\rho \cdot \text{d}z}-{\partial }_x(B{A}_m{\partial }_x\widehat{u})-{\partial }_z(B{K}_{\text{h}}{\partial }_z\widehat{u})=0$

and

${\partial }_{x}B\widehat{u}+{\partial }_{z}B\widehat{w}=0$

giving an equation for the surface elevation:

$B_{z=\varsigma }{\partial }_t\varsigma +{\partial }_x{\displaystyle {\int }_{-H}^{\varsigma }B\widehat{u}}\cdot \text{d}z=0$

The heat and salt-balance equations are

${\partial }_{t}B\widehat{T}+{\partial }_{x}B\widehat{u}\widehat{T}+{\partial }_{z}B\widehat{w}\widehat{T}-{\partial }_x(B{A}_{\text{h}}{\partial }_x\widehat{T})-{\partial }_z(B{K}_{\text{h}}{\partial }_z\widehat{T})=B\widehat{I}$

${\partial }_{t}B\widehat{S}+{\partial }_{x}B\widehat{u}\widehat{S}+{\partial }_{z}B\widehat{w}\widehat{S}-{\partial }_x(B{A}_{\text{h}}{\partial }_x\widehat{S})-{\partial }_z(B{K}_{\text{h}}{\partial }_z\widehat{S})=0$

The boundary conditions are essentially the same as those for the 3D equations except that terms involving y and ∂ _y are equated to zero. A sigma-coordinate version of these equations is given by Ford et al. (1990).

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Oceanic Dynamics

Zhihua Zhang , John C. Moore , in Mathematical and Physical Fundamentals of Climate Change, 2015

12.2 Inertial motion

The Cartesian coordinate system is the coordinate system used most commonly in studies of oceanic dynamics. The standard convention is that x is to the east, y is to the north, and z is up. The f-plane is a Cartesian coordinate system in which the Coriolis parameter is assumed to be constant. The β -plane is also a Cartesian coordinate systems in which the Coriolis parameter is assumed to vary linearly with latitude.

If the water on the sea surface moves only under the influence of Coriolis force, no other force acts on the water, then such a motion is called to an inertial motion.

By (11.34),

$\{\begin{array}{l}\frac{\partial v_1}{\partial t}+v_1\frac{\partial v_1}{\partial x}+v_2\frac{\partial v_1}{\partial y}+v_3\frac{\partial v_1}{\partial z}=-\frac{1}{\rho }\frac{\partial p}{\partial x}+f{v}_2+F_x,\hfill \\ \frac{\partial v_2}{\partial t}+v_1\frac{\partial v_2}{\partial x}+v_2\frac{\partial v_2}{\partial y}+v_3\frac{\partial v_2}{\partial z}=-\frac{1}{\rho }\frac{\partial p}{\partial y}-f{v}_1+F_y,\hfill \\ \frac{\partial v_3}{\partial t}+v_1\frac{\partial v_3}{\partial x}+v_2\frac{\partial v_3}{\partial y}+v_3\frac{\partial v_3}{\partial z}=-\frac{1}{\rho }\frac{\partial p}{\partial z}-g+F_z,\hfill \end{array}$

where the velocity v = (v ₁, v ₂, v ₃), the friction F = (F _x ,F _y ,F _y ), and the Coriolis parameter f = 2Ω sinϕ. The first two equations are the horizontal momentum equations and the third equation is the vertical momentum equation.

Dropping the quadratic terms in the material derivatives, we obtain the horizontal momentum equations as

$\{\begin{array}{l}\frac{\partial v_1}{\partial t}=-\frac{1}{\rho }\frac{\partial p}{\partial x}+f{v}_2+F_x,\hfill \\ \frac{\partial v_2}{\partial t}=-\frac{1}{\rho }\frac{\partial p}{\partial y}-f{v}_1+F_y.\hfill \end{array}$

If only Coriolis force acts on the water, there must be no horizontal pressure gradient,

$\frac{\partial p}{\partial x}=\frac{\partial p}{\partial y}=0,$

and so

(12.1) $\{\begin{array}{l}\frac{\partial v_1}{\partial t}=f{v}_2+F_x,\hfill \\ \frac{\partial v_2}{\partial t}=-f{v}_1+F_y.\hfill \end{array}$

For a frictionless ocean, F _x = F _y = 0. So (12.1) reduces to the two coupled, first-order, linear, differential equations

$\{\begin{array}{l}\frac{\partial v_1}{\partial t}=f{v}_2,\hfill \\ \frac{\partial v_2}{\partial t}=-f{v}_1.\hfill \end{array}$

This system of equations can be solved with standard techniques as follows.

Solving the second equation for v ₁ gives

$v_1=-\frac{1}{f}\frac{\partial v_2}{\partial t}.$

Inserting it into the first equation gives

$-\frac{1}{f}\frac{{\partial }^2{v}_2}{\partial t^2}=f{v}_2.$

Therefore, the inertial motion satisfies the system of equations

$\{\begin{array}{l}{v}_1+\frac{1}{f}\frac{\partial v_2}{\partial t}=0,\hfill \\ \frac{{\partial }^2{v}_2}{\partial t^2}+f^2{v}_2=0.\hfill \end{array}$

This system of equations has the solution

$\{\begin{array}{l}{v}_1=V\sin ft,\hfill \\ v_2=V\cos ft,\hfill \\ V^2=v_1^2+v_2^2.\hfill \end{array}$

The solution is a parameter equation for a circle with diameter D _i = 2V and period $T_{\text{i}}=\frac{2\pi }{f}=\frac{T_{\text{sd}}}{2\sin \varphi }$ , where $T_{\text{sd}}=\frac{2\pi }{\text{Ω}}$ is a sidereal day. The period T _i is called the inertial period. The current described by it is called an inertial current or an inertial oscillation.

Inertial currents are the commonest currents in the ocean. They have been observed at all depths in the ocean and at all latitudes. The motions are transient and decay in a few days. Oscillations at different depths or at different nearby sites are usually incoherent.

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Spacecraft Dynamics

Yoshiaki Ohkami , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

II.A Definition of Coordinate Systems and Representation of Vectors

A dexterous Cartesian coordinate is assumed to be attached to an arbitrary space system, and it is generically expressed by the three unit vectors e ₁, e ₂, and e ₃. These unit vectors form an orthogonal and normal system given by the following relations:

(1) $\{\begin{array}{ccc}\hfill {\mathbf{e}}_{\text{i}}\cdot {\mathbf{e}}_{\text{j}}& =\hfill & {\delta }_{\text{<mi mathvariant="italic" is="true">ij</mi>}},\hfill \\ \hfill {\mathbf{e}}_{\text{i}+2}& =\hfill & {\mathbf{e}}_{\text{i}}\times {\mathbf{e}}_{\text{i}+1}.\end{array}$

In addition to this fundamental relation, the coordinate system is conveniently expressed by the vector array denoted {e}, and

(2) $\begin{array}{ccc}\hfill \left\{\mathbf{e}\right\}=\left\{\begin{array}{c}\hfill {\mathbf{e}}_1\\ \hfill {\mathbf{e}}_2\\ \hfill {\mathbf{e}}_3\end{array}\right\},& \hfill {\left\{\mathbf{e}\right\}}^{\text{T}}=\left\{{\mathbf{e}}_1,{\mathbf{e}}_2,{\mathbf{e}}_3\right\}.\hfill & \hfill \end{array}$

Applying Eq. (2) to the unit vectors, the following relations result.

(3) $\begin{array}{cccc}\hfill {\mathbf{e}}_1={\left\{\mathbf{e}\right\}}^{\text{T}}\left[\begin{array}{c}\hfill 1\hfill \\ \hfill 0\hfill \\ \hfill 0\hfill \end{array}\right],& \hfill {\mathbf{e}}_1={\left\{\mathbf{e}\right\}}^{\text{T}}\left[\begin{array}{c}\hfill 0\hfill \\ \hfill 1\hfill \\ \hfill 0\hfill \end{array}\right],\hfill & {\mathbf{e}}_1={\left\{\mathbf{e}\right\}}^{\text{T}}\left[\begin{array}{c}\hfill 0\hfill \\ \hfill 0\hfill \\ \hfill 1\hfill \end{array}\right].\hfill & \hfill \hfill \end{array}$

A general vector, A, is then expressed by

(4) $\mathbf{A}={\{\mathbf{e}\}}^{\text{T}}\text{A},\text{A}=\left[\begin{array}{c}{\text{A}}_1\\ {\text{A}}_2\\ {\text{A}}_3\end{array}\right]\in {\text{R}}^{3\times 1}.$

where A is a 3   ×   3 matrix given by

$\text{A}=\left[{\text{A}}_1,{\text{A}}_2,{\text{A}}_3\right].$

Equation (4) gives a general and rigorous representation of a vector since it includes the vector basis and the component corresponding to the direction defined by the basis. It is noted that {e}, or sometimes {e} ^T , is called the "vector array" or "vectorix," and this notation is very useful in the analysis of mechanically complex spacecraft or sensor kinematics as well as coordinate transformations.

Consider two coordinates designated by vector bases, {e} and {u}; then

(5) $1.\left\{\mathbf{e}\right\}·{\left\{\mathbf{e}\right\}}^{\text{T}}={\text{U}}_3,$

(6) $2.\left\{\mathbf{e}\right\}·{\left\{\mathbf{u}\right\}}^{\text{T}}=[{\mathbf{e}}_{\text{i}}·{\mathbf{u}}_{\text{j}}]=\text{C},$

where U ₃ denotes a 3   ×   3 unit matrix and C a 3   ×   3 coordinate transform matrix. Based on these relations, the coordinate transformations are given by

(7a) $\left\{\mathbf{e}\right\}=\text{C}\left\{\mathbf{u}\right\},$

(7b) ${\left\{\mathbf{e}\right\}}^{\text{T}}={\left\{\mathbf{u}\right\}}^{\text{T}}{\text{C}}^{\text{T}},$

(7c) $\left\{\mathbf{u}\right\}={\text{C}}^{\text{T}}\left\{\mathbf{e}\right\},$

(7d) ${\left\{\mathbf{u}\right\}}^{\text{T}}={\left\{\mathbf{u}\right\}}^{\text{T}}{\text{C}}^{\text{T}}.$

Noting the fundamental properties of e _i and u _j given by Eq. (1), the transformation matrix C is an orthonormal matrix with the following properties (Fig. 1):

FIGURE 1. Coordinate transformation diagram.

(8a) $\text{C}{\text{C}}^{\text{T}}={\text{C}}^{\text{T}}\text{C}={\text{U}}_3,$

(8b) ${\text{C}}^{-1}={\text{C}}^{\text{T}}.$

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Atmospheric Dynamics

Zhihua Zhang , John C. Moore , in Mathematical and Physical Fundamentals of Climate Change, 2015

11.9.3 Component form of the navier-stokes equation

By using Cartesian coordinates, we take unit vectors i pointing eastward and j pointing northward, and k pointing upward at a point on Earth's surface.

Consider small incremental distances:

$\begin{array}{l}\text{d}x=r\cos \varphi d\text{λ}\text{in}\text{the}\text{eastward(zone)}\text{direction,}\hfill \\ \text{dy}=r\text{d}\varphi \text{in}\text{the}\text{northward}\text{(meridional)}\text{direction,}\hfill \\ \text{dz}=\text{d}r\text{in}\text{the}\text{vertical}\text{direction,}\hfill \end{array}$

where φ, λ, and z are latitude, longitude, and the vertical distance from Earth's surface, respectively, and r = a + z, where a is Earth's radius.

Let the velocity v = v ₁ i+ v ₂ j + v ₃ k. Notice that i,j, and k change with time.

Then

$\frac{\text{D}\mathbf{v}}{\text{D}t}=\frac{\text{D}{v}_1}{\text{D}t}\mathbf{i}+v_1\frac{\text{D}\mathbf{i}}{\text{D}t}+\frac{\text{D}{v}_2}{\text{D}t}\mathbf{j}+v_2\frac{\text{D}\mathbf{j}}{\text{D}t}+\frac{\text{D}{v}_3}{\text{D}t}\mathbf{k}+v_3\frac{\text{D}\mathbf{k}}{\text{D}t}.$

First, we compute the material derivatives of the unit vector k.

Notice that $\mathbf{k}=\frac{\mathbf{r}}{r}$ , where r is the magnitude of position vector r. then

$\frac{\text{D}\mathbf{k}}{\text{D}t}=\frac{\text{D}}{\text{D}t}\left(\frac{\mathbf{r}}{r}\right)=\frac{1}{r}\frac{\text{D}\mathbf{r}}{\text{D}t}-\frac{\mathbf{r}}{r^2}\frac{\text{D}\mathbf{r}}{\text{D}t}=\frac{1}{r}\left(\frac{\text{D}\mathbf{r}}{\text{D}t}-\mathbf{k}\frac{\text{D}r}{\text{D}t}\right).$

By the definition of the material derivative of the position vector given in Section 11.7,

$\frac{\text{D}\mathbf{k}}{\text{D}t}=\frac{1}{r}\left(\frac{\text{D}\mathbf{r}}{\text{D}t}-\mathbf{k}\frac{\text{D}r}{\text{D}t}\right).$

By (11.22), (11.23) and dz = dr,

$\frac{\text{D}\mathbf{k}}{\text{D}t}=\frac{1}{r}(\mathbf{v}-v_3\mathbf{k})=\frac{v_1\mathbf{i}+v_2\mathbf{j}}{r}.$

Next, we compute the material derivative of the unit vector j.

Let the constant angular velocity Ω = Ω (j cos φ + k sin φ). It follows from $\frac{D\mathbf{\Omega }}{\text{D}t}=0$ that

$\cos \varphi \frac{\text{D}\mathbf{j}}{\text{D}t}+\mathbf{j}\frac{\text{Dcos}\varphi }{\text{D}t}+\sin \varphi \frac{\text{D}\mathbf{k}}{\text{D}t}+\mathbf{k}\frac{\text{Dsin}\varphi }{\text{D}t}=0.$

This is equivalent to

$\frac{\text{D}\mathbf{j}}{\text{D}t}=\mathbf{j}\tan \varphi \frac{\text{D}\varphi }{\text{D}t}-\tan \varphi \frac{\text{D}\mathbf{k}}{\text{D}t}-\mathbf{k}\frac{\text{D}\varphi }{\text{D}t}=(\mathbf{j}\tan \varphi -\mathbf{k})\frac{\text{D}\varphi }{\text{D}t}-\tan \varphi \frac{v_1\mathbf{i}+v_2\mathbf{j}}{r}.$

By the definition of the material derivative, (11.23), and dy = rdϕ, we get

$\begin{array}{l}\frac{\text{D}\varphi }{\text{D}t}=\frac{\partial \varphi }{\partial t}+v_1\frac{\partial \varphi }{\partial x}+v_2\frac{\partial \varphi }{\partial y}+v_3\frac{\partial \varphi }{\partial z}\hfill \\ =\frac{\partial \varphi }{\partial t}+\frac{\partial \varphi }{\partial x}\frac{\text{d}x}{\text{d}t}+\frac{\partial \varphi }{\partial y}\frac{\text{d}y}{\text{d}t}+\frac{\partial \varphi }{\partial z}\frac{\text{d}z}{\text{d}t}=\frac{\text{d}\varphi }{\text{d}t}=\frac{1}{r}\frac{\text{d}y}{\text{d}t}=\frac{v_2}{r}.\hfill \end{array}$

Therefore,

$\frac{\text{D}\mathbf{j}}{\text{D}t}=(\mathbf{j}\tan \varphi -\mathbf{k})\frac{v_2}{r}-\tan \varphi \frac{v_1\mathbf{i}+v_2\mathbf{j}}{r}=-\frac{v_1\tan \varphi }{r}\mathbf{i}-\frac{v_2}{r}\mathbf{k}.$

Finally, we compute the material derivative of the unit vector i.

Notice that i = j × k. Applying property (iii) of the material derivative given in Section 11.7, we get

$\frac{\text{D}\mathbf{i}}{\text{D}t}=\frac{D(\mathbf{j}\times \mathbf{k})}{Dt}=\frac{\text{D}\mathbf{j}}{Dt}\times \mathbf{k}+\mathbf{j}\times \frac{\text{D}\mathbf{k}}{\text{D}t}=\frac{-v_1\tan \varphi \mathbf{i}-v_2\mathbf{k}}{r}\times \mathbf{k}+\mathbf{j}\times \frac{v_1\mathbf{i}+v_2\mathbf{j}}{r}.$

By the orthogonality of the unit vectors,

$\mathbf{i}\times \mathbf{k}=-\mathbf{j},\mathbf{k}\times \mathbf{k}=0,\mathbf{j}\times \mathbf{i}=-\mathbf{k},\mathbf{j}\times \mathbf{j}=0,$

and so

$\frac{\text{D}\mathbf{i}}{\text{D}t}=\frac{v_1\tan \varphi \mathbf{j}-v_1\mathbf{k}}{r}.$

Summarizing these results, we get

$\begin{array}{ll}\frac{\text{D}\mathbf{v}}{\text{D}t}=\hfill & \frac{\text{D}{v}_1}{\text{D}t}\mathbf{i}+v_1\frac{v_1\tan \varphi \mathbf{j}-v_1\mathbf{k}}{r}+\frac{\text{D}{v}_2}{\text{D}t}\mathbf{j}+v_2\frac{-v_1\tan \varphi \mathbf{i}-v_2\mathbf{k}}{r}+\frac{{\text{Dv}}_3}{\text{D}t}\mathbf{k}\hfill \\ \hfill & +v_3\frac{v_1\mathbf{i}+v_2\mathbf{j}}{r}.\hfill \end{array}$

The Coriolis term, the vector product of 2Ω and v, is given by

$2\mathbf{\Omega }\times \mathbf{v}=2\Omega (v_{3}cos\varphi -v_{2}sin\varphi )\mathbf{i}+2\Omega v_1\sin \varphi \mathbf{j}-2\Omega v_1\cos \varphi \mathbf{k},$

and the pressure gradient term is given by

$-\frac{\nabla p}{\rho }=-\frac{1}{\rho }\left(\frac{\partial p}{\partial x}\mathbf{i}+\frac{\partial p}{\partial y}\mathbf{j}+\frac{\partial p}{\partial z}\mathbf{k}\right).$

Disregard the centripetal acceleration Ω × (Ω × r). Let

${\mathbf{F}}_v=F_x\mathbf{i}+F_y\mathbf{j}+F_z\mathbf{k}.$

Then the Navier-Stokes equation (11.32) is written in the form

$\begin{array}{lll}\frac{\text{D}{v}_1}{\text{D}t}\mathbf{i}\hfill & +v_1\frac{v_1\tan \varphi \mathbf{j}-v_1\mathbf{k}}{r}\hfill & +\frac{\text{D}{v}_2}{\text{D}t}\mathbf{j}+v_2\frac{-v_1\tan \varphi \mathbf{i}-v_2\mathbf{k}}{r}+\frac{{\text{Dv}}_3}{\text{D}t}\mathbf{k}\hfill \\ \hfill & +v_3\frac{v_1\mathbf{i}+v_2\mathbf{j}}{r}=-\frac{1}{\rho }\hfill & \left(\frac{\partial p}{\partial x}\mathbf{i}+\frac{\partial p}{\partial y}\mathbf{j}+\frac{\partial p}{\partial z}\mathbf{k}\right)-2\text{Ω}(v_3\cos \varphi -v_2\sin \varphi )\mathbf{i}\hfill \\ \hfill & \hfill & -2\text{Ω}{v}_1\sin \varphi \mathbf{j}+2\text{Ω}{v}_1\cos \varphi \mathbf{k}-g\mathbf{k}+F_x\mathbf{i}+F_x\mathbf{j}+F_x\mathbf{k}.\hfill \end{array}$

Collecting the terms in i,j, and k, we can write the Navier-Stokes equation (11.32) in the component form:

$\{\begin{array}{l}\frac{\text{D}{v}_1}{\text{D}t}=-\frac{1}{\rho }\frac{\partial p}{\partial x}+\left(2\Omega +\frac{v_1}{r\cos \varphi }\right)(v_2\sin \varphi -v_3\cos \varphi )+F_x,\hfill \\ \frac{\text{D}{v}_2}{\text{D}t}=-\frac{1}{\rho }\frac{\partial p}{\partial y}-\frac{v_2{v}_3}{r}-\left(2\Omega +\frac{v_1}{r\cos \varphi }\right)v_1\sin \varphi +F_y,\hfill \\ \frac{\text{D}{v}_3}{\text{D}t}=-\frac{1}{\rho }\frac{\partial p}{\partial z}+\frac{v_1^2+v_2^2}{r}+2\Omega v_1\cos \varphi -g+F_z,\hfill \end{array}$

where the first two equations are called the horizontal momentum equations and the third equation is called the vertical momentum equation.

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Revision exercises

Hubert Chanson , in Hydraulics of Open Channel Flow (Second Edition), 2004

Notation

x, y, z	Cartesian coordinates
r,θ,z	polar coordinates
$\frac{\partial }{\partial x}$	partial differentiation with respect to the x-coordinate
$\frac{\partial }{\partial y},\frac{\partial }{\partial z}$	partial differential (Cartesian coordinate)
$\frac{\partial }{\partial r},\frac{\partial }{\partial \theta }$	partial differential (polar coordinate)
$\frac{\partial }{\partial t}$	partial differential with respect of the time t
$\frac{\text{D}}{\text{D}t}$	absolute derivative
δij	identity matrix element δii = 1 and δij = 0 (for i different of j)
N!	N-factorial: N!= 1 × 2 × 3 × 4 × … ×(N − 1)× N

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An Introduction to Atmospheric Gravity Waves

In International Geophysics, 2002

1.2.2 Wave Scales

We shall use a Cartesian coordinate system ( x, y, z) with x and y in the horizontal plane and z in the vertical direction. The coordinates have unit vectors $(\widehat{x},\widehat{y},\widehat{z})$ . Unless otherwise noted, the horizontal directions of wave motion will be along the x-axis. The wavelength l is the distance between successive crests of a wave, as illustrated in Fig. 1.8. In this book, we shall consider only waves with horizontal wavelengths less than 1000 km so that the effects of the effects of the Earth's curvature can be neglected (Hines, 1968). Accordingly, we can consider horizontal planes as being flat. The mathematical description of a wave involves trigonometric functions, for example, sin(2πx/l), etc., and it is convenient to define the wavenumber as

Figure 1.8. A wave with wavelength l moves to the right with speed c. The wave crest moves from point a to b in time τ, which is the period of oscillation of the wave as seen by a stationary observer.

(1.1) $\mathcal{K}=\frac{2\pi }{\ell }.$

We can think of the wavenumber as 2π times the number of wave oscillations per unit length, or wavelength per unit radian. The wavenumber is a fundamental property of a wave. We label the wavenumbers in the x-, y-, and z-directions as k, l, and m, respectively. These wavenumbers are defined as

(1.2) $\begin{array}{ccc}k=\frac{2\pi }{{\lambda }_x}& l=\frac{2\pi }{{\lambda }_y}& m=\frac{2\pi }{{\lambda }_z},\end{array}$

where λ_x, λ_y, and λ_z are the wavelengths of the wave in the x-, y-, and z-directions, respectively. The wave vector, $\overrightarrow{k}$ , defines the direction of wave propagation and is given by

(1.3) $\overrightarrow{\mathcal{K}}=k\widehat{x}+l\widehat{y}+m\widehat{z}.$

The wave period, τ, is the time required for the fluid particles to make one oscillation. If the wave is moving, then the wave period is the time required for successive wave crests to pass a stationary observer, as illustrated in Fig. 1.8. For waves with periods less than a few hours, the effects of the Earth's rotation (the Coriolis force, see Appendix A) can be ignored. The wave frequency, ω, is 2π times the number of wave oscillations per unit time, i.e.,

(1.4) $\omega =\frac{2\pi }{\tau }$

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An Introduction to Atmospheric Gravity Waves

Carmen J. Nappo , in International Geophysics, 2012

1.2.3 Wave Scales

We shall use a Cartesian coordinate system ( x, y, z) with xand y in the horizontal plane, and z in the vertical direction. The coordinates have unit vectors $(\hat{x}\text{,}\hat{y}\text{,}\hat{z})$ . Unless otherwise noted, the horizontal directions of wave motion will be along the x-axis. The wavelength $\lambda$ is the distance between successive crests ⁴ of a wave as illustrated in Fig. 1.13. In this book, we shall consider mostly waves with horizontal wavelengths less than 1000   km so that the effects of the earth's curvature may be neglected (Hines, 1968). Accordingly, we can consider horizontal planes as being flat. The mathematical description of a wave involves trigonometric functions, for example, $\sin (2\pi x/\lambda )$ , etc., and it is convenient to define the wavenumber as

Fig. 1.13. A wave with wavelength $\lambda$ moves to the right with speed c. The wave crest moves from point a to b in time $\tau$ which is the period of oscillation of the wave as seen by a stationary observer.

(1.1) $\kappa =\frac{2\pi }{\lambda }\text{.}$

We can think of the wavenumber as $2\pi$ times the number of wave oscillations per unit length, or wavelength per unit radian. The wavenumber is a fundamental property of a wave. We will label the wavenumbers in the x-, y-, and z-directions as k, l, and m, respectively. These wavenumbers are defined as

(1.2) $k=\frac{2\pi }{{\lambda }_x}\text{,}l=\frac{2\pi }{{\lambda }_y}\text{,}m=\frac{2\pi }{{\lambda }_z}\text{,}$

where ${\lambda }_x\text{,}{\lambda }_y$ , and ${\lambda }_z$ are the wavelengths of the wave in the x-, y-, and z-directions, respectively. The wave vector, $\overrightarrow{\kappa }$ , points in the direction of the traveling wave, and is given by

(1.3) $\overrightarrow{\kappa }=k\hat{x}+l\hat{y}+m\hat{z}\text{.}$

The wave period, $\tau$ , is the time required for the fluid particles to make one oscillation. If the wave is moving in a still medium, then the wave period $\tau$ is the time required for successive wave crests to pass a stationary observer, as illustrated in Fig. 1.13. For waves with periods less than a few hours, the effects of the earth's rotation (the Coriolis force, see the Appendix) can be ignored. The wave frequency, $\omega$ , is $2\pi$ times the number of wave oscillations per unit time or radians per second i.e.,

(1.4) $\omega =\frac{2\pi }{\tau }\text{.}$

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International Handbook of Earthquake and Engineering Seismology, Part A

Klaus Mosegaard , Albert Tarantola , in International Geophysics, 2002

2.3 Homogeneous Probability Distributions

In some parameter spaces, there is an obvious definition of distance between points, and therefore of volume. For instance, in the 3D Euclidean space the distance between two points is just the Euclidean distance (which is invariant under translations and rotations). Should we choose to parametrize the position of a point by its Cartesian coordinates { x, y, z}, the volume element in the space would be dV(x, y, z) = dx dy dz, while if we choose to use geographical coordinates, the volume element would be $dV\left(r,\theta ,\phi \right)=r^{2}sin\theta drd\vartheta d\phi$ .

Definition. The homogeneous probability distribution is the probability distribution that assigns to each region of the space a probability proportional to the volume of the region.

Then, which probability density represents such a homogeneous probability distribution? Let us give the answer in three steps.

•

If we use Cartesian coordinates {x, y, z}, as we have dV(x, y, z) = dx dy dz, the probability density representing the homogeneous probability distribution is constant: f(x, y, z) = κ.

•

If we use geographical coordinates {r, θ, φ}, as we have $dV\left(r,\theta ,\phi \right)=r^{2}sin\theta drd\theta d\phi$ , the probability density representing the homogeneous probability distribution is $g\left(r,\theta ,\phi \right)=k{r}^{2}sin\theta$ .

•

Finally, if we use an arbitrary system of coordinates {u, ν, w}, in which the volume element of the space is dV(u, ν, w) = ν(u, ν, w) du dν dw, the homogeneous probability distribution is represented by the probability density h (u, ν, w) = κν (u, νv, w).

This is obviously true, since if we calculate the probability of a region A of the space, with volume V(A), we get a number proportional to V(A).

From these observations we can arrive at conclusions that are of general validity. First, the homogeneous probability distribution over some space is represented by a constant probability density only if the space is flat (in which case rectilinear systems of coordinates exist) and if we use Cartesian (or rectilinear) coordinates. The other conclusions can be stated as rules:

Rule 1.

The probability density representing the homogeneous probability distribution is easily obtained if the expression of the volume element dV(u ₁, u ₂,…) = ν(u ₁, u ₂,…) du ₁ du ₂ … of the space is known, as it is then given by h(u ₁, u ₂, …) = κν(u ₁, u ₂,…), where k is a proportionality constant (that may have physical dimensions).

Rule 2.

If there is a metric g _ij (u ₁, u ₂,…) in the space, then the volume element is given by $dV\left(u_1,u_2,\dots \right)=\sqrt{det\text{g}}\left(u_1,u_2,\dots \right)d{u}_{1}d{u}_2\cdots$ , i.e., we have $\upsilon \left(u_1,u_2,\dots \right)=\sqrt{det\text{g}\left(u_1,u_2,\dots \right)}$ . The probability density representing the homogeneous probability distribution is, then, $h\left(u_1,u_2,7\dots \right)=\kappa \sqrt{det\text{g}\left(u_1,u_2,\dots \right)}$ .

Rule 3.

If the expression of the probability density representing the homogeneous probability distribution is known in one system of coordinates, then it is known in any other system of coordinates, through the Jacobian rule.

Indeed, in the expression above, $g\left(r,\theta ,\phi \right)=k{r}^2\sin \theta$ , we recognize the Jacobian between the geographical and the Cartesian coordinates (where the probability density is constant).

For short, when we say the homogeneous probability density we mean the probability density representing the homogeneous probability distribution. One should remember that, in general, the homogeneous probability density is not constant.

Let us now examine 'positive parameters,' like a temperature, a period, or a seismic wave propagation velocity. One of the properties of the parameters we have in mind is that they occur in pairs of mutually reciprocal parameters:

Period T = 1/v; Frequency v = 1/T

Resistivity ρ = 1/σ; Conductivity σ = 1/ρ

Temperature T = 1/(κβ); Thermodynamic parameter β = 1/(κT)

Mass density ρ = 1/ℓ; Lightness ℓ = 1/ρ

Compressibility γ = 1/κ; Bulk modulus (uncompressibility) κ = 1/γ

Wave velocity c = 1/n; Wave slowness n = 1/c.

When working with physical theories, one may freely choose one of these parameters or its reciprocal.

Sometimes these pairs of equivalent parameters come from a definition, like when we define frequency v as a function of the period T, by v = 1/T. Sometimes these parameters arise when analyzing an idealized physical system. For instance, Hooke's law, relating stress σ_{ℓ j} to strain ε_{ℓ j} can be expressed as ${\sigma }_{ij}=c_{ij}{\varepsilon }_{kl}$ , thus introducing the stiffness tensor c _{i jkℓ}, or as ${\varepsilon }_{ij}=d_{ij}{\sigma }_{kl}$ , thus introducing the compliance tensor d _{i jkℓ}, the inverse of the stiffness tensor. Then the respective eigenvalues of these two tensors belong to the class of scalars analyzed here.

Let us take, as an example, the pair conductivity–resistivity (which may be thermal, electric, etc.). Assume we have two samples in the laboratory S ₁ and S ₂ whose resistivities are respectively ρ₁ and ρ₂. Correspondingly, their conductivities are σ₁ = 1/ρ₁ and σ₂ = 1/ρ₂. How should we define the 'distance' between the 'electrical properties' of the two samples? As we have $|{\rho }_2-{\rho }_1|\ne |{\sigma }_2-{\sigma }_1|$ , choosing one of the two expressions as the 'distance' would be arbitrary. Consider the following definition of 'distance' between the two samples:

(7) $D\left(S_1,S_2\right)=\left|log\frac{{\rho }_2}{{\rho }_1}\right|=\left|log\frac{{\sigma }_2}{{\sigma }_1}\right|.$

This definition (i) treats symmetrically the two equivalent parameters ρ and σ and, more importantly, (ii) has an invariance of scale (what matters is how many 'octaves' we have between the two values, not the plain difference between the values). In fact, it is the only definition of distance between the two samples S ₁ and S ₂ that has an invariance of scale and is additive (i.e., D(S ₁, S ₂) + D(S ₂, S ₃) = D(S ₁, S ₃)).

Associated to the distance $D\left(x_1,x_2\right)=|log\left(x_2/x_1\right)|$ is the distance element (differential form of the distance)

(8) $dL\left(x\right)=\frac{dx}{x}.$

This being a 'one-dimensional volume,' we can now apply Rule 1 above to get the expression of the homogeneous probability density for such a positive parameter:

(9) $f\left(x\right)=\frac{k}{x}.$

Defining the reciprocal parameter y = 1/x and using the Jacobian rule, we arrive at the homogeneous probability density for y:

(10) $g\left(y\right)=\frac{k}{y}.$

These two probability densities have the same form: the two reciprocal parameters are treated symmetrically. Introducing the logarithmic parameters

(11) $x^{*}log\frac{x}{x_0};y^{*}=log\frac{y}{y_0},$

where x ₀ and y ₀ are arbitrary positive constants, and using the Jacobian rule, we arrive at the homogeneous probability densities:

(12) $f^{\prime }\left(x^{*}\right)=k;g^{\prime }\left(y^{*}\right)=k.$

This shows that the logarithm of a positive parameter (of the type considered above) is a 'Cartesian' parameter. In fact, it is the consideration of Eqs. (12), together with the Jacobian rule, that allows full understanding of the (homogeneous) probability densities (9) and (10).

The association of the probability density f(u) = k/u with positive parameters was first made by Jeffreys (1939). To honor him, we propose to use the term Jeffreys parameters for all the parameters of the type considered above. The 1/u probability density was advocated by Jaynes (1968), and a nontrivial use of it was made by Rietsch (1977) in the context of inverse problems.

Rule 4.

The homogeneous probability density for a Jeffreys quantity u is f(u) = κ/u.

Rule 5.

The homogeneous probability density for a 'Cartesian parameter' u (like the logarithm of a Jeffreys parameter, an actual Cartesian coordinate in an Euclidean space, or the Newtonian time coordinate) is f(u) = κ. The homogeneous probability density for an angle describing the position of a point in a circle is also constant.

If a parameter u is a Jeffreys parameter with the homogeneous probability density f(u) = κ/u, then its inverse, its square, and, in general, any power of the parameter is also a Jeffreys parameter, as it can easily be seen using the Jacobian rule.

Rule 6.

Any power of a Jeffreys quantity (including its inverse) is a Jeffreys quantity.

It is important to recognize when we do not face a Jeffreys parameter. Among the many parameters used in the literature to describe an isotropic linear elastic medium we find parameters like the Lamé's coefficients λ and μ, the bulk modulus κ, the Poisson ratio σ, etc. A simple inspection of the theoretical range of variation of these parameters shows that the first Lamé parameter λ and the Poisson ratio σ may take negative values, so they are certainly not Jeffreys parameters. In contrast, Hooke's law ${\sigma }_{ij}=c_{ijkl}{\varepsilon }^{kl}$ , defining a linearity between stress σ _ij and strain ε _ij , defines the positive definite stiffness tensor c _{i jkℓ} or, if we write ${\varepsilon }_{ij}=d_{ijkl}{\sigma }^{kl}$ , defines its inverse, the compliance tensor d _{i jkℓ}. The two reciprocal tensors c _{i jkℓ} and d _{i jkℓ} are 'Jeffreys tensors.' This is a notion whose development is beyond the scope of this paper, but we can give the following rule.

Rule 7.

The eigenvalues of a Jeffreys tensor are Jeffreys quantities. ⁶

As the two (different) eigenvalues of the stiffness tensor c _{i jkℓ} are λ _κ = 3 _κ (with multiplicity 1) and λ_μ = 2μ (with multiplicity 5), we see that the incompressibility modulus κ and the shear modulus μ are Jeffreys parameters⁷ (as are any parameters proportional to them, or any powers of them, including the inverses). If, for some reason, instead of working with κ and μ, we wish to work with other elastic parameters, for instance, the Young modulus Y and the Poisson ratio σ, or the two elastic wave velocities, then the homogeneous probability distribution must be found using the Jacobian of the transformation (see Appendix H).

Some probability densities have conspicuous 'dispersion parameters,' like the σ's in the normal probability density $f\left(x\right)=kexp\left(-\frac{{\left(x-x_0\right)}^2}{2{\sigma }^2}\right)$ , in the log-normal probability $g\left(X\right)=\frac{k}{X}exp\left(-\frac{{\left(logX/X_0\right)}^2}{2{\sigma }^2}\right)$ or in the Fisher probability density (Fisher, 1953) $h\left(\vartheta ,\phi \right)=\kappa sin\theta exp\left(cos\theta /{\sigma }^2\right)$ . A consistent probability model requires that when the dispersion parameter σ tends to infinity, the probability density tends to the homogeneous probability distribution. For instance, in the three examples just given, f(x) → κ, g(X) → κ/X, and h (θ, φ) → κ sin θ, which are the respective homogeneous probability densities for a Cartesian quantity, a Jeffreys quantity, and the geographical coordinates on the surface of the sphere. We can state the following rule.

Rule 8.

If a probability density has some 'dispersion parameters,' then, in the limit where the dispersion parameters tend to infinity, the probability density must tend to the homogeneous one.

As an example, using the normal probability density $f\left(x\right)=\kappa exp\left(-\frac{{\left(x-x_0\right)}^2}{2{\sigma }^2}\right)$ , for a Jeffreys parameter is not consistent. Note that it would assign a finite probability to negative values of a positive parameter that, by definition, is positive. More technically, this would violate our Postulate 1. Using the log-normal probability density for a Jeffreys parameter is consistent.

There is a problem of terminology in the Bayesian literature. The homogeneous probability distribution is a very special distribution. When the problem of selecting a 'prior' probability distribution arises in the absence of any information, except the fundamental symmetries of the problem, one may select as prior probability distribution the homogeneous distribution. But enthusiastic Bayesians do not call it 'homogeneous,' but 'noninformative.' We cannot recommend using this terminology. The homogeneous probability distribution is as informative as any other distribution, it is just the homogeneous one (see Appendix D).

In general, each time we consider an abstract parameter space, each point being represented by some parameters x = {x ¹, x ² … x ⁿ }, we will start by solving the (sometimes nontrivial) problem of defining a distance between points that respects the necessary symmetries of the problem. Only exceptionally this distance will be a quadratic expression of the parameters (coordinates) being used (i.e., only exceptionally our parameters will correspond to 'Cartesian coordinates' in the space). From this distance, a volume element dV (x) = ν (x) d x will be deduced, from where the expression f (x) = κν (x) of the homogeneous probability density will follow. Sometimes, we can directly define the volume element, without the need of a distance. We emphasize the need of defining a distance—or a volume element—in the parameter space, from which the notion of homogeneity will follow. With this point of view, we slightly depart from the original work by Jeffreys and Jaynes.

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Electromagnetic Fields in Horizontally Stratified Media

Michael S. Zhdanov , in Foundations of Geophysical Electromagnetic Theory and Methods (Second Edition), 2018

5.3.1 Spatial Frequency-Domain (SFD) Representation of the Electromagnetic Field in a Horizontally Layered Medium

We begin with a Cartesian coordinate system with its origin on the surface of the earth and with the z axis directed downwards. At a given point, we will assume the earth consists of N homogeneous layers, each with a conductivity ${\sigma }_j$ and a thickness $d_j$ $(j=1,2,3,...,N)$ . The model is bounded by a nonconducting upper half-space at $z=0$ . The magnetic permeability, μ, is assumed to have the value for free space, ${\mu }_0$ . The source of the field is taken to be a current system circulating at high altitude with the density ${\mathbf{j}}^Q$ (Fig. 5.6). The field varies in time by the factor $e^{-i\omega t}$ . However, the field varies so slowly that we can neglect displacement currents; that is, we are examining only a quasi-stationary model of the field.

Figure 5.6. Model used in the problem of layered-earth excitation by an arbitrary system of magnetospheric currents.

In accord with Eq. (4.123), outside the region in which source currents are flowing, in the atmosphere and within the j-th layer of the earth, the magnetic and electric fields satisfy the equations:

(5.157) $\begin{array}{cc}\hfill {\mathrm{\nabla }}^2\mathbf{H}+k_j^2\mathbf{H}& =0,\hfill \\ \hfill {\mathrm{\nabla }}^2\mathbf{E}+k_j^2\mathbf{E}& =0,\hfill \end{array}$

where $j=0,1,2,....,N$ and $k_j^2=i\omega {\mu }_0{\sigma }_j$ .

Boundary conditions at each boundary between two layers include continuity of all components of the magnetic field H and of the tangential components of the electric field E:

(5.158) $H_{x,y,z{|}_{z=z_j-0}}=H_{x,y,z{|}_{z=z_j+0}},$

(5.159) $E_{x,y{|}_{z=z_j-0}}=E_{x,y{|}_{z=z_j+0}},$

where $j=1,2,...,N-1$ and

$z_j=\sum _{\ell =1}^j{d}_{\ell },z_0=0.$

The vertical component of the electric field exhibits a discontinuity at each boundary specified as:

(5.160) ${\sigma }_j{E}_z{|}_{z=z_j-0}={\sigma }_{j+1}{E}_z{|}_{z=z_j+0}.$

In addition, on the strength of Eq. (4.120)

$\begin{array}{cc}\hfill div\mathbf{H}& =0,\hfill \\ \hfill div\mathbf{E}& =0,\hfill \end{array}$

we can claim the continuity of the vertical derivatives of the vertical components of the electric and magnetic fields at each boundary:

(5.161) ${\frac{\partial H_z}{\partial z},\frac{\partial E_z}{\partial z}|}_{z=z_j-0}={\frac{\partial H_z}{\partial z},\frac{\partial E_z}{\partial z}|}_{z=z_j+0}.$

At great depths, the fields E and H tend to zero. In studying the electromagnetic field in this horizontally layered model, it is most convenient to work in the spatial wave number domain.

We can express the magnetic and electric fields as Fourier integrals:

(5.162) $\begin{array}{cc}\hfill \mathbf{H}& =\frac{1}{4{\pi }^2}{\iint }_{-\infty }^{\infty }\mathbf{h}e^{-i(k_{x}x}{}^{+k_{y}y}{}^{)}d{k}_{x}d{k}_y,\hfill \\ \hfill \mathbf{E}& =\frac{1}{4{\pi }^2}{\iint }_{-\infty }^{\infty }\mathbf{e}e^{-i(k_{x}x}{}^{+k_{y}y}{}^{)}d{k}_{x}d{k}_y,\hfill \end{array}$

where $k_x,k_y$ are spatial frequencies along the x and y axes: $\mathbf{h}=\mathbf{h}(k_x,k_y,z)$ , $\mathbf{e}=\mathbf{e}(k_x,k_y,z)$ are the spatial spectrums of the magnetic and electric fields, respectively, and related to H and E through the inverse Fourier transforms:

(5.163) $\begin{array}{cc}\hfill \mathbf{h}& ={\iint }_{-\infty }^{\infty }\mathbf{H}e^{i(k_{x}x}{}^{+k_{y}y}{}^{)}d{k}_{x}d{k}_y,\hfill \\ \hfill \mathbf{e}& ={\iint }_{-\infty }^{\infty }\mathbf{E}e^{i(k_{x}x}{}^{+k_{y}y}{}^{)}d{k}_{x}d{k}_y.\hfill \end{array}$

On the basis of Eq. (5.157) and the properties of the spatial spectrums (5.56) and (5.57), the Helmholtz equation is satisfied in each layer and in the atmosphere:

(5.164) $\begin{array}{cc}\hfill {\mathbf{h}}^{″}& =n_j^2\mathbf{h},\hfill \\ \hfill {\mathbf{e}}^{″}& =n_j^2\mathbf{e},\hfill \end{array}$

where the primes designate differentiation with respect to z, $n_j={(k_x^2+k_y^2-k_j^2)}^{1/2}$ , and we choose that expression for the root for which

(5.165) $Re(n_j)>0.$

In accord with Eq. (5.158) through Eq. (5.162):

(5.166) $h_{x,y,z}{|}_{z=z_j-0}=h_{x,y,z}{|}_{z=z_j+0},$

(5.167) $e_{x,y}{|}_{z=z_j-0}=e_{x,y}{|}_{z=z_j+0},$

(5.168) ${\sigma }_j{e}_z{|}_{z=z_j-0}={\sigma }_{j+1}{e}_z{|}_{z=z_j+0},$

(5.169) $h_z^{\prime }{|}_{z=z_j-0}=h_z^{\prime }{|}_{z=z_j+0},$

(5.170) $e_z^{\prime }{|}_{z=z_j-0}=e_z^{\prime }{|}_{z=z_j+0}.$

The spatial spectrums h and e approach zero as z becomes infinite.

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Absorbing boundary conditions

David Pardo , ... Victor M. Calo , in Modeling of Resistivity and Acoustic Borehole Logging Measurements Using Finite Element Methods, 2021

7.5 PML in Cartesian coordinate system

The PML construction process in Cartesian coordinates and the corresponding weak formulation is straightforward to extend to the multi-dimensional case. Section 7.3.2 describes the construction in a 1D case. As before, the PML construction assumes the analyticity of the solution in the direction of PML, extends the equations to the complex domain, performs the analytical continuation of the solution in the complex space, and defines a complex mapping that expresses the complex stretching of the coordinates. Formally, we can define a smooth bijective map

(7.16) $\tilde{\mathbf{T}}:{\mathbb{R}}^n\ni \mathit{x}\to \tilde{\mathit{x}}\in {\mathbb{C}}^n(\text{alternatively}{\tilde{x}}^i={\tilde{x}}^i(\mathit{x})={\tilde{x}}^i(x^j)),$

where n denotes the dimension of the space. The corresponding Jacobian matrix and its determinant are

(7.17) $\tilde{\mathbf{J}},{(\tilde{J})}_{\cdot j}^i=\frac{\partial {\hat{x}}^i}{\partial x^j},\tilde{J}=\left|\tilde{\mathbf{J}}\right|=\det (\tilde{\mathbf{J}}).$

This definition of a complex mapping covers all possible cases of partial stretchings in less than n variables, as well as the axial complexification of particular coordinates, i.e., ${\tilde{x}}^i={\tilde{x}}^i(x^i)$ , where stretching of one coordinate depends solely on that coordinate. The latter case is the most frequently encountered in the literature, and the resulting method is called the Uniaxial PML. The additional dependence of the stretched coordinate on other coordinates is the key idea of the Multiaxial-PML (M-PML) proposed first by Meza-Fajardo and Papageorgiou [237].

Next, we rewrite the differential equations to transfer the complex coordinate differentiation into the differentiation with respect to the real ones. To do so, we use the appropriate Piola transforms defined in the previous section for parametric finite elements. It is enough to substitute the map (7.4) and its Jacobian (7.5) with definitions (7.16) and (7.17), respectively, and finally use transformations given in (7.7). Below, we write them explicitly

(7.18)

as well as the transformations for a volume and surface elements

$\text{d}\tilde{\mathit{x}}=\tilde{J}\text{d}\mathit{x},\text{d}\tilde{\mathrm{\Gamma }}=\left|\tilde{J}{\tilde{\mathbf{J}}}^{-T}\mathit{n}\right|\text{d}\mathrm{\Gamma }.$

In a Cartesian coordinate system, due to the constant basis vectors, the calculation of the gradient of a vector field (which contributes later to the strain tensor; components of the vector field are elements of energy space $H^1$ ) is particularly simple, namely

${\mathbf{\nabla }}_{\tilde{x}}\tilde{\mathbf{u}}=\frac{\partial \tilde{\mathbf{u}}}{\partial {\tilde{x}}^j}\otimes {\mathbf{e}}_j=\frac{\partial {\tilde{u}}^i}{\partial {\tilde{x}}^j}{\mathbf{e}}_i\otimes {\mathbf{e}}_j=\frac{\partial u^i}{\partial x^k}\frac{\partial x^k}{\partial {\tilde{x}}^j}{\mathbf{e}}_i\otimes {\mathbf{e}}_j={({\tilde{J}}_x^{-1})}_{\cdot j}^k\frac{\partial u^i}{\partial x^k}{\mathbf{e}}_i\otimes {\mathbf{e}}_j.$

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