Chapter 10, Vector calculus Video Solutions, Mathematical Methods for Physics and Engineering: A Comprehensive Guide

Chapter Questions

Problem 1

Evaluate the integral
$$
\int\left[\mathbf{a}(\dot{\mathbf{b}} \cdot \mathbf{a}+\mathbf{b} \cdot \mathbf{a})+\mathbf{a}(\mathbf{b} \cdot \mathbf{a})-2(\mathbf{a} \cdot \mathbf{a}) \mathbf{b}-\dot{\mathbf{b}}|\mathbf{a}|^{2}\right] d t
$$
in which $\dot{\mathbf{a}}, \dot{\mathbf{b}}$ are the derivatives of $\mathbf{a}, \mathbf{b}$ with respect to $t$.

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Problem 2

At time $t=0$, the vectors $\mathbf{E}$ and $\mathbf{B}$ are given by $\mathbf{E}=\mathbf{E}_{0}$ and $\mathbf{B}=\mathbf{B}_{0}$, where the fixed unit vectors $\mathbf{E}_{0}$ and $\mathbf{B}_{0}$ are orthogonal. The equations of motion are
$$
\begin{aligned}
&\frac{d \mathbf{E}}{d t}=\mathbf{E}_{0}+\mathbf{B} \times \mathbf{E}_{0} \\
&\frac{d \mathbf{B}}{d t}=\mathbf{B}_{0}+\mathbf{E} \times \mathbf{B}_{0}
\end{aligned}
$$
Find $\mathbf{E}$ and $\mathbf{B}$ at a general time $t$, showing that after a long time the directions of $\mathbf{E}$ and $\mathbf{B}$ have almost interchanged.

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Problem 3

The general equation of motion of a (non-relativistic) particle of mass $m$ and charge $q$ when it is placed in a region where there is a magnetic field $\mathbf{B}$ and an electric field $\mathbf{E}$ is
$$
m \ddot{\mathbf{r}}=q(\mathbf{E}+\dot{\mathbf{r}} \times \mathbf{B})
$$
here $\mathbf{r}$ is the position of the particle at time $t$ and $\dot{\mathbf{r}}=d \mathbf{r} / d t$ etc. Write this as three separate equations in terms of the Cartesian components of the vectors involved.

For the simple case of crossed uniform fields $\mathbf{E}=E \mathbf{i}, \mathbf{B}=B \mathbf{j}$ in which the particle starts from the origin at $t=0$ with $\dot{\mathbf{r}}=v_{0} \mathbf{k}$, find the equations of motion and show the following:
(a) if $v_{0}=E / B$ then the particle continues its initial motion;
(b) if $v_{0}=0$ then the particle follows the space curve given in terms of the parameter $\xi$ by
$$
x=\frac{m E}{B^{2} q}(1-\cos \xi), \quad y=0, \quad z=\frac{m E}{B^{2} q}(\xi-\sin \xi)
$$
Interpret this curve geometrically and relate $\xi$ to $t$. Show that the total distance travelled by the particle after time $t$ is
$$
\frac{2 E}{B} \int_{0}^{t}\left|\sin \frac{B q t^{\prime}}{2 m}\right| d t^{\prime}
$$

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Problem 4

Use vector methods to find the maximum angle to the horizontal at which a stone may be thrown so as to ensure that it is always moving away from the thrower.

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Problem 5

If two systems of coordinates with a common origin $O$ are rotating with respect to each other, the measured accelerations differ in the two systems. Denoting by $\mathbf{r}$ and $\mathbf{r}^{\prime}$ position vectors in frames $O X Y Z$ and $O X^{\prime} Y^{\prime} Z^{\prime}$ respectively, the connection between the two is
$$
\ddot{\mathbf{r}}^{\prime}=\ddot{\mathbf{r}}+\dot{\omega} \times \mathbf{r}+2 \omega \times \dot{\mathbf{r}}+\omega \times(\omega \times \mathbf{r})
$$
where $\omega$ is the angular velocity vector of the rotation of $O X Y Z$ with respect to $O X^{\prime} Y^{\prime} Z^{\prime}$ (taken as fixed). The third term on the RHS is known as the Coriolis acceleration, whilst the final term gives rise to a centrifugal force.

Consider the application of this result to the firing of a shell of mass $m$ from a stationary ship on the steadily rotating earth, working to the first order in $\omega\left(=7.3 \times 10^{-5} \mathrm{rad} \mathrm{s}^{-1}\right)$. If the shell is fired with velocity $\mathbf{v}$ at time $t=0$ and only reaches a height that is small compared to the radius of the earth, show that its acceleration, as recorded on the ship, is given approximately by
$$
\ddot{\mathbf{r}}=\mathbf{g}-2 \omega \times(\mathbf{v}+\mathbf{g} t)
$$
where $m \mathbf{g}$ is the weight of the shell measured on the ship's deck.
The shell is fired at another stationary ship (a distance $\mathbf{s}$ away) and $\mathbf{v}$ is such that the shell would have hit its target had there been no Coriolis effect.
(a) Show that without the Coriolis effect the time of flight of the shell would have been $\tau=-2 \mathbf{g} \cdot \mathbf{v} / g^{2}$
(b) Show further that when the shell actually hits the sea it is off target by approximately
$$
\frac{2 \tau}{g^{2}}[(\mathbf{g} \times \omega) \cdot \mathbf{v}](\mathbf{g} \tau+\mathbf{v})-(\omega \times \mathbf{v}) \tau^{2}-\frac{1}{3}(\omega \times \mathbf{g}) \tau^{3}
$$
(c) Estimate the order of magnitude $\Delta$ of this miss for a shell for which $v=300$ $\mathrm{m} \mathrm{s}^{-1}$, firing close to its maximum range ( $\mathbf{v}$ makes an angle of $\pi / 4$ with the vertical) in a northerly direction, whilst the ship is stationed at latitude $45^{\circ}$ North.

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Problem 6

Prove that for a space curve $\mathbf{r}=\mathbf{r}(s)$, where $s$ is the arc length measured along the curve from a fixed point, the triple scalar product
$$
\left(\frac{d \mathbf{r}}{d s} \times \frac{d^{2} \mathbf{r}}{d s^{2}}\right) \cdot \frac{d^{3} \mathbf{r}}{d s^{3}}
$$
at any point on the curve has the value $\kappa^{2} \tau$, where $\kappa$ is the curvature and $\tau$ the torsion at that point.

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Problem 7

For the twisted space curve $y^{3}+27 a x z-81 a^{2} y=0$, given parametrically by
$$
x=a u\left(3-u^{2}\right), \quad y=3 a u^{2}, \quad z=a u\left(3+u^{2}\right)
$$
show the following:
(a) that $d s / d u=3 \sqrt{2} a\left(1+u^{2}\right)$, where $s$ is the distance along the curve measured from the origin;
(b) that the length of the curve from the origin to the Cartesian point $(2 a, 3 a, 4 a)$ is $4 \sqrt{2 a}$
(c) that the radius of curvature at the point with parameter $u$ is $3 a\left(1+u^{2}\right)^{2}$;
(d) that the torsion $\tau$ and curvature $\kappa$ at a general point are equal;
(e) that any of the Frenet-Serret formulae that you have not already used directly are satisfied.

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Problem 8

The shape of the curving slip road joining two motorways that cross at right angles and are at vertical heights $z=0$ and $z=h$ can be approximated by the space curve
$$
\mathbf{r}=\frac{\sqrt{2} h}{\pi} \ln \cos \left(\frac{z \pi}{2 h}\right) \mathbf{i}+\frac{\sqrt{2} h}{\pi} \ln \sin \left(\frac{z \pi}{2 h}\right) \mathbf{j}+z \mathbf{k}
$$
Show that the radius of curvature $\rho$ of the curve is $(2 h / \pi) \operatorname{cosec}(z \pi / h)$ at height $z$ and that the torsion $\tau=-1 / \rho$. (To shorten the algebra, set $z=2 h \theta / \pi$ and use $\theta$ as the parameter.)

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Problem 9

In a magnetic field, field lines are curves to which the magnetic induction $\mathbf{B}$ is everywhere tangential. By evaluating $d \mathbf{B} / d s$, where $s$ is the distance measured along a field line, prove that the radius of curvature at any point on a line is given by
$$
\rho=\frac{B^{3}}{|\mathbf{B} \times(\mathbf{B} \cdot \nabla) \mathbf{B}|}
$$

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Problem 10

(a) Using the parameterization $x=u \cos \phi, y=u \sin \phi, z=u \cot \Omega$, find the sloping surface area of a right circular cone of semi-angle $\Omega$ whose base has radius $a$. Verify that it is equal to $\frac{1}{2} \times$ perimeter of the base $\times$ slope height.
(b) Using the same parameterization as in (a) for $x$ and $y$, and an appropriate choice for $z$, find the surface area between the planes $z=0$ and $z=Z$ of the paraboloid of revolution $z=\alpha\left(x^{2}+y^{2}\right)$

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Problem 11

(a) Parameterising the hyperboloid
$$
\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=1
$$
by $x=a \cos \theta \sec \phi, y=b \sin \theta \sec \phi, z=c \tan \phi$, show that an area element on its surface is
$$
d S=\sec ^{2} \phi\left[c^{2} \sec ^{2} \phi\left(b^{2} \cos ^{2} \theta+a^{2} \sin ^{2} \theta\right)+a^{2} b^{2} \tan ^{2} \phi\right]^{1 / 2} d \theta d \phi
$$
(b) Use this formula to show that the area of the curved surface $x^{2}+y^{2}-z^{2}=a^{2}$ between the planes $z=0$ and $z=2 a$ is
$$
\pi a^{2}\left(6+\frac{1}{\sqrt{2}} \sinh ^{-1} 2 \sqrt{2}\right)
$$

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Problem 12

For the function
$$
z(x, y)=\left(x^{2}-y^{2}\right) e^{-x^{2}-y^{2}}
$$
find the location(s) at which the steepest gradient occurs. What are the magnitude and direction of that gradient? (The algebra involved is easier if plane polar coordinates are used.)

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Problem 13

Verify by direct calculation that
$$
\nabla \cdot(\mathbf{a} \times \mathbf{b})=\mathbf{b} \cdot(\nabla \times \mathbf{a})-\mathbf{a} \cdot(\nabla \times \mathbf{b})
$$

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Problem 14

(a) Simplify
$$
\nabla \times \mathbf{a}(\nabla \cdot \mathbf{a})+\mathbf{a} \times[\nabla \times(\nabla \times \mathbf{a})]+\mathbf{a} \times \nabla^{2} \mathbf{a}
$$
(b) By explicitly writing out the terms in Cartesian coordinates prove that
$$
[\mathbf{c} \cdot(\mathbf{b} \cdot \nabla)-\mathbf{b} \cdot(\mathbf{c} \cdot \nabla)] \mathbf{a}=(\nabla \times \mathbf{a}) \cdot(\mathbf{b} \times \mathbf{c})
$$
(c) Prove that $\mathbf{a} \times(\nabla \times \mathbf{a})=\nabla\left(\frac{1}{2} a^{2}\right)-(\mathbf{a} \cdot \nabla) \mathbf{a}$.

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Problem 15

Evaluate the Laplacian of the function
$$
\psi(x, y, z)=\frac{z x^{2}}{x^{2}+y^{2}+z^{2}}
$$
(a) directly in Cartesian coordinates, and (b) after changing to a spherical polar coordinate system. Verify that, as they must, the two methods give the same result.

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Problem 16

Verify that $(10.42)$ is valid for each component separately when $\mathbf{a}$ is the Cartesian vector $x^{2} y \mathbf{i}+x y z \mathbf{j}+z^{2} y \mathbf{k}$, by showing that each side of the equation is equal to $z \mathbf{i}+(2 x+2 z) \mathbf{j}+x \mathbf{k}$

Manik P.

Numerade Educator

Problem 17

The (Maxwell) relationship between a time-independent magnetic field $\mathbf{B}$ and the current density $\mathbf{J}$ (measured in S.I. units in $\mathrm{A} \mathrm{m}^{-2}$ ) producing it,
$$
\nabla \times \mathbf{B}=\mu_{0} \mathbf{J}
$$
can be applied to a long cylinder of conducting ionised gas which, in cylindrical polar coordinates, occupies the region $\rho<a$.
(a) Show that a uniform current density $(0, C, 0)$ and a magnetic field $(0,0, B)$, with $B$ constant $\left(=B_{0}\right)$ for $\rho>a$ and $B=B(\rho)$ for $\rho<a$, are consistent with this equation. Obtain expressions for $C$ and $B(\rho)$ in terms of $B_{0}$ and $a$, given that $\mathbf{B}$ is continuous at $\rho=a$
(b) The magnetic field can be expressed as $\mathbf{B}=\nabla \times \mathbf{A}$, where $\mathbf{A}$ is known as the vector potential. Show that a suitable $\mathbf{A}$ can be found which has only one non-vanishing component, $A_{\phi}(\rho)$, and obtain explicit expressions for $A_{\phi}(\rho)$ for both $\rho<a$ and $\rho>a$. Like $\mathbf{B}$, the vector potential is continuous at $\rho=a$
(c) The gas pressure $p(\rho)$ satisfies the hydrostatic equation $\nabla p=\mathbf{J} \times \mathbf{B}$ and vanishes at the outer wall of the cylinder. Find a general expression for $p$

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Problem 18

(a) For cylindrical polar coordinates $\rho, \phi, z$ evaluate the derivatives of the three unit vectors with respect to each of the coordinates, showing that only $\partial \hat{\mathbf{e}}_{\rho} / \partial \phi$ and $\partial \hat{\mathbf{e}}_{\phi} / \partial \phi$ are non-zero.
(i) Hence evaluate $\nabla^{2} \mathbf{a}$ when $\mathbf{a}$ is the vector $\hat{\mathbf{e}}_{\rho}$, i.e. a vector of unit magnitude everywhere directed radially outwards from the $z$-axis.
(ii) Note that it is trivially obvious that $\nabla \times \mathbf{a}=\mathbf{0}$ and hence that equation $(10.41)$ requires that $\dot{\nabla}(\nabla \cdot \mathbf{a})=\nabla^{2} \mathbf{a}$.
(iii) Evaluate $\nabla(\nabla \cdot \mathbf{a})$ and show that the latter equation holds, but that
$$
[\nabla(\nabla \cdot \mathbf{a})]_{\rho} \neq \nabla^{2} a_{\rho}
$$
(b) Rework the same problem in Cartesian coordinates (where, as it happens, the algebra is more complicated).

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Problem 19

Maxwell's equations for electromagnetism in free space (i.e. in the absence of charges, currents and dielectric or magnetic media) can be written
(i) $\nabla \cdot \mathbf{B}=0$
(ii) $\nabla \cdot \mathbf{E}=0$
(iii) $\nabla \times \mathbf{E}+\frac{\partial \mathbf{B}}{\partial t}=\mathbf{0}$
(iv) $\nabla \times \mathbf{B}-\frac{1}{c^{2}} \frac{\partial \mathbf{E}}{\partial t}=\mathbf{0}$.
A vector $\mathbf{A}$ is defined by $\mathbf{B}=\nabla \times \mathbf{A}$, and a scalar $\phi$ by $\mathbf{E}=-\nabla \phi-\partial \mathbf{A} / \partial t$. Show that if the condition
(v) $\nabla \cdot \mathbf{A}+\frac{1}{c^{2}} \frac{\partial \phi}{\partial t}=0$
is imposed (this is known as choosing the Lorenz gauge), then both $\mathbf{A}$ and $\phi$ satisfy the wave equations
(vi) $\nabla^{2} \phi-\frac{1}{c^{2}} \frac{\partial^{2} \phi}{\partial t^{2}}=0$,
(vii) $\quad \nabla^{2} \mathbf{A}-\frac{1}{c^{2}} \frac{\partial^{2} \mathbf{A}}{\partial t^{2}}=\mathbf{0}$
The reader is invited to proceed as follows.
(a) Verify that the expressions for $\mathbf{B}$ and $\mathbf{E}$ in terms of $\mathbf{A}$ and $\phi$ are consistent with (i) and (iii).
(b) Substitute for $\mathbf{E}$ in (ii) and use the derivative with respect to time of $(v)$ to eliminate $\mathbf{A}$ from the resulting expression. Hence obtain (vi).
(c) Substitute for $\mathbf{B}$ and $\mathbf{E}$ in (iv) in terms of $\mathbf{A}$ and $\phi .$ Then use the divergence of (v) to simplify the resulting equation and so obtain (vii).

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Problem 20

For a description in spherical polar coordinates with axial symmetry of the flow of a very viscous fluid, the components of the velocity field $\mathbf{u}$ are given in terms of the stream function $\psi$ by
$$
u_{r}=\frac{1}{r^{2} \sin \theta} \frac{\partial \psi}{\partial \theta}, \quad u_{\theta}=\frac{-1}{r \sin \theta} \frac{\partial \psi}{\partial r}
$$
Find an explicit expression for the differential operator $E$ defined by
$$
E \psi=-(r \sin \theta)(\nabla \times \mathbf{u})_{\phi}
$$
The stream function satisfies the equation of motion $E^{2} \psi=0$ and, for the flow of a fluid past a sphere, takes the form $\psi(r, \theta)=f(r) \sin ^{2} \theta$. Show that $f(r)$ satisfies the (ordinary) differential equation
$$
r^{4} f^{(4)}-4 r^{2} f^{\prime \prime}+8 r f^{\prime}-8 f=0
$$

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Problem 21

Paraboloidal coordinates $u, v, \phi$ are defined in terms of Cartesian coordinates by
$$
x=u v \cos \phi, \quad y=u v \sin \phi, \quad z=\frac{1}{2}\left(u^{2}-v^{2}\right)
$$
Identify the coordinate surfaces in the $u, v, \phi$ system. Verify that each coordinate surface $(u=$ constant, say) intersects every coordinate surface on which one of the other two coordinates $(v$, say $)$ is constant. Show further that the system of coordinates is an orthogonal one and determine its scale factors. Prove that the $u$-component of $\nabla \times \mathbf{a}$ is given by
$$
\frac{1}{\left(u^{2}+v^{2}\right)^{1 / 2}}\left(\frac{a_{\phi}}{v}+\frac{\partial a_{\phi}}{\partial v}\right)-\frac{1}{u v} \frac{\partial a_{v}}{\partial \phi}
$$

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Problem 22

Non-orthogonal curvilinear coordinates are difficult to work with and should be avoided if at all possible, but the following example is provided to illustrate the content of section $10.10$

In a new coordinate system for the region of space in which the Cartesian coordinate $z$ satisfies $z \geq 0$, the position of a point $\mathbf{r}$ is given by $\left(\alpha_{1}, \alpha_{2}, R\right)$, where $\alpha_{1}$ and $\alpha_{2}$ are respectively the cosines of the angles made by $\mathbf{r}$ with the $x$ - and $y-$ coordinate axes of a Cartesian system and $R=|\mathbf{r}| .$ The ranges are $-1 \leq \alpha_{i} \leq 1$, $0 \leq R<\infty$
(a) Express $\mathbf{r}$ in terms of $\alpha_{1}, \alpha_{2}, R$ and the unit Cartesian vectors $\mathbf{i}, \mathbf{j}, \mathbf{k}$.
(b) Obtain expressions for the vectors $\mathbf{e}_{i}\left(=\partial \mathbf{r} / \partial \alpha_{1}, \ldots\right)$ and hence show that the scale factors $h_{i}$ are given by
$$
h_{1}=\frac{R\left(1-\alpha_{2}^{2}\right)^{1 / 2}}{\left(1-\alpha_{1}^{2}-\alpha_{2}^{2}\right)^{1 / 2}}, \quad h_{2}=\frac{R\left(1-\alpha_{1}^{2}\right)^{1 / 2}}{\left(1-\alpha_{1}^{2}-\alpha_{2}^{2}\right)^{1 / 2}}, \quad h_{3}=1
$$
(c) Verify formally that the system is not an orthogonal one.
(d) Show that the volume element of the coordinate system is
$$
d V=\frac{R^{2} d \alpha_{1} d \alpha_{2} d R}{\left(1-\alpha_{1}^{2}-\alpha_{2}^{2}\right)^{1 / 2}}
$$
and demonstrate that this is always less than or equal to the corresponding expression for an orthogonal curvilinear system.
(e) Calculate the expression for $(d s)^{2}$ for the system, and show that it differs from that for the corresponding orthogonal system by
$$
\frac{2 \alpha_{1} \alpha_{2} R^{2}}{1-\alpha_{1}^{2}-\alpha_{2}^{2}} d \alpha_{1} d \alpha_{2}
$$

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Problem 23

Hyperbolic coordinates $u, v, \phi$ are defined in terms of Cartesian coordinates by
$$
x=\cosh u \cos v \cos \phi, \quad y=\cosh u \cos v \sin \phi, \quad z=\sinh u \sin v
$$
Sketch the coordinate curves in the $\phi=0$ plane, showing that far from the origin they become concentric circles and radial lines. In particular, identify the curves $u=0, v=0, v=\pi / 2$ and $v=\pi .$ Calculate the tangent vectors at a general point, show that they are mutually orthogonal and deduce that the appropriate scale factors are
$$
h_{u}=h_{v}=\left(\cosh ^{2} u-\cos ^{2} v\right)^{1 / 2}, \quad h_{\phi}=\cosh u \cos v
$$
Find the most general function $\psi(u)$ of $u$ only that satisfies Laplace's equation $\nabla^{2} \psi=0$

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Problem 24

In a Cartesian system, $A$ and $B$ are the points $(0,0,-1)$ and $(0,0,1)$ respectively. In a new coordinate system a general point $P$ is given by $\left(u_{1}, u_{2}, u_{3}\right)$ with $u_{1}=\frac{1}{2}\left(r_{1}+r_{2}\right), u_{2}=\frac{1}{2}\left(r_{1}-r_{2}\right), u_{3}=\phi ;$ here $r_{1}$ and $r_{2}$ are the distances $A P$ and $B P$ and $\phi$ is the angle between the plane $A B P$ and $y=0$.
(a) Express $z$ and the perpendicular distance $\rho$ from $P$ to the $z$-axis in terms of $u_{1}, u_{2}, u_{3}$
(b) Evaluate $\partial x / \partial u_{i}, \partial y / \partial u_{i}, \partial z / \partial u_{i}$, for $i=1,2,3$.
(c) Find the Cartesian components of $\hat{\mathbf{u}}_{j}$ and hence show that the new coordinates are mutually orthogonal. Evaluate the scale factors and the infinitesimal volume element in the new coordinate system.
(d) Determine and sketch the forms of the surfaces $u_{i}=$ constant.
(e) Find the most general function $f$ of $u_{1}$ only that satisfies $\nabla^{2} f=0$

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