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Problems
for Appendix D
Index
Notation for Vectors and Tensors
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A.
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B.
D.1 A geo-stationary satellite orbits the earth at
radius 41000
km in the equatorial plane, and is positioned at 0o
longitude. A satellite dish located in
Providence, Rhode
Island
(Longitude  ,
Latitude  ) is to be pointed at the satellite. In this problem, you will calculate the
angles  and  to position the satellite. Let {i,j,k}
be a Cartesian basis with origin at the center of the earth, k pointing
to the North Pole and i pointing towards the intersection of the
equator (0o latitude) and the Greenwich meridian (0o longitude). Define a
spherical-polar coordinate system  with basis vectors  in the usual way. Take the earth’s radius as 6000 km.
D.1.1
Write down the
values of  for Providence, Rhode
Island
D.1.2
Write down the
position vector of the satellite in the Cartesian {i,j,k}coordinate
system
D.1.3
Hence, find the
position vector of the satellite relative to the center of the earth in the  basis located at Providence, Rhode
Island.
D.1.4
Find the
position vector SP of the satellite relative to Providence, Rhode
Island, in
terms of basis vectors  located at Providence, Rhode
Island.
D.1.5
Find the
components of a unit vector parallel to SP, in terms of basis vectors  located at Providence, Rhode
Island.
D.1.6
Hence,
calculate the angles  and
D.2 Calculate the gradient and divergence of the
following vector fields
D.2.1
D.2.2
D.2.3
D.3 Show that the components of the gradient of a vector
field in spherical-polar coordinates is
D.4 In this problem you will derive the expression given
in Appendix D for the gradient operator associated with polar coordinates.
D.4.1
Consider a
scalar field . Write down an expression for the change df in f due to an infinitesimal change in the three coordinates ,
to first order in .
D.4.2
Write down an
expression for the change in position vector  due to an infinitesimal change in the three
coordinates ,
to first order in ,
expressing your answer as components in the  basis.
D.4.3
Hence, find
expressions for  in terms of
D.4.4
Finally,
substitute the result of (c) into the result of (a) to obtain an expression
relating df to . Rearrange the result into the form ,
and hence deduce the expression for the gradient operator.
D.5 In this problem you will derive the expression given
in Appendix D for the gradient operator associated with polar coordinates.
D.5.1
Consider a
scalar field . Write down an expression for the change df in f due to an infinitesimal change in the three coordinates ,
to first order in .
D.5.2
Write down an
expression for the change in position vector  due to an infinitesimal change in the three
coordinates ,
to first order in ,
expressing your answer as components in the  basis.
D.5.3
Hence, find
expressions for  in terms of
D.5.4
Finally,
substitute the result of (c) into the result of (a) to obtain an expression
relating df to . Rearrange the result into the form ,
and hence deduce the expression for the gradient operator.
D.6 Show that the components of the divergence of a
tensor field S in spherical-polar coordinates are
D.7 Show that the components of the gradient of a vector
field in cylindrical-polar coordinates are
D.8 Consider a rigid sphere, as shown in the
figure. The sphere is rotated through
an angle  about k
and has instantaneous angular velocity  about the k direction. Let  denote the spherical-polar coordinates of a
point in the sphere prior to deformation, and let  denote the spherical-polar basis vectors
associated with this point. Let  denote the spherical-polar coordinates of
the same point after deformation, and let  denote the corresponding basis vectors.Â
D.8.1
Write down the
deformation mapping relating  to
D.8.2
Write down the
velocity field in the sphere in terms of  and ,
expressing your answer as components in
D.8.3
Find the
spatial velocity gradient  as a function of ,
expressing your answer as components in .
D.8.4
Show that the
deformation gradient can be expressed as  and find a similar expression for .
(its enough just to show that material fibers parallel to the basis vectors
in the undeformed solid are parallel to the basis vectors in the deformed
solid)
D.8.5
Use the result
of (d) and (e) to verify that
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