Problems for Appendix D

 

Index Notation for Vectors and Tensors

 

 

 

 

A.    

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. 

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  

 

 

 

 

(c) A.F. Bower, 2008
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