p.15
Problem: If and are vectors, then , , are three numbers whose values are defined for any set of axes. Are they the components of a vector?
Solution: If are components of a vector they should transform like the components of a vector. See Table 2 p. 13
A transformation with
shows that are not the components of a vector.
Answer: No, because they do not transform like vector components.
p.22
p. 31
Problem:
Electrical conductivity tensor with axes , ,
is transformed to a new set of axes , , with the following angles
, , , .
Draw up a transformation table similar to (11) on p.9 and check that the sum of the squares of for each row and column is 1.
Solution: It follows from the given angles that . consists of the direction cosines of the angles:
Checking the columns and rows
Problem: Transform to and interpret the result.
Solution: A second-rank tensor transforms according to (22) on p.11: :
This can be reduced because we have some zero components in the electrical conductivity tensor :
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now it is just a matter of calculating the components
or written in array notation
Interpretation: The new set of axes is the principle axes of the tensor.
Problem: Radial vector with direction cosines in axes . Find the electrical conductivity in that direction with an analytical expression.
Solution: Get the vector components in the principle axes : Using the transformation for polar vector components leads to . Using equation (32) on p.25 gives us .
Problem: An electric field is applied in direction . Calculate the components of and the current density along the axes.
Solution: The components of the electric field are :
The components of the current density are :
Problem: determine the magnitude and direction of the current density .
Solution:
Magnitude:
Direction: . It lies in the , plane with angles and .
Problem: Repeat [6] and [7] but with the axes.
Solution:
Components:
Magnitude:
Direction: with angles and .