Consider vectors A = 5e1 + (-3)e2 + 2e3 and B = 5e2 + 10e3, and tensor T = AB + BA
Transform the tensor into a coordinate system rotated 45 degrees about the z-axis. Show the components of both the original and rotated tensor.
Work:
original tensor: ai*ei
column 1: 5e1, -3e2, 2e3
column 2: 0e1, 5e2, 10e3
transformed tensor: ai = lij*aj
column 1: abar1, abar2, abar3
column 2: bbar1, bbar2, bbar3
By cross-multiplication:
AB = (5e1 - 3e2 + 2e3)*(0e1 + 5e2 + 10e3)
AB = (25e1e2 + 50e1e3 = 15e2e2 + 20e3e3 + 10e3e2 + 20e3e3)
BA = (0e1 + 5e2 + 10e3)*(5e1 - 3e2 + 2e3)
BA = (25e2e1 - 15e2e2 + 10e2e3 +50e3e1 - 30e3e2 + 20e3e3)
Solution:
T = AB + BA = 25e1e2 + 50 e1e3 -30e2e2 + 10e3e2 + 60e3e3 + 25e2e1 + 10e2e3 + 50e3e1 - 30e3e2)