Homework #3

    The coordinate system in the figure below has been transformed so that all three axes form equal angles with the original \(z\)-axis, and the new \(x'\)-axis is directly above the original \(x\)-axis.

    Note - this orientation corresponds to the orientation of many atomic crystals in face centered cubic (FCC) metals undergoing tension in the Z-direction. Common FCC metals are aluminum, copper, nickel, and steels at very high temperatures. Steel at room temperature is body centered cubic (BCC).

  1. What set of Roe angles will produce this transformation?

  2. What single rotation axis and angle will produce the transformation?

  3. A 2-D problem: Given \( \quad {\bf v} = (5,9) \) and \(\quad {\bf A} = \left[ \matrix { 5 & 2 \\ 2 & 3 } \right] \quad \), apply a 180° coordinate rotation to both and show that the signs of all the components change on one but not the other. Any insight on this?

  4. Show that    \( \nabla \cdot \nabla || {\bf x} || = 2 / || {\bf x} || \)