Appendix D Coordinate systems Rectangular coordinate system Coordinate variables l=x,-0<x< Vector algebra =XAr+yAy+2A A·B=AxBx+AyBy+A2B A×B=A,A,A Br B, B. Dyadic representation Rarvy + Taxi+ gay&+yayy y+yay 2+ a= Xay yay+ ia,= axx+avy+a2 i a arx+ axyy +axzZ a, arx+ ayyy +azz +ayx+azz =axi+ayyy+axy Z ax2 i+avy+azz @2001 by CRC Press LLCAppendix D Coordinate systems Rectangular coordinate system Coordinate variables u = x, −∞ < x < ∞ (D.1) v = y, −∞ < y < ∞ (D.2) w = z, −∞ < z < ∞ (D.3) Vector algebra A = xˆ Ax + yˆ Ay + zˆ Az (D.4) A · B = Ax Bx + Ay By + Az Bz (D.5) A × B = xˆ yˆ zˆ Ax Ay Az Bx By Bz (D.6) Dyadic representation a¯ = xˆaxxxˆ + xˆaxyyˆ + xˆaxzzˆ + + yˆayxxˆ + yˆayyyˆ + yˆayzzˆ + + zˆazxxˆ + zˆazyyˆ + zˆazzzˆ (D.7) a¯ = xaˆ  x + yaˆ  y + zaˆ  z = axxˆ + ayyˆ + azzˆ (D.8) a x = axxxˆ + axyyˆ + axzzˆ (D.9) a y = ayxxˆ + ayyyˆ + ayzzˆ (D.10) a z = azxxˆ + azyyˆ + azzzˆ (D.11) ax = axxxˆ + ayxyˆ + azx zˆ (D.12) ay = axyxˆ + ayyyˆ + azy zˆ (D.13) az = axzxˆ + ayzyˆ + azzzˆ (D.14)