7-46. Show that (A×B)·(C×D) =
∣∣
∣∣A·C A·D
B·C B·D∣∣
∣∣7-47. Let F =F(x, y, z) and G=G(x, y, z ) denote continuous functions which are
everywhere differentiable. Show that
(a) ∇(F+G) = ∇F+∇G (b) ∇(F G ) = F∇G+G∇F (c) ∇(
F
G)
=G∇F−F∇G
G^27-48. (Least-squares)
(a) Assume that (x 1 , y 1 ),(x 2 , y 2 ),... ,(xi, yi),... , (xN, yN) are N known distinct data
points that are plotted in the x, y plane along with a sketch of the straight
line
y=αx +βwhere αand β are constants to be determined. The situation is illustrated in
figure 7-28.
Figure 7-28. Linear least-squares fit.
Each data point (xi, yi) has associated with it an error Ei which is defined as the
difference between the y value of the line and the y value of the data point. For
example, the error associated with the data point (xi, yi)can be written
Ei=Ei(α, β ) ={yof line
}
−{yof data point
}Ei=Ei(α, β ) = (αx i+β)−yi.