GRADIENT, DIVERGENCE and CURL
76
(e) A x V O _ (2yz i - x2Yj + xz2k) x
(ai aj +
+
_a4) a
k)
Y
i
j
k
2yz
-x2y
xz2
ax
ay
az
a0 2 2 (-xY az - xz .ay)i
_
- (6x y2z2 + 2x3z5) i
a
+
a
a
a )k
(xz2 ax - 2yz az)j
+
(2yz ay + x 2y
(4x2yz5 - 12x2y2z3) j
+
(4x2yz4 + 4x3y2z3) k
+
ax
= A x Vq .
Comparison with (d) illustrates the result (A x V)
INVARIANCE
38. Two rectangular xyz and x'y'z' coordinate systems having the same origin are rotated with respect to each other. Derive the transformation equations between the coordinates of a point in the two systems. Let r and r' be the position vectors of any point P in the two systems (see figure on page 58). Then since
r=rf,
x+ y'j' + z' k'
(1)
=
xi + yj
Now for any vector A we have (Problem 20, Chapter 2), A
j'
+
i'
=
+
(A-k') k'
Then letting A = i, j, k in succession, (2)
i
=
i
=
k
=
(k . i') i'
(k j') j'
+
+
1111'
+
121jf
+
131k'
(j- k') k'
=
1121'
+
122 it
+
132k'
(k k') k'
=
113 i'
+
123 j'
+
133k'
Substituting equations (2) in (1) and equating coefficients of i', j', k' we find (3)
x' = 111x + 112Y + 113z,
y' = 121 x + 122 Y + 123 z ,
z' = 131X + 132Y + 133Z
the required transformation equations.
39. Prove
i' =
111 i + 112 j + 113 k
jf = 121i +122j +123k k' = 1311 + 132j + 133k For any vector A we have
A = (A i) i + (A. j) j + (A k) k .
Then letting A = i', j', k' in succession, 1'
_
jr
=
k'
_
(i' i) i
+
(i' j) j
+
(i' k) k
=
111i + 112i + 113k 121 i
(k' i) i
+
(k' j) i
+
(k' k) k
=
131 1
+ 122 j + 123 k + 132 i + 133 k