If r is the position vector, a is any vector, and m is any constant vector, find
ID: 1369844 • Letter: I
Question
If r is the position vector, a is any vector, and m is any constant vector, find the following using these identities, but without resorting to any particular set of coordinates. Show all work.
a) div (mxr)
b) curl (mxr)
c) grad (m.r/r³)
d) div (rnr) [where r represents unit vector not position vector]
e) curl (rnr) [where r represents unit vector not position vector]
f) If fa = grad(g), show that a.(curl a) = 0, regardless of the functions f and g, except for f=0. (Hint: since the final expression requires you to know curl (a), start by taking the curl of both sides of the original expression.)
Explanation / Answer
a) div (mxr) = 0, because cross product between constant and vector is zero.
b) curl (mxr) = 0, because cross product between constant and vector is zero.
c) grad (m.r/r³) = m {d/dr[r/r3]} = m [d/dx {[x i + y j + z k] / [x2 + y2 + z2]3/2}]
Answer: m[[x2 + y2 + z2]3/2 - 2x ([x2 + y2 + z2]1/2) } /[x2 + y2 + z2]3] i^
+ m[[x2 + y2 + z2]3/2 - 2y ([x2 + y2 + z2]1/2) } /[x2 + y2 + z2]3] j^
+ m[[x2 + y2 + z2]3/2 - 2z ([x2 + y2 + z2]1/2) } /[x2 + y2 + z2]3] j^
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