Here are some amusing exercises. I've given away some
of the answers.
1) Explain why multiplying by scalars is essentially a
wedge product.
Answer: Clifford Algebra has its roots in Grassman's
theory of _geometric extent_.
-- Scalars have no geometric extent
-- Vectors have geometric extent in one direction
-- Bivectors have geometric extent in two directions
-- et cetera.
The wedge is the "paint brushing" operator. If vector
P describes the spatial extent of the brush, and we
drag it in the direction of vector Q, it paints the
bivector P/\Q.
If we have a brush with no spatial extent (a sharp,
pointlike brush) represented by the scalar s, and drag
it in the direction of vector Q, it paints something
with extent in only one direction, namely the vector
sQ = s/\Q
2) Explain why the dot product of a scalar with anything
else is zero.
Partial answer: Restrict consideration to the product of
a scalar (s) with a vector (V). We want to uphold the rule
MV = M.V + M/\V
for any vector V and any multivector M, so in the particular
case where M is a scalar, we have
sV = s.V + s/\V
and since we have already decided that sV = s/\V that doesn't
leave a lot of choices for s.V.
See also equations 18 and 20 in Harke.
3a) Prove that
del . s = 0
for any scalar (i.e. grade=0 object) s.
3b) Prove that
del . (del . V) = 0
for any vector (i.e. grade=1 object) V.
3c) Prove that
del . (del . B) = 0
for any bivector (i.e. grade=2 object) B.
3d) [optional, extra credit] Prove that
del . (del . M) = 0
for any multivector M whatsoever.
4) Let J be the 4-vector representing the density & current
of electric charges. Explain why the equation
del . J = 0
represents conservation of charge.