Problem 3. Consider the Cartesian (x, y, z) and spherical (r, 0, 0) coordinate systems shown below. We can see
that:
The unit vectors are related by:
er
eg
eo
Or, they can be inverted to obtain:
=
=
=
T
x
=
= r sin cos
Y
r sin
z = r cose
x, ex
ex =
sin cos de, + cos
ey
sin sin pe, + cos
ez = cose, sin eg
Show that the gradient of a scalar, a, is give by:
да
Ər
√x² + y² + 2²
sin
cos pe
+ sin
cos cos pe
+ cos
- sin de + cos de
Va er
+
1 да
r 20
z, e₂
Ф
sin o
Ө
eo +
sin pey + cos lez
sin pey - sinfez
cos peg - sin de
sin pee + cos de
1
r sin
r, e,
ed
да
do
-eo
y, eyn