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area of the region
R
goes to zero, and we call this limit
)=
d
Φ
E
(
t
,
P
,
λ
dA
,
(26.70)
the irradiance at
P
.
Before proceeding further with this approach, we ask that you carefully review
the definition of the derivative. Typically when we write
df
/
dx
, we require that
f
1
h
(
f
(
x
+
h
)
be a function of a variable
x
, and that
0;
this limit is called the derivative. In the formulation above, there's no “variable”
A
, and
Φ
is certainly not a function of
A
. We can repair this problem by saying,
“Let
f
(
r
)
be the power arriving at a disk of radius
r
about
P
in the surface; the
area of that disk is
−
f
(
x
))
have a limit as
h
→
r
2
)
, which represents
the power arriving at the disk, per area. We then define
d
Φ
r
2
, and we can define
g
(
r
)=
f
(
r
)
π
/
(
π
dA
(
P
)
to be
g
(
0
)
.”
But one then must ask, “Would the result have been the same if I'd used a family
of shrinking squares rather than disks? What about other shapes? And is
g
obvi-
ously differentiable?” And for each new concept defined by a “derivative” like this
one, one has to reconsider the corresponding questions. The integral formulation
we have pursued makes one single assumption—the existence of an integrable
spectral radiance function
L
—and everything else follows from that.
Having made this critique of the “derivative” formulation, we should also men-
tion its advantages. One of these is that when you write
/
E
=
d
Φ
dA
(26.71)
you know that if you want to compute the power,
Φ
, you'll need to compute an
integral,
Φ=
EdA
(26.72)
=
d
Φ
dA
dA
,
(26.73)
with the obvious notion that “the
dA
s cancel.” In our experience, the presence
of various cosines makes this sort of computation fraught with peril. Our stu-
dents routinely draw false conclusions by being insufficiently precise about what
they mean in such derivations. On the other hand, once you have some expe-
rience with this notation, and have gotten past the usual mistakes, it's a great
convenience.
To continue with the standard derivative description, the radiant exitance is
also defined as a derivative of power with respect to area,
M
=
d
Φ
dA
,
(26.74)
where this time
Φ
means the power
leaving
the surface rather than arriving
there.
The radiant intensity (a term we haven't previously defined, and will not men-
tion again) is the derivative of flux with respect to solid angle,
I
=
d
Φ
d
,
(26.75)
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