What is meant by $d(x,y)$ in Fubini’s theorem? According to Wolfram MathWorld, Fubini’s theorem takes a multiple integral over a region $R=\{(x,y):x\in[a,b]\wedge y\in[c,d]\}$ and turns it into an iterated integral by the relationship
$$\iint_Rf(x,y)\,d(x,y)=\int_a^b\int_c^df(x,y)\,dy\,dx$$
I have never seen the differential $d(x,y)$ before. Normally, I see something of the nature $dA=dx\,dy$.
Could someone explain what $d(x,y)$ means and whence it came?
 A: Basically, $dA$, $d(x,y)$, $dxdy$ are all the same thing. It is actually better to write $dA$ or $d(x,y)$ in general to avoid confusion. But some authors write $dA = dxdy$.  
If you know measure theory, $\int f(x,y) \, d(x,y)$ means integration with respect to the product measure on $\mathbb{R}^2$. $d(x,y)$ is the differential of the product measure. 
You can also think about it in terms of Riemman integration (if you prefer that to measure theory). I recommend Chapter 14 of Apostol's Mathematical Analysis. It has an excellent treatment of Riemann integration in multiple variables. Let me give you some idea here:
Think of cutting $R$ up into $m$ rectangles. Let $\Delta A_i$ be the area of the $i$-th rectangle. Pick a sample point $(x_i^{\ast},y_i^{\ast})$ in each rectangle. If $f$ is positive, the volume trapped by the graph $z=f(x,y)$ above $R$ is approximately
$$
\sum_{i=1}^{m} f(x_i^{\ast},y_i^{\ast}) dA_i.
$$ 
If $f$ is not always positive, this sum represents a signed volume (just like a signed area in 1-D Calculus). If we take an appropriate limit as $m \to \infty$, we obtain the integral 
\begin{align}\label{1}\tag{1}
\int_R f(x,y) dxdy = \int_R f(x,y) d(x,y) = \int_R f(x,y) dA = \lim_{m \to \infty} \sum_{i=1}^{m} f(x_i^{\ast},y_i^{\ast}) \Delta A_i.  
\end{align}
On the other hand, the interated integral in terms of Riemann sums is 
\begin{align}
\label{2}\tag{2}
\int_{c}^{d} \int_{a}^{b} f(x,y) dxdy 
&= \int_{c}^{d} \left( \int_{a}^{b} f(x,y) dx \right) dy
= \int_{c}^{d} \left( \lim_{n \to \infty} \sum_{j=1}^{n} f(x_j^{\ast},y) \Delta x \right) dy \\
&= \lim_{N \to \infty} \sum_{k=1}^{N} \left( \lim_{n \to \infty} \sum_{j=1}^{n} f(x_j^{\ast},y_k^{\ast}) \Delta x \right) \Delta y.
\end{align}
Fubini's theorem is saying that these two expressions ((1) and (2)) are equal under certain conditions.
