Are $d^3r$, $d^3\vec{r}$, and $d^3{\bf r}$ the same as $dV$, the volume element? I've seen the term $d^3r$, $d^3\vec{r}$, and $d^3{\bf r}$ being used instead of $dV$. Are they exactly the same? Do they have a different connotation?
 A: Writing $\int\ldots d^3r$ for an integral over volume is just wrong. The $r$ is written to look like a scalar, which doesn't make sense.
Writing $d^3\vec{r}$ or $d^3{\bf r}$ is the same thing. Writing a vector with an arrow or in boldface means exactly the same thing.
The difference between writing $dV$ and $d^3{\bf r}$ is simply that with the latter, you could use ${\bf r}$ in the integrand. For instance, $\int {\bf r}\cdot {\bf h}\;d^3{\bf r}$ makes sense and makes it clear what is going on, whereas $\int {\bf r}\cdot {\bf h}\;dV$ is weird-looking and doesn't really make it clear that the integration is over a varying ${\bf r}$. We can have integrals like
$$\int \frac{d^3{\bf r}_1d^3{\bf r}_2}{|{\bf r}_1-{\bf r}_2|}$$
for something like the energy of a uniform charge distribution.
A: Actually they are the same.
But the first ones in terms of 'r' gives a better intuition of what variable we are varying.
A: In order to perform in integration over a certain volume, you can write in a general way $$ \text{volume} =\int \text{d} V. \tag{1}$$
If you do your calculations in three-dimensional space, you can write this in an equivalent way: $$ \text{volume} =\int \text{d} V = \int \text{d}^3 r= \int \text{d}^3 \textbf r= \int \text{d}^3 \vec r, \tag{2}$$
where $\vec r$ and $\textbf r$ can be used to emphasize that $r$ has more components than one (although this is already covered by the "$\text d^3$" ($\to$ matter of personal taste). 
The advantage of (2) over (1) is that you can distinguish between several variables. Let's look at an example: Say you want to integrate the function $f(\vec x,\vec y)$ over the volume. Something like  $$ ''\!\!\int\text dV\, f(\vec x,\vec y)\;'' \tag{3} $$
does not make too much sense. It would be better to clearly indicate which variable you want to integrate over, i.e. something like this: 
$$ \int\text d^3\vec y\; f(\vec x,\vec y) \tag{4}  $$
