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I have a formula in a paper that I want to write out by hand, and it contains two "D"s, a normal D, $D$ in latex, and a 'mathcal', caligraphic D, $\mathcal{D}$ in latex.
What are some common/standard/convenient ways to write out a mathcal D by hand, so that it is obviously different from a 'normal' D?
You can try to imitate my calligraphy, follow the one in the other answers or develop your own.
http://www.fontlab.com/font-editor/fontographer/
To give a slightly flippant answer: put an extra loop somewhere. It doesn't really matter where you put the loop, a loop is necessary and sufficient for your audience to think it's calligraphic.
Per OP's request I'm posting my comment as an answer.
I would write $D$ in the same way which you use in handwriting when text consists only of capital letters (resembling $\mathrm{D}$), such as in the following picture, which I downloaded from here
And for $\mathcal D$ I would use the usual way for handwritten capital $D$, such as in the following picture, which I found here.
How curious; when I was younger lots of people were struggling to find a way to get the things they knew how to write on paper or on the blackboard into print in a recognisable way, now it is the other way around. Anyway, in writing there is no absolute uniformity and different people write the same thing in many different ways (when I started teaching in France, I had to get used to the fact that students here write $z$ in a way that to me looks perfectly like a $y$, but they write $y$ just a tad differently). Just make sure you write your $\mathcal D$ in a way that looks "handwritten" and somewhat resembles a "D", but sufficiently different from how you write $D$ (and much depends on what that is). Personally I tend to make a little loop at the bottom left of $\mathcal D$ to suggest it is written in one continued stroke (which is indeed they way I write it), while $D$ has two separate strokes.
I got this from my Topology professor:
For letters with a leading vertical line, make a mathcal with a prominent down-stroke. Start at the top of the line, go down in an arc, then come up to the left, so the vertical line of the D is kind of an elongated oval instead of a line. Then complete the D normally. Add some flourishes strategically to make it clear what letter it is, that part will vary with your individual handwriting; just practice a bit until it looks right.
This works for D, P, R, K, B, M, and N really well. It also kind of works for E, F, and T.
$\mathcal{D}$ is calligraphic, not gothic. $\mathfrak{D}$ is gothic. I learned to write calligraphic letters when I was 6 or 7 years old, but it is very common to switch to "normal" upper-case letters when growing up. I suggest you should look for your old exercise book :-)
Although you say two dees, you actually specify three: roman, italic, and calligraphic. I write roman letters (in mathematics) in a boxy upright angular style, with serifs where possible; italic letters are written in my natural style; and script or calligraphic letters are decorated with twirly or looped ends.
