∂

The "curly d" symbol is used in Mathematics for 2 separate purposes.

1. For partial derivatives. If you have a real function of more than one variable, say f:R³→R, you can look at derivatives of f(x,y,z) with respect to any of the 3 variables. ∂f/∂y, for instance, is defined as follows: for any real x,z, define f_x,z(y)=f(x,y,z). Then f_x,z:R→R is a plain ordinary real function, and we can look at its derivative df_x,z/dy at any point y; define ∂f/∂y=df_x,z/dy. Note that ∂f/∂x, ∂f/∂y, and ∂f/∂z may all be defined at a point (x,y,z) but f may still not be differentiable at (x,y,z). But if f is differentiable, the gradient will satisfy ∇f=(∂f/∂x,∂f/∂y,∂f/∂z).

   Used in this way, ∂f (and ∂y) will never appear on their own -- there is no concept of "differential" appearing here.

   You may say "no big deal" upon hearing this. You will be right, but hey! It's traditional!

2. For the boundary. If A⊂X is a subset of a topological space X, ∂A is the boundary of A: it is defined by ∂A=cl(A)\int(A), where cl(A) is the closure of X, and int(X) is the interior of X. (In a topological space where the topology is described by converging sequences,) any point b∈∂X has a sequence of points s_n∈X converging to it and a sequence of points t_n∉X converging to it. This is easy to see: ∂A is the intersection of the closure of A with the closure of its complement. (In general, you need to use nets instead of sequences in the above formulation.)

   ∂A exactly describes what we would intuitively consider the surface of A.

IMPORTANT NOTE

No, it's not a delta. Small delta looks like this: δ, while our tail twists the other way: ∂. It really is a "curly d".

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