Partial differential

The character ∂ is a stylized cursive d mainly used as a mathematical symbol, usually to denote a partial derivative such as . It is also used for boundary of a set, the boundary operator in a chain complex, and the conjugate of the Dolbeault operator on smooth differential forms over a complex manifold. It should be distinguished from other similar-looking symbols such as lowercase Greek letter delta or the lowercase Latin letter eth.

History

The symbol was introduced originally in 1770 by Nicolas de Condorcet, who used it for a partial differential, and adopted for the partial derivative by Adrien-Marie Legendre in 1786.
It represents a specialized cursive type of the letter d, just as the integral sign originates as a specialized type of a long s.
Use of the symbol was discontinued by Legendre, but it was taken up again by Carl Gustav Jacob Jacobi in 1841, whose usage became widely adopted.

Names and coding

The symbol is variously referred to as "partial", "curly d" or "Jacobi's delta", or as "del". It may also be pronounced simply "dee", "partial dee", "doh", "dow" or "die".
The Unicode character is accessed by HTML entities ∂ or ∂, and the equivalent LaTeX symbol is accessed by \partial.

Uses

∂ is also used to denote the following:

The Jacobian.
The boundary of a set in topology.
The boundary operator on a chain complex in homological algebra.
The boundary operator of a differential graded algebra.
The conjugate of the Dolbeault operator on complex differential forms.
The boundary ∂ of a set of vertices S in a graph is the set of edges leaving S, which defines a cut.