What is the difference between Δ, δ, d and ∂
when used in math and physics?
I have never quite understood the difference between Δ, δ, d and ∂ when it comes different
equations and formulas. I've understood that has to do with derivatives and infitesimal changes
and whatnot, but I would like to get a better understanding of the differences.
'Δ' -- means a change in some variable. This makes it a difference operation: Δt = t2 - t1.
'd' -- means an 'infinitesimal' change, or a "differential form." It's kind of like a limit as Δt -> 0, but
it is compatible with relative rates or different kinds of limits, so that the notion of a derivative is
preserved (in the form dy/dx, for example.)
'∂' -- means a partial differential. It's basically the same as d, except it also tells you that there are
other related variables that are being held constant. In other words, it is never a complete picture.
It's typically used in partial derivatives -- derivatives that are only in one dimension of a larger
dimensional space.
'δ' is just a lowercase Δ, and its meaning depends on context. Usually it means an 'infinitesimal'
change like d, but when you don't want to talk about differential forms or derivatives using the
other notations.
So, they all mean roughly the same thing by themselves (except for Δ, which is a simple
difference.) Together, they help provide a context and subtle details about the problems being
addressed. Most of these are a matter of convention.
∇ meaning
Del, or nabla, is an operator used in mathematics, in particular in vector calculus, as
a vector differential operator, usually represented by the nabla symbol ∇. When applied to
a function defined on a one-dimensional domain, it denotes its standard derivative as defined
in calculus. When applied to a field (a function defined on a multi-dimensional domain), it may
denote the gradient (locally steepest slope) of a scalar field (or sometimes of a vector field, as in
the Navier–Stokes equations), the divergence of a vector field, or the curl (rotation) of a vector
field, depending on the way it is applied.
Strictly speaking, del is not a specific operator, but rather a convenient mathematical notation for
those three operators, that makes many equations easier to write and remember. The del symbol
can be interpreted as a vector of partial derivative operators, and its three possible meanings—
gradient, divergence, and curl—can be formally viewed as the product with a scalar, a dot
product, and a cross product, respectively, of the del "operator" with the field. These formal
products do not necessarily commute with other operators or products. These three uses,
detailed below, are summarized as:
