The differential element or just differential of a quantity refers to an infinitesimal change in said quantity, and is defined as the limit of a change in quantity as the change approaches zero.

dx=\lim_{\Delta x\to0}\Delta x

Differentials are useful in the definitions of both derivatives and integrals; for example, the derivative of y with respect to x is defined as

\frac{dy}{dx}=\lim_{\Delta x\to0}\frac{\Delta y}{\Delta x}

When transforming coordinates, the value of a differential element is computed using the determinant of the Jacobian matrix.

\prod_{i=1}^n dx_i=\frac{\part(x_1,\ldots,x_n)}{\part(u_1,\ldots,u_n)}\prod_{i=1}^n du_i=
\dfrac{\part x_1}{\part u_1}&\cdots&\dfrac{\part x_n}{\part u_1}\\
\dfrac{\part x_1}{\part u_n}&\cdots&\dfrac{\part x_n}{\part u_n}
\end{vmatrix}\prod_{i=1}^n du_i

Formulae for differential elements

Line elements

Area elements

Surface elements

Volume elements