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= \begin{vmatrix} \dfrac{\part x_1}{\part u_1}&\cdots&\dfrac{\part x_n}{\part u_1}\\ \vdots&\ddots&\vdots\\ \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

  • $ ds=\|\vec{r'}\|dt=\sqrt{\left(\tfrac{dx}{dt}\right)^2+\left(\tfrac{dy}{dt}\right)^2 + \left(\tfrac{dz}{dt}\right)^2}dt $ (scalar line element)
  • $ d\vec r=\vec{r'}dt=\frac{\vec{r'}}{\|\vec{r'}\|}ds $ (tangential vector element)
  • $ d\vec n=\frac{\vec T'}{\left\|\vec T'\right\|}ds $ (normal vector element)

Area elements

Surface elements

  • $ dS=\big\|\vec{r}_u\times\vec{r}_v\big\|dA $ (scalar surface element)
  • $ (\vec{r}_u\times\vec{r}_v)dA=\frac{\vec{r}_u\times\vec{r}_v}{\big\|\vec{r}_u\times\vec{r}_v\big\|}dS $ (normal vector surface element)

Volume elements

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