Jacobians: $\fcolorbox{white}{yellow}{ If $y=f(x)$, then $dy = f'(x) dx$}$.
If $x=x(s,t), \m y=y(s,t)$, then $\fcolorbox{white}{yellow}{$dA = dxdy = |J| dsdt$}$, where $J=$ Jacobian:
$\mm$ $J = \frac{\p (x,y)}{\p(s, t)} = \left| \begin{array}{cc} \frac{\p x}{\p s} & \frac{\p x}{\p t} \\ \frac{\p y}{\p s} & \frac{\p y}{\p t} \end{array}\right|$
3-d formula is similar: $\m$ If $u=u(r,s,t), \m v=v(r,s,t), \m w = w(r,s,t)$, then
$\mm$ $\fcolorbox{white}{yellow}{$dV = dudvdw = |J| dr dsdt$}$, where $J=$ Jacobian:
$\mm$ $J = \frac{\p (u,v,w)}{\p(r, s, t)} = \left|
\begin{array}{ccc} \frac{\p u}{\p r} & \frac{\p u}{\p s} & \frac{\p u}{\p t} \\
\frac{\p v}{\p r} & \frac{\p v}{\p s} & \frac{\p v}{\p t} \\
\frac{\p w}{\p r} & \frac{\p w}{\p s} & \frac{\p w}{\p t} \end{array}\right|$