The diff_table command is basically the inverse facility of PDEtools[declare]: it permits entering (input) expressions and their derivatives using compact mathematical notation without using macros or aliases. The notation implemented by diff_table is the jet notation also used by the DifferentialAlgebra package and represents a remarkable saving in redundant typing on input. diff_table also works with anticommutative variables set using the Physics package.

$\mathrm{with}\left(\mathrm{PDEtools}\,\mathrm{diff\_table}\right)\:$

$U≔\mathrm{diff\_table}\left(u\left(x\,y\,t\right)\right)\:$

$V≔\mathrm{diff\_table}\left(v\left(x\,y\,t\right)\right)\:$

$\mathrm{e1}≔U\left[y,t\right]+V\left[x,x\right]+U\left[x\right]U\left[y\right]+U\left[\right]U\left[x,y\right]$

$\mathrm{e1}≔u\left(x\,y\,t\right)\left(\frac{{\partial }^2}{\partial x\partial y}u\left(x\,y\,t\right)\right)+\left(\frac{\partial }{\partial x}u\left(x\,y\,t\right)\right)\left(\frac{\partial }{\partial y}u\left(x\,y\,t\right)\right)+\frac{{\partial }^2}{\partial t\partial y}u\left(x\,y\,t\right)+\frac{{\partial }^2}{\partial x^2}v\left(x\,y\,t\right)$

$\mathrm{PDEtools}\left[\mathrm{declare}\right]\left(u\left(x\,y\,t\right)\,v\left(x\,y\,t\right)\right)$

$u\left(x\,y\,t\right)\mathrm{will\; now\; be\; displayed\; as}u$

$v\left(x\,y\,t\right)\mathrm{will\; now\; be\; displayed\; as}v$

$\mathrm{e2}≔V\left[t\right]+U\left[x\right]+U\left[x,x,y\right]+U\left[x\right]V\left[\right]+U\left[\right]V\left[x\right]$

$\mathrm{e2}≔u{v}_x+u_{x}v+u_x+u_{x,x,y}+v_t$

$\mathrm{lprint}\left(\right)$

u(x,y,t)*diff(v(x,y,t),x)+diff(u(x,y,t),x)*v(x,y,t)+diff(u(x,y,t),x)+diff(diff(diff(u(x,y,t),x),x),y)+diff(v(x,y,t),t)

$\mathrm{show}$

$u\left(\frac{\partial v}{\partial x}\right)+\left(\frac{\partial u}{\partial x}\right)v+\frac{\partial u}{\partial x}+\frac{{\partial }^{3}u}{\partial x^2\partial y}+\frac{\partial v}{\partial t}$