The line_int command computes the solution to the equation $\nabla \left(F\right)=G$ as a line integral when the partial derivatives of F are given as a list DF. For example, if $\mathrm{DF}=\[\mathrm{Fx},\mathrm{Fy},...\]$, where $\mathrm{Fx},...$ are partial derivatives of $F$ with respect to $x,..,$ then line_int([Fx, Fy,...], [x,y,...]) will return $F$ up to an additive constant.

For a line integral to make sense (return an answer which is correct), DF must be a total derivative. By default, line_int does not check whether DF is true or false. It is, however, possible to force line_int to check $\mathrm{DF}$ before doing any further computations, by including the word check in the calling sequence as the third or fourth argument.

By default line_int does not apply any simplification to the integrands before evaluating the line integral. However, in many cases (especially if the dimension of the line integral is greater than 2), it may be convenient to simplify the integrand before performing each integration: Enter a simplification procedure as the third or fourth argument of the command.

This function is part of the DEtools package, and so it can be used in the form line_int(..) only after executing the command with(DEtools). However, it can always be accessed through the long form of the command by using DEtools[line_int](..).

$\mathrm{with}\left(\mathrm{DEtools}\right)\:$

$\mathrm{vars}≔\left[x\,y\,z\right]$

$\mathrm{vars}≔\left[x\,y\,z\right]$

$\mathrm{eq}≔F\left(\mathrm{op}\left(\mathrm{vars}\right)\right)$

$\mathrm{eq}≔F\left(x\,y\,z\right)$

$\mathrm{DF}≔\mathrm{map2}\left(\mathrm{diff}\,\mathrm{eq}\,\mathrm{vars}\right)$

$\mathrm{DF}≔\left[\frac{\partial }{\partial x}F\left(x\,y\,z\right)\,\frac{\partial }{\partial y}F\left(x\,y\,z\right)\,\frac{\partial }{\partial z}F\left(x\,y\,z\right)\right]$

$\mathrm{line\_int}\left(\mathrm{DF}\,\mathrm{vars}\right)$

$F\left(x\,y\,z\right)$

$\mathrm{eq}≔\mathrm{line\_int}\left(\left[\mathrm{Fx}\left(x\,y\right)\,\mathrm{Fy}\left(x\,y\right)\right]\,\left[x\,y\right]\right)$

$\mathrm{eq}≔\int \mathrm{Fy}\left(x\,y\right)\ⅆy+\int \mathrm{Fx}\left(x\,y\right)\ⅆx+\int \left(\int -\frac{\partial }{\partial x}\mathrm{Fy}\left(x\,y\right)\ⅆy\right)\ⅆx$

$\mathrm{DF}≔\left[y^2\,2x^{2}y\right]$

$\mathrm{DF}≔\left[y^2\,2x^{2}y\right]$

$\mathrm{line\_int}\left(\mathrm{DF}\,\left[x\,y\right]\,\mathrm{check}\right)$

Error, (in `ODEtools/line_int`) The given first argument, [y^2, 2*x^2*y], is not a total derivative