At the heart of the Simple units environment is the new command Units[TestDimensions]. This command verifies dimensional correctness of one or more expressions with units. By default, it returns true if there is a valid assignment of dimensions to all subexpressions of expr, and false otherwise. Validity is tested for all subexpressions occurring in expr.

This command can be useful to verify the dimensional validity of, for example, a set of equations involving some quantities with units.

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

$\mathrm{eq1}≔F=ma$

$\mathrm{eq1}≔F=ma$

$\mathrm{eq2}≔F=m^{2}a$

$\mathrm{eq2}≔F=m^{2}a$

$\mathrm{TestDimensions}\left(\mathrm{eq1}\,\left\{F::⟦N⟧\,a::\left(⟦\frac{m}{s^2}⟧\right)\,m::⟦\mathrm{kg}⟧\right\}\right)$

$\mathrm{true}$

$\mathrm{TestDimensions}\left(\mathrm{eq2}\,\left\{F::⟦N⟧\,a::\left(⟦\frac{m}{s^2}⟧\right)\,m::⟦\mathrm{kg}⟧\right\}\right)$

$\mathrm{false}$

If you know the dimension of only some of the quantities occurring in a set of equations, you can derive some things about the dimensions of the other symbols and of composite subexpressions. Sometimes you can fully derive the dimension of some quantities, other times there is not enough information.

$\mathrm{TestDimensions}\left(\mathrm{eq1}\,\left\{F::⟦N⟧\,m::⟦\mathrm{kg}⟧\right\}\,\mathrm{output}=\mathrm{units}\right)$

$\left\{F::⟦N⟧\,a::\left(⟦\frac{m}{s^2}⟧\right)\,m::⟦\mathrm{kg}⟧\,\left(ma\right)::⟦N⟧\right\}$

If there are multiple valid assignments of dimensions to the symbols in an expression (in other words, if there is not enough information to fully determine the dimensions of some quantities), then Maple will make an arbitrary choice.

$\mathrm{TestDimensions}\left(\mathrm{eq1}\,\left\{F::⟦N⟧\right\}\,\mathrm{output}=\mathrm{units}\right)$

$\left\{F::⟦N⟧\,a::1\,m::⟦N⟧\,\left(ma\right)::⟦N⟧\right\}$

Instead of expressing the dimension by using a unit of the given expression, Maple can also return the name of the dimension (as it would be returned by convert/dimensions with the base option).

$\mathrm{TestDimensions}\left(\mathrm{eq1}\,\left\{F::⟦N⟧\,m::⟦\mathrm{kg}⟧\right\}\,\mathrm{output}=\mathrm{dimensions}\right)$

$\left\{F::\left(\frac{\mathrm{mass}\mathrm{length}}{{\mathrm{time}}^2}\right)\,a::\left(\frac{\mathrm{length}}{{\mathrm{time}}^2}\right)\,m::\mathrm{mass}\,\left(ma\right)::\left(\frac{\mathrm{mass}\mathrm{length}}{{\mathrm{time}}^2}\right)\right\}$

In this case, if there is ambiguity in the given set of expressions, Maple will determine an expression representing the fully general dimension for any subexpression. These expressions are mostly useful for programmer access.

$\mathrm{result}≔\mathrm{TestDimensions}\left(\mathrm{eq1}\,\left\{F::⟦N⟧\right\}\,\mathrm{output}=\mathrm{dimensions}\right)$

$\mathrm{result}≔\left\{F::\left(\frac{\mathrm{mass}\mathrm{length}}{{\mathrm{time}}^2}\right)\,a::\left({\mathrm{length}}^{{\mathrm{\_t58}}_{1,1}}{\mathrm{mass}}^{{\mathrm{\_t58}}_{1,2}}{\mathrm{time}}^{{\mathrm{\_t58}}_{1,3}}{\mathrm{electric\_current}}^{{\mathrm{\_t58}}_{1,4}}{\mathrm{thermodynamic\_temperature}}^{{\mathrm{\_t58}}_{1,5}}{\mathrm{amount\_of\_substance}}^{{\mathrm{\_t58}}_{1,6}}{\mathrm{luminous\_intensity}}^{{\mathrm{\_t58}}_{1,7}}{\mathrm{currency}}^{{\mathrm{\_t58}}_{1,8}}{\mathrm{amount\_of\_information}}^{{\mathrm{\_t58}}_{1,9}}{\mathrm{logarithmic\_gain}}^{{\mathrm{\_t58}}_{1,10}}\right)\,m::\left({\mathrm{length}}^{1-{\mathrm{\_t58}}_{1,1}}{\mathrm{mass}}^{1-{\mathrm{\_t58}}_{1,2}}{\mathrm{time}}^{-2-{\mathrm{\_t58}}_{1,3}}{\mathrm{electric\_current}}^{-{\mathrm{\_t58}}_{1,4}}{\mathrm{thermodynamic\_temperature}}^{-{\mathrm{\_t58}}_{1,5}}{\mathrm{amount\_of\_substance}}^{-{\mathrm{\_t58}}_{1,6}}{\mathrm{luminous\_intensity}}^{-{\mathrm{\_t58}}_{1,7}}{\mathrm{currency}}^{-{\mathrm{\_t58}}_{1,8}}{\mathrm{amount\_of\_information}}^{-{\mathrm{\_t58}}_{1,9}}{\mathrm{logarithmic\_gain}}^{-{\mathrm{\_t58}}_{1,10}}\right)\,\left(ma\right)::\left(\frac{\mathrm{mass}\mathrm{length}}{{\mathrm{time}}^2}\right)\right\}$

By manipulating the result, we can see that the dimension of $ma$ is indeed what is claimed.

$\mathrm{result\_as\_equations}≔\mathrm{map}\left(\mathrm{lhs}=\mathrm{rhs}\,\mathrm{result}\right)$

$\mathrm{result\_as\_equations}≔\left\{F=\frac{\mathrm{mass}\mathrm{length}}{{\mathrm{time}}^2}\,a={\mathrm{length}}^{{\mathrm{\_t58}}_{1,1}}{\mathrm{mass}}^{{\mathrm{\_t58}}_{1,2}}{\mathrm{time}}^{{\mathrm{\_t58}}_{1,3}}{\mathrm{electric\_current}}^{{\mathrm{\_t58}}_{1,4}}{\mathrm{thermodynamic\_temperature}}^{{\mathrm{\_t58}}_{1,5}}{\mathrm{amount\_of\_substance}}^{{\mathrm{\_t58}}_{1,6}}{\mathrm{luminous\_intensity}}^{{\mathrm{\_t58}}_{1,7}}{\mathrm{currency}}^{{\mathrm{\_t58}}_{1,8}}{\mathrm{amount\_of\_information}}^{{\mathrm{\_t58}}_{1,9}}{\mathrm{logarithmic\_gain}}^{{\mathrm{\_t58}}_{1,10}}\,m={\mathrm{length}}^{1-{\mathrm{\_t58}}_{1,1}}{\mathrm{mass}}^{1-{\mathrm{\_t58}}_{1,2}}{\mathrm{time}}^{-2-{\mathrm{\_t58}}_{1,3}}{\mathrm{electric\_current}}^{-{\mathrm{\_t58}}_{1,4}}{\mathrm{thermodynamic\_temperature}}^{-{\mathrm{\_t58}}_{1,5}}{\mathrm{amount\_of\_substance}}^{-{\mathrm{\_t58}}_{1,6}}{\mathrm{luminous\_intensity}}^{-{\mathrm{\_t58}}_{1,7}}{\mathrm{currency}}^{-{\mathrm{\_t58}}_{1,8}}{\mathrm{amount\_of\_information}}^{-{\mathrm{\_t58}}_{1,9}}{\mathrm{logarithmic\_gain}}^{-{\mathrm{\_t58}}_{1,10}}\,ma=\frac{\mathrm{mass}\mathrm{length}}{{\mathrm{time}}^2}\right\}$

$\mathrm{m\_dim}≔\mathrm{eval}\left(m\,\mathrm{result\_as\_equations}\right)$

$\mathrm{m\_dim}≔{\mathrm{length}}^{1-{\mathrm{\_t58}}_{1,1}}{\mathrm{mass}}^{1-{\mathrm{\_t58}}_{1,2}}{\mathrm{time}}^{-2-{\mathrm{\_t58}}_{1,3}}{\mathrm{electric\_current}}^{-{\mathrm{\_t58}}_{1,4}}{\mathrm{thermodynamic\_temperature}}^{-{\mathrm{\_t58}}_{1,5}}{\mathrm{amount\_of\_substance}}^{-{\mathrm{\_t58}}_{1,6}}{\mathrm{luminous\_intensity}}^{-{\mathrm{\_t58}}_{1,7}}{\mathrm{currency}}^{-{\mathrm{\_t58}}_{1,8}}{\mathrm{amount\_of\_information}}^{-{\mathrm{\_t58}}_{1,9}}{\mathrm{logarithmic\_gain}}^{-{\mathrm{\_t58}}_{1,10}}$

$\mathrm{a\_dim}≔\mathrm{eval}\left(a\,\mathrm{result\_as\_equations}\right)$

$\mathrm{a\_dim}≔{\mathrm{length}}^{{\mathrm{\_t58}}_{1,1}}{\mathrm{mass}}^{{\mathrm{\_t58}}_{1,2}}{\mathrm{time}}^{{\mathrm{\_t58}}_{1,3}}{\mathrm{electric\_current}}^{{\mathrm{\_t58}}_{1,4}}{\mathrm{thermodynamic\_temperature}}^{{\mathrm{\_t58}}_{1,5}}{\mathrm{amount\_of\_substance}}^{{\mathrm{\_t58}}_{1,6}}{\mathrm{luminous\_intensity}}^{{\mathrm{\_t58}}_{1,7}}{\mathrm{currency}}^{{\mathrm{\_t58}}_{1,8}}{\mathrm{amount\_of\_information}}^{{\mathrm{\_t58}}_{1,9}}{\mathrm{logarithmic\_gain}}^{{\mathrm{\_t58}}_{1,10}}$

$\mathrm{expand}\left(\mathrm{m\_dim}\mathrm{a\_dim}\right)=\mathrm{eval}\left(ma\,\mathrm{result\_as\_equations}\right)$

$\frac{\mathrm{mass}\mathrm{length}}{{\mathrm{time}}^2}=\frac{\mathrm{mass}\mathrm{length}}{{\mathrm{time}}^2}$