>

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

>

$\mathrm{Setup}\left(\mathrm{mathematicalnotation}\=\mathrm{true}\right)$

$\left[{\mathrm{mathematicalnotation}}{\=}{\mathrm{true}}\right]$
 (1) 
To understand what Assume does it is necessary to concisely review what assume does and focus the difference. Consider a generic variable $x$. Nothing is known about it
>

$\mathrm{about}\left(x\right)$

x:
nothing known about this object
 
Each variable has associated a number that depends on the session, and the computer (internally) uses this number to refer to the variable:
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$\mathrm{addressof}\left(x\right)$

${18446884044863444414}$
 (2) 
When using the assume command to place assumptions on a variable, this number, associated to it, changes, for example:
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$\mathrm{assume}\left(0<x\phantom{\rule[0.0ex]{0.5em}{0.0ex}}\mathbf{and}\phantom{\rule[0.0ex]{0.5em}{0.0ex}}x<\frac{1\mathrm{\π}}{2}\right)$

>

$\mathrm{addressof}\left(x\right)$

${18446884044731636414}$
 (3) 
Indeed, the variable $x$ got redefined and renamed, it is not anymore the variable $x$ referenced in (2):
>

$\mathrm{about}\left(x\right)$

Originally x, renamed x~:
is assumed to be: RealRange(Open(0),Open(1/2*Pi))
 
To undo assumptions placed using the assume command one reassigns the variable $x$ to itself:
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$x\u2254\'x\'$

Check the numerical address: it is again equal to (2):
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$\mathrm{addressof}\left(x\right)$

${18446884044863444414}$
 (5) 
The key observation here is that variables that receive assumptions using assume get redefined each time you place assumptions on them. Although this redefinition of variables is convenient in some contexts, it also means two things:
1.

All the equations/expressions, entered before placing the assumptions on $x$ using assume, involve a variable $x$ that is different than the one that exists after placing the assumptions. So, these previous expressions cannot be reused, because they involve a different variable.

2.

Because, after placing the assumptions using assume, $x$ refers to a different variable, programs that depend on the $x$ that existed before placing the assumptions (e.g. the spacetime metric g_, general relativity tensors, Physics:Vectors and VectorCalculus commands) do not recognize the new $x$ redefined by assume.

To addressed these issues, Assume places the assumptions without redefining the variables, and allows for placing assumptions in addition to previously placed assumptions also without redefining the variables. For example, before placing assumptions this simplification attempt accomplishes nothing:
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$\mathrm{simplify}\left(\mathrm{arccos}\left(\mathrm{cos}\left(x\right)\right)\right)$

${\mathrm{arccos}}{}\left({\mathrm{cos}}{}\left({x}\right)\right)$
 (6) 
Let's assume now that $0<x<\frac{\pi}{2}$
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$\mathrm{Assume}\left(0<x\phantom{\rule[0.0ex]{0.5em}{0.0ex}}\mathbf{and}\phantom{\rule[0.0ex]{0.5em}{0.0ex}}x<\frac{1\mathrm{\π}}{2}\right)$

$\left\{{x}{::}\left({0}{\,}\frac{{\mathrm{\pi}}}{{2}}\right)\right\}$
 (7) 
For convenience, the Assume command also echoes the assumptions you place. Now, the address of $x$ is still the same as in (2) before placing the assumption
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$\mathrm{addressof}\left(x\right)$

${18446884044863444414}$
 (8) 
So the variable did not get redefined. The system however knows about the assumption  Assume uses all the machinery of the assume command:
>

$\mathrm{about}\left(x\right)$

x:
is assumed to be: RealRange(Open(0),Open(1/2*Pi))
 
Hence, expressions entered before placing assumptions can be reused. For example, you can reuse (6), we now have
>

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

To clear the assumptions on $x$, you can use Assume(x = x), or Assume(clear = {x, ..}) in the case of many variables being cleared in one go or, in the case of a single variable being cleared, also
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$\mathrm{Assume}\left(\mathrm{clear}\=x\right)$

>

$\mathrm{about}\left(x\right)$

x:
nothing known about this object
 
In summary, the Assume implements the concept of an assuming of extended action, where assumptions can be turned ON and OFF at any moment without changing the variables involved.
Assume includes the functionality of the additionally command, also without redefining the variables receiving additional assumptions. For that purpose add the keyword additionally anywhere in the calling sequence. For example:
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$\mathrm{Assume}\left(x::\mathrm{positive}\right)$

$\left\{{x}{::}\left({0}{\,}{\mathrm{\infty}}\right]\right\}$
 (11) 
>

$\mathrm{about}\left(x\right)$

x:
is assumed to be: RealRange(Open(0),infinity)
 
>

$\mathrm{Assume}\left(\mathrm{additionally}\,x<1\right)$

$\left\{{x}{::}\left({0}{\,}{1}\right)\right\}$
 (12) 
>

$\mathrm{about}\left(x\right)$

x:
is assumed to be: RealRange(Open(0),Open(1))
 
The variable $x$ always refer to the same object, itself, the same as in (2) before placing any assumption
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$\mathrm{addressof}\left(x\right)$

${18446884044863444414}$
 (13) 