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g s
g s
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This is an example of how operators do not in general commute. That is: if $x$ and $y$ are variables, $xy=yx$, but if $f$ and $g$ are operators, $fg$ does not generally equal $gf$. An operator is a set of instructions for what to do to the expression that follows it. Consider as a simple example f ="add$f =$"add 5" and g =$g =$ "multiply by 10". Then $fgx = 10x+5$ and $gfx = 10x + 50$. If we want to reverse the operators, we need a third operator, which has the effect of undoing the consequence of the order reversal. Suppose we started with $gx$ and wanted to operate on x$x$ with f$f$. In this case, we could introduce h =$h =$ "subtract 4545". Then $fgx = hgfx$.

Or we could introduce an operator that undid g$g$, using $g^{-1} $="divide by 10" and. Then we can use the identity $gfg^{-1}gx=gfx$.

Here, "convert Cartesian to polar" and "take the time derivative" are operators. Newton's mechanics are formulated in Cartesian, so if we want to operate with the time derivative and get Newtonian results, we need either a Cartesian coordinate expression, or a third operator. That is: either "convert polar to Cartesian" as $ g^{-1}$ or "undo the consequence of operating on 'convert Cartesian to Polar' with 'take the time derivative'" as $h$.

This is an example of how operators do not in general commute. That is: if $x$ and $y$ are variables, $xy=yx$, but if $f$ and $g$ are operators, $fg$ does not generally equal $gf$. An operator is a set of instructions for what to do to the expression that follows it. Consider as a simple example f ="add 5" and g = "multiply by 10". Then $fgx = 10x+5$ and $gfx = 10x + 50$. If we want to reverse the operators, we need a third operator, which has the effect of undoing the consequence of the order reversal. Suppose we started with $gx$ and wanted to operate on x with f. In this case, we could introduce h = "subtract 45. Then $fgx = hgfx$.

Or we could introduce an operator that undid g, $g^{-1} $="divide by 10" and use the identity $gfg^{-1}gx=gfx$.

Here, "convert Cartesian to polar" and "take the time derivative" are operators. Newton's mechanics are formulated in Cartesian, so if we want to operate with the time derivative and get Newtonian results, we need either a Cartesian coordinate expression, or a third operator. That is: either "convert polar to Cartesian" as $ g^{-1}$ or "undo the consequence of operating on 'convert Cartesian to Polar' with 'take the time derivative'" as $h$.

This is an example of how operators do not in general commute. That is: if $x$ and $y$ are variables, $xy=yx$, but if $f$ and $g$ are operators, $fg$ does not generally equal $gf$. An operator is a set of instructions for what to do to the expression that follows it. Consider as a simple example $f =$"add 5" and $g =$ "multiply by 10". Then $fgx = 10x+5$ and $gfx = 10x + 50$. If we want to reverse the operators, we need a third operator, which has the effect of undoing the consequence of the order reversal. Suppose we started with $gx$ and wanted to operate on $x$ with $f$. In this case, we could introduce $h =$ "subtract 45". Then $fgx = hgfx$.

Or we could introduce an operator that undid $g$, using $g^{-1} $="divide by 10". Then we can use the identity $gfg^{-1}gx=gfx$.

Here, "convert Cartesian to polar" and "take the time derivative" are operators. Newton's mechanics are formulated in Cartesian, so if we want to operate with the time derivative and get Newtonian results, we need either a Cartesian coordinate expression, or a third operator. That is: either "convert polar to Cartesian" as $ g^{-1}$ or "undo the consequence of operating on 'convert Cartesian to Polar' with 'take the time derivative'" as $h$.

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g s
g s
  • 14.1k
  • 2
  • 21
  • 43

This is an example of how operators do not in general commute. That is: if $x$ and $y$ are variables, $xy=yx$, but if $f$ and $g$ are operators, $fg$ does not generally equal $gf$. An operator is a set of instructions for what to do to the expression that follows it. Consider as a simple example f ="add 5" and g = "multiply by 10". Then $fgx = 10x+5$ and $gfx = 10x + 50$. If we want to reverse the operators, we need a third operator, which has the effect of undoing the consequence of the order reversal. Suppose we started with $gx$ and wanted to operate on x with f. In this case, we could introduce h = "subtract 45. Then $fgx = hgfx$.

Or we could introduce an operator that undid g, $g^{-1} $="divide by 10" and use the identity $gfg^{-1}gx=gfx$.

Here, "convert Cartesian to polar" and "take the time derivative" are operators. Newton's mechanics are formulated in Cartesian, so if we want to operate with the time derivative and get Newtonian results, we need either a Cartesian coordinate expression, or a third operator. That is: either "convert polar to Cartesian" as $ g^{-1}$ or "undo the consequence of operating on 'convert Cartesian to Polar' with 'take the time derivative'" as $h$.