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April 11th, 2015, 07:51 AM   #1
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Math Focus: Calculus
Fake multivariable function notation

So let's say I had a multivariable function, random example:
$\displaystyle f(x,t)=x^2 + x\cdot t + \sin{t} $

But what if x was actually a function of t itself.
Would the following notation be permitted?

$\displaystyle f(x(t),t)=x^2(t) + x(t)\cdot t + \sin{t} $

And also, does that imply the following?

$\displaystyle 1^\circ \; f(x(t-t_0),t)=x^2(t-t_0) + x(t-t_0)\cdot t + \sin{t} \\$
$\displaystyle 2^\circ \; f(x(t),t-t_0)=x^2(t-t_0) + x(t-t_0)\cdot (t-t_0) + \sin{(t-t_0)} $

The first one makes sense, but the second one seems a bit strange.
Is there a better way to write this down?
Thanks.
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April 11th, 2015, 10:10 AM   #2
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The first interpretation would be correct.
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