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I am having trouble wrapping my head around this problem

given a function $f=h(g_1,g_2,g_3)$ if $h=x^2-yz$ and $f=h(x+y, y^2, x+z)$ then is it correct to apply the pointwise vector coordinate function and get $f=(x+y)^2-y^2(x+z)$

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Assuming $h(x,y,z)=x^2-yz$ then yes it is correct. –  Maesumi Feb 26 '13 at 3:39
    
Is there an explanation of what this means, expressing f in terms of h like this (physics or geometrically maybe) –  knk Feb 26 '13 at 17:53
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All you are doing is "substitution". In this problem it is not clear what the "meaning" is, as no context is given.

In a standard substitution you typically change the item being measured. For example if you had $y=\sqrt{a^2-x^2}$ and you let $x=a\sin \theta$ then $y$ becomes $a\cos\theta$. Instead of measuring a length $x$ to get $y$ now you are measuring angle $\theta$ to get $y$.

In your problem $g_1$ and $g_2$ and $g_3$ specify your original input.

For $h$ you may want to use different lettering than $x,y,z$, e.g. $h(u,v,w)=u^2-vw$. Then say $u(x,y,z)=x+y,v(x,y,z)=y^2,w(x,y,z)=x+z$, and then $f(x,y,z)=h(u(x,y,z),v(x,y,z),w(x,y,z))$. Now it looks more convoluted but perhaps the role each item plays is less confusing.

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