I have to prove that if $f(x,y,z)=f_{a}(x,y,z)+f_b(x,y,z)+f_c(x,y,z)$ is a conservative vector field and and $g(x,y,z)=g_{a}(x,y,z)+g_b(x,y,z)+g_c(x,y,z)$ is also a conservative vector field, then $(cf+dg)(x,y,z)$ is conservative.

  • $\begingroup$ Sorry, meant to say prove that it is also conservative just for clarification. $\endgroup$ – user126006 Feb 4 '14 at 2:57
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    $\begingroup$ While you're editing, do you mean $f = (f_a,f_b,f_c)$? These should be vector-valued functions. Here's a hint: What does conservative mean? Write that down for each of $f$ and $g$ and see if you can deduce what you want for $cf+dg$. $\endgroup$ – Ted Shifrin Feb 4 '14 at 3:09

I think that if they are real valued functions it would change the nature of the answer because then I think it becomes a proof of curl(f)+curl(g)=curl(f+g), which is easy enough to prove.


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