Dual vector

  • #1
648
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[tex]f(\vec{x}+\epsilon \vec{y})-f(\vec{x})=\epsilon \mbox{d}f_{\vec{x}}(\vec{y})+O(\epsilon^2)[/tex].
Is ##\mbox{d}f_{\vec{x}}(\vec{y})## dual vector and why? Is it because ##\mbox{d}## is linear transformation? Also why equality
[tex]f(\vec{x}+\epsilon \vec{y})-f(\vec{x})=\epsilon \mbox{d}f_{\vec{x}}(\vec{y})+O(\epsilon^2)[/tex]
is correct?
 

Answers and Replies

  • #2
martinbn
Science Advisor
2,211
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To get a meaningful answer you need to provide more context.

The equality is correct because that is how ##df## is defined. And it is a dual vector because it takes vectors as arguments and gives a number as a result, and it is linear in the argument (the ##\vec{y}## in you expression).
 

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