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Imagine an ideal spring fixed by one end at left side of a wall and stretched horizontally up to L by the right side. Taking an horizontal axis increasing in right direction, having versor $\vec i $, with origin in spring resting position we have:

$$ \vec F = - k x \vec i$$ $$ \vec ds = dx \vec i$$

Work is given by: $\int_0^L \vec F \cdot \vec ds $ = $\int_0^L - k x dx $ = - k L^2 /2

Question is: in the expression: $ \vec ds = dx \vec i$, the displacement dx should be considered a positive real number (component of $ \vec ds$), but I know from mathematics that dx is just a symbol, not a real number. Here instead dx is used to represent an infinitesimal positive displacement. How could this make sense? And can displacement dx actually be used as a positive real number, so as if for example compressing the spring on the other side we would have $ \vec ds = - dx \vec i$ ?

Giack_89
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  • https://math.stackexchange.com/questions/4504726/work-of-weight-force Isn’t this the same? – insipidintegrator Aug 03 '22 at 11:51
  • This? https://math.stackexchange.com/questions/4504831/is-dx-at-least-in-integrals-a-positive-real-number?noredirect=1&lq=1 – insipidintegrator Aug 03 '22 at 11:51
  • @insipidintegrator similar but not exactly the same... this is the simplest case possible... hope i will receive more answers... – Giack_89 Aug 03 '22 at 12:01
  • @insipidintegrator not completely because they just say: it's a symbol, or it's a limit number which cannot be treated as a number. Ok but then how can you explain the usage (it seems) as number in physics? (hence this question here). It seems everyone is just avoiding the problem.. – Giack_89 Aug 03 '22 at 12:09

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