but why? A vector has magnitude and direction. if you make an increase to that vector in a different direction than the original vector, even if it is infinitesimall, the resultant direction changes. so why or how can they say this?

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but why? A vector has magnitude and direction. if you make an increase to that vector in a different direction than the original vector, even if it is infinitesimall, the resultant direction changes. so why or how can they say this?

Is this a book on non-standard analysis?

You my want to look at In Elementary Calculus: An Infinitesimal Approach

In Elementary Calculus: An Infinitesimal Approach by Jerome Keisler in chapter 3, there is good proof of this. The chapters and whole book is a free down-load at H. Jerome Keisler Home Page.

Since you are talking about differentiation I assume that your vector $\vec R = \vec R(t)$ and perhaps represents the position vector of a moving particle. Unless the particle is moving directly away from the origin, you wouldn't expect $\vec R(t+\Delta t) - \vec R(t)$ to be parallel to $\vec R(t)$, would you?

but why? A vector has magnitude and direction. if you make an increase to that vector in a different direction than the original vector, even if it is infinitesimall, the resultant direction changes. so why or how can they say this?

YOU are saying exactly the same thing as the quote!

This was in a book for theory of elasticity, when introducing necessary math concepts. so then you are saying, a very small increase to a vector, for differentiation purposes, does not necessarily need to be in the same direction with the original vector correct?Since you are talking about differentiation I assume that your vector $\vec R = \vec R(t)$ and perhaps represents the position vector of a moving particle. Unless the particle is moving directly away from the origin, you wouldn't expect $\vec R(t+\Delta t) - \vec R(t)$ to be parallel to $\vec R(t)$, would you?

Yes.This was in a book for theory of elasticity, when introducing necessary math concepts. so then you are saying, a very small increase to a vector, for differentiation purposes, does not necessarily need to be in the same direction with the original vector correct?

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