# vector differentiation

#### ketanco

Under the topic of differentiation of vectors, it says "an infinitesimall increment dR of a vector R, does not need to be collinear with vector R".
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?

#### Plato

MHF Helper
Under the topic of differentiation of vectors, it says "an infinitesimall increment dR of a vector R, does not need to be collinear with vector R". 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?
WHAT says?
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.

1 person

#### Walagaster

MHF Helper
Under the topic of differentiation of vectors, it says "an infinitesimall increment dR of a vector R, does not need to be collinear with vector R".
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?
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?

1 person

#### HallsofIvy

MHF Helper
I notice you are from Turkey so perhaps this is just a language problem. You quote "an infinitesimal increment dR of a vector R, does not need to be collinear with vector R" but then you say " 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."

YOU are saying exactly the same thing as the quote!

1 person

#### ketanco

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?
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?

#### Walagaster

MHF Helper
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?
Yes.