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I have two vectors $\vec{A}$ and $\vec{B}$ as shown below:

enter image description here

The point at the origin of vector $\vec{B}$ has coordinates $(x,y)$. The angle between the two vectors is $\theta$.

Now in my physics book there is an expression $\dfrac{\partial(\cos\theta)}{\partial x}$. How does this expression makes sense?

At point $(x,y)$, there is a vector $\vec{B}$. But at point $(x+dx,y)$, there should be no vector.

By changing $dx$ (i.e. by moving the point from $(x,y)$ to $(x+dx,y)$) we only change the point, not the whole vector. If someone says that the whole vector needs to be moved, how can we prove it mathematically?

Edit: Simplified version of a part of the treatise enter image description here enter image description here enter image description here

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  • $\begingroup$ Could you include more information about the context of your textbook’s assertion? And could you clarify what the name of the textbook is? It’s important to know whether this book looks at infinitesimals in the $\Bbb R$ way or in the nonstandard way. $\endgroup$ – Chase Ryan Taylor Sep 7 '17 at 16:14
  • $\begingroup$ It uses infinitesimals just like all physics books (saying like "there is an infinitesimal change in $x$". One thing I didn't mention: It deals with the work done by vector $\vec{B}$ in the field of vector $\vec{A}$ $\endgroup$ – Joe Sep 7 '17 at 16:20
  • $\begingroup$ By work, we are meant to move the whole vector $\vec{B}$. But the expression $\dfrac{\partial(\cos\theta)}{\partial x}$ only says about changing the point, not the whole vector. $\endgroup$ – Joe Sep 7 '17 at 16:24
  • $\begingroup$ $\vec{A}$ and $\vec{B}$ are current elements. $\endgroup$ – Joe Sep 7 '17 at 16:33
  • $\begingroup$ You need to expound upon that in the context of the question (and maybe migrate it to Physics SE). $\endgroup$ – Chase Ryan Taylor Sep 7 '17 at 16:33

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