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This is for a homework problem, but I'm asking just a conceptual question. Is it possible to use the del operator on a standard vector? That's what I'm being asked to do, but I thought that was supposed to be used for fields, not single vectors.

For example, the vector 3.5x+6y+4z is given, and it asks for the Divergence and Curl. But I would assume both of these to be 0 or a zero vector because there is no derivative of the components of the vector. Another asks for the gradient of a vector <5,6,3> if its in rectangular, cylindrical, or spherical(Which also doesn't make much sense to me), but theres no taking dx of 5 or dy of 6, its all just 0.

It seems odd that half of the questions asked of me are ending in just 0 for everything, so Im worried Im misunderstanding how the del operator is supposed to work. Are these values supposed to all just be 0?

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Obviously, if you have spatially constant vector fields, their spatial rates of change will be 0 in any direction. Application of the divergence and curl produce 0 or the zero vector, respectively. Sort of like how a constant function $f(x)=c$ in single variable calculus has a derivative of 0 everywhere.

Added: Using x,y,z to represent the standard Cartesian basis vectors instead of $\hat{i},\hat{j},\hat{k}$ is a sure recipe for trouble. How do you represent the scalar field $f(x,y,z)=3.5x + 6y + 4z$?

Addendum 2: Note that for coordinate systems in which the basis vectors are a function of space (e.g. cylindrical)

$$ \hat{e}_r = \cos(\theta) \hat{e}_1 + \sin(\theta) \hat{e}_2 \\ \hat{e}_\theta = -sin(\theta) \hat{e}_1 + \cos(\theta) \hat{e}_2 \\ \hat{e}_z = \hat{e}_3 $$

If you have a vector field with (cylindrical) components $5,6,3$ everywhere, it is not really a constant vector field at all because the basis vectors change from point to point

(Notice the contrast with rectangular cartesian, where a vector field with components $5,6, 3$ everywhere is a constant vector field.)

In fact, if you transform the components of this vector back to cartesian, the vector $5,6,3$ has cartesian components which are functions of space. E.g. its x component is $5 \cos(\theta) \hat{e}_1 - 6 \sin(\theta) \hat{e}_2 + \hat{e}_3$, where $\theta=\tan^{-1}(y/x)$

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  • $\begingroup$ I think you should address the part of the question that reads "if it's in rectangular, cylindrical, or spherical" as well. $\endgroup$
    – Muphrid
    Oct 9 '14 at 3:48
  • $\begingroup$ @Muphrid thanks, I've added something about coordinate systems in which the basis vectors are not constant $\endgroup$ Oct 9 '14 at 4:01

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