# Gradient/Curl/Divergence of a Single Vector

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?

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 $\theta=\tan^{-1}(y/x)$

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