# Exterior derivative of a vector field

The exterior derivative of a scalar function is

$$d f(x,y,z) = ( \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy + \frac{\partial f}{\partial z} dz )$$

Am I correct in assuming then that

$$d\left( F_x(x,y,z) e_x + F_y(x,y,z) e_y + F_z(x,y,z) e_z \right)$$

would be

$$\left( \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \right) dx \wedge dy + \left( \frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x} \right) dz \wedge dx + \left( \frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z} \right) dy \wedge dz$$

We only talk about exterior derivatives of differential $$k$$-forms, not vector fields. However, what we can do is the following: given a vector field $$F: \Bbb{R}^3 \to \Bbb{R}^3$$, $$F = (F_x, F_y, F_z)$$, we can consider the following one-form: \begin{align} \omega &= F_x \, dx + F_y \, dy + F_z \, dz \end{align} And yes, the exterior derivative of the one-form $$\omega$$ is indeed the thing you wrote down: \begin{align} d\omega &= \left(\dfrac{\partial F_y}{\partial x} - \dfrac{\partial F_x}{\partial y}\right) dx \wedge dy + \left(\frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x} \right) dz \wedge dx + \left(\frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z} \right) dy \wedge dz \end{align}
Just some fun extra tidbits: if you know some vector calculus, the above expression probably looks pretty familiar, almost like the curl of $$F$$, though not quite. If you want to somehow get the curl of $$F$$ from here, you need to look at the "Hodge star" operator, which assigns to the above $$2$$-form $$d\omega$$ a certain $$1$$-form $$\alpha$$, namely \begin{align} \alpha &= \left(\dfrac{\partial F_y}{\partial x} - \dfrac{\partial F_x}{\partial y}\right) dz + \left(\frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x} \right) dy + \left(\frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z} \right) dx \end{align} then from here, you can get a vector field, $$G$$, (pretty much by replacing $$dx$$ with $$e_x$$, $$dy$$ with $$e_y$$ and $$dz$$ with $$e_z$$), \begin{align} G:= \left( \frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z}\right) e_x + \left(\frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x} \right) e_y + \left(\dfrac{\partial F_y}{\partial x} - \dfrac{\partial F_x}{\partial y}\right) e_z, \end{align} and this is precisely the curl of $$F$$
$$d(F_xdx + F_ydy + F_zdz) = dF_x \wedge dx + dF_y \wedge dy + dF_z \wedge dz$$ and use that $$dF = \frac{\partial F}{\partial x}dx + \frac{\partial F}{\partial y}dy + \frac{\partial F}{\partial z}dz$$ according to your first claim.