For example. I want to find the exterior derivative of
$$\omega=(-3x-2y-z)dx +(2x^2+3y^3+4z^4)dy + (x+2y+3z)dz$$
Here's what I did:
$$ \begin{align} d\omega &= \left[\frac{\partial}{\partial x}(-3x-2y-z)\ dx+\frac{\partial}{\partial x}(2x^2+3y^3+4z^4)\ dy+\frac{\partial}{\partial x}(x+2y+3z)\ dz\right]dx \\ &+ \left[\frac{\partial}{\partial y}(-3x-2y-z)\ dx+\frac{\partial}{\partial y}(2x^2+3y^3+4z^4)\ dy+\frac{\partial}{\partial y}(x+2y+3z)\ dz\right]dy \\ &+ \left[\frac{\partial}{\partial z}(-3x-2y-z)\ dx+\frac{\partial}{\partial z}(2x^2+3y^3+4z^4)\ dy+\frac{\partial}{\partial z}(x+2y+3z)\ dz\right]dz \\ &= (-3\ dx +4x\ dy +1\ dz)\ dx +(-2\ dx+9y^2\ dy+2\ dz)\ dy+(-1\ dx+16z^3\ dy+3\ dz)\ dz \\ &= 4x\ dy\ dx+ dz\ dx-2\ dx\ dy+2\ dz\ dy-1\ dx\ dz+16z^3\ dy\ dz \\ &= -4x\ dx\ dy-2\ dx\ dy-1\ dx\ dz-1\ dx\ dz-2\ dy\ dz+16z^3\ dy\ dz \\ &= (-4x-2)\ dx\ dy-2\ dx\ dz+(16z^3-2)\ dy\ dz \end{align} $$
However, Maple says this:
Maple says I'm off by a factor of -1? What am I doing wrong?
Thank you!
Your order of operations is wrong. If $$ \omega = u\ dx + v\ dy + w\ dz $$
then $$ \begin{align} d\omega &= \frac{\partial u}{\partial x} \ dx\wedge dx + \frac{\partial v}{\ \partial x} \ dx \wedge dy + \frac{\partial w}{\partial x}\ dx \wedge dz \\ &\ + \frac{\partial u}{\partial y} \ dy\wedge dx + \frac{\partial v}{\partial y} \ dy \wedge dy + \frac{\partial w}{\partial y}\ dy \wedge dz \\ &\ + \frac{\partial u}{\partial z} \ dz\wedge dx + \frac{\partial v}{\partial z} \ dz \wedge dy + \frac{\partial w}{\partial z}\ dz \wedge dz \end{align} $$
You had it written as $$ d\omega = \left(\frac{\partial u}{\partial x} \ dx + \frac{\partial v}{\partial x} \ dy + \frac{\partial w}{\partial x} \ dz \right) \wedge dx + \dots $$
which caused the sign error