What Does This Notation Mean ("derivative" of a 1-form)? On page 47 of Helgason's book Differential Geometry, Lie Groups, and Symmetric Spaces, he uses the notation $$\frac{\partial \omega}{\partial t}$$ where $\omega$ is a 1-form on a manifold and $t$ is one of the coordinates of a chart. (If you look on the page, you will see that $\omega$ actually has super and subscript indices $i$ and $j$ because it is a connection 1-form.)  He uses this repeatedly on that page so I don't think its a typo, but I can not find any explanation of this notation and I've never encountered it before. Can someone tell me what it means?  Does it have a coordinate invariant meaning when $t$ is thought of as a real-valued function on the manifold?
 A: To Carl and @JesseMadnick: This notation has intrinsic meaning only because we are working with the product manifold $M\times I$ and $\frac{\partial}{\partial t}$ has a global meaning, independent of local coordinates on $M$. Recall that we say a $k$-form $\phi$ on $M\times I$ is horizontal for the projection $\pi\colon M\times I\to M$ if $\iota_{\partial/\partial t}\phi = 0$ (i.e., if $\phi(\frac{\partial}{\partial t},X_1,\dots,X_{k-1}) = 0$ for any vector fields $X_i$ on $M\times I$).
Then, given a $k$-form $\omega$ on $M\times I$, by $\dfrac{\partial \omega}{\partial t}$ we mean the unique horizontal $k$-form so that $$d\omega = dt\wedge \frac{\partial \omega}{\partial t} + \text{terms not involving }dt\,.$$
In particular, if you choose local coordinates $x$ on $M$, we can write $\omega = \sum_{|I|=k} f_I(x,t)dx^I + \sum_{|J|=k-1} g_J(x,t)dx^J\wedge dt$ (using increasing multi-indices, as usual), and then $\dfrac{\partial \omega}{\partial t} = \sum \dfrac{\partial f_I(x,t)}{\partial t}dx^I$. One can check this is well-defined.
A: Dear Carl:  Happy to respond to your question.  The derivative with respect to t is only used for the forms omega bar defined in (5')  on page 46.  They are 1-forms in the da(i) and the coefficients  are functions  g of (t, a(1),   , a(m)).   The derivatives of these omega bars are given by the usual derivative with respect to t  of  these coefficients  g. The equations (6) and (7)  then boil down to differential equations for these functions g.
Helgason
