How can I make sense of the equation $\mathrm{d}=\mathrm{d}_{\mathbb R^3}+\mathrm{d}t\wedge\partial_t$? Is it valid for 1-forms only?

Question

I need to understand the equation $\mathrm{d}=\mathrm{d}_{\mathbb R^3}+\mathrm{d}t\wedge\partial_t$ (especially the term $\mathrm{d}t\wedge\partial_t$). The author claims that it is valid in Minkowski Space. But I don't understand how to make sense of that formula, if anything only for 1-forms.

Answer 1

I interpret the identity as $\mathrm{d}f = \mathrm{d}\mathbf{x}\cdot\nabla f + \mathrm{d}t\,\partial_t f$ for a function $f(\mathbf{x}, t),$ i.e. $\mathrm{d}_{\mathbf{R}^3} = \mathrm{d}\mathbf{x}\cdot\nabla$ and $\mathrm{d}t \wedge \partial_t = \mathrm{d}t \, \partial_t$.