I am working on a problem that requires me to compute curl(V) div(V) where V: R4 -> R3, (coordinates are (t,x,y,z) - where x,y,z are time dependent variable) and basis of V =[Vx Vy Vz].
I know I can get curl(V) by taking differential of 1-form and div(V) by taking differential of 2-form.
In this case, is 1-form can be written as Vxdx + Vydy + Vzdz or
Vxdx + Vxdt + Vydy + Vydt + Vzdz + Vzdt?
Similarly, is 2-form can be written as Vxdy ∧ dz + Vydz ∧ dx + Vzdx ∧ dy or
Vxdy ∧ dz + Vxdy ∧ dt+ Vxdt ∧ dz + Vydz ∧ dx + Vydz ∧ dt + Vydz ∧ dt + Vzdx ∧ dy + Vzdt ∧ dy +Vzdx ∧ dt??
The basis of the vector space of differential $k$-forms in $n$-dimensions consists of all elements of the form: $$ dx_{i_1} \wedge dx_{i_2} \wedge \ldots \wedge dx_{i_k} $$ where $1 \leq i_1 < i_2 < \cdots < i_k \leq n$. Thus, a generic differential $k$-form in $n$-dimensions can be written as $$ \sum_{1 \leq i_1 < i_2 < \cdots < i_k \leq n} a_{i_1 i_2 \cdots i_k} dx_{i_1} \wedge dx_{i_2} \wedge \ldots \wedge dx_{i_k} $$ where the $a_{i_1 i_2 \cdots i_k}$ are real valued functions of $\mathbb{R}^n$.
That said, my guess is that, in your case, they're asking you to calculate the div, grad, and curl as you usually would in $3$-dimensions. You'll get an answer that depends on time, but which won't involve $dt$. Unless you are doing very advanced math, curl is only defined as an operation which takes (possibly time-dependent) vector fields in $\mathbb{R}^3$ to (possibly time-dependent) vector fields in $\mathbb{R}^3$. So, in your case, I think they want the first of the two expressions you wrote, but I can't be sure without more context.