I have the following question:
Let $i, j, k$ denote the unit vectors on $x, y, z$ axes of cartesian coordinates, respectively. Calculate the line integral $\int_c A \cdot d\textbf{r} $ and the surface integral $\int_S ( \bigtriangledown \times \textbf{A})\cdot d\textbf{S}$ for the vector field $\textbf{A} = z^2\textbf{i}+2x\textbf{j}+3y^2\textbf{k}$ where $S$ is the following surface and $C$ is its boundary.
$S: x+y+z=1, (x\geq0,y\geq0,z\geq0)$
In regards to the line integral question, most questions I have come across give you the $\textbf{r}$ but in this case its not given, so am I meant to use something like $\textbf{r} = (t,t^2,t^3)$ to complete it ? also what boundaries are meant to be used since nothing is really given ?
For the surface integral question i tried parametizing by using $(z = 1-x-y) , (y = 1-x), (x=1)$ but I got a really long equation that doesnt seem right, but seems like the only way to me, unless im missing something ? Also with the surface integral equation given in the question seems kindof weird since I usually use $\int\textbf{A}\cdot(r_x\times r_3)dydx$, are these equivalent or are you meant to use their formula ?