I am calculating 3A of Tfy-0.1064 in Aalto University. I realized here that I am misunderstanding something in vector calculus: the thing market in green particularly.
I know
$$\nabla\times E= -\frac{\partial B}{\partial t}$$
so
$$E=\int (\nabla\times B)\cdot dA=\int B\cdot dl.$$
The magnetic flux on a square-loop of side 0.2m is by a solenoid. I am stuck with the dot-product. I cannot understand what $dA$ is or $dl$ is in the integral over the magnetic field $B$. $\int dA=0.04m^2$ and $\int dl=0.8m$ but what about $\int B\cdot dl$ or $\int (\nabla\times B)\cdot dA$?
Select the XY plane along the square-loop then $\frac{dB}{dt}=\frac{d B_z}{dt}=35\frac{mT}{s}$. Now $\nabla\times B=\hat j \partial_z B_x+\hat i \partial_z B_y$ (calculation) so $(\nabla\times B)\cdot dA=\alpha \partial_z B_x+\beta \partial_z B_y$ where $\alpha,\beta$ are some scalars such that $dA=\hat i \alpha +\hat j \beta$ but $\alpha,\beta$ still unknown to me (can you see it?). $dA$ corresponds to the area of the square to which the solenoid induce the current.
But what is $dA$ in vector form?
What is $dl$ in vector form where $l$?
$l$ is apparently the perimeter of the square-loop, yes? Can I select it as a circle $l=\cos(\theta)\hat i+\sin(\theta)\bar j$?
I cannot calculate the dot product before learning to parametrize the shapes such as the square with vectors. How can I express aka parametrize the $dA$ and $dl$ in order to calculate (using dot-product) the electric field induced by the change in magnetic flux $\frac{dB}{dt}=35mT/s$ of solenoid on the square-loop?
Or shortly, how can I parametrize the shapes? If I could select arbitrary path, then I could use cylinder coordinates with some radius $r$ and some angle $\theta$ but can I do the integration in a way that the square was a circle?