I have the same question in here: https://physics.stackexchange.com/questions/741463/why-can-these-two-expression-be-equal-in-the-curl-and-divergence
But honestly i am not sure that this question should belong to mathematics or physics,so i post this question here and there,hoping someone can explain to me!thankyou!
1.The expression of curl of $\vec A$ is $\nabla\times \vec A$,and the calculation of $\nabla\times \vec A$ is as belowed
$\nabla\times \vec A= \begin{vmatrix} \vec u_x & \vec u_y & \vec u_z \\ \frac{\partial }{\partial x} & \frac{\partial }{\partial y} & \frac{\partial }{\partial z} \\ A_x & A_y & A_z \\ \end{vmatrix} $
2.The definition of curl of $\vec A$ is $\lim_{S\to 0} \frac{ \vec{u}_n \oint_c\vec{A}\cdot d \vec{l_c}}{\Delta S}$
So if we combine 1. and 2. ,we can know
$ \begin{vmatrix} \vec u_x & \vec u_y & \vec u_z \\ \frac{\partial }{\partial x} & \frac{\partial }{\partial y} & \frac{\partial }{\partial z} \\ A_x & A_y & A_z \\ \end{vmatrix} =\lim_{S\to 0} \frac{ \vec{u}_n \oint_c\vec{A}\cdot d \vec{l_c}}{\Delta S}$
i want to ask why can they be equal?why can the curl of A be written as $\nabla\times \vec A$?
I have the same question in the divergence $\nabla\cdot \vec A=\lim_{\Delta v\to 0} \frac{ \oint_s\vec{A}\cdot d\vec s}{\Delta v}$
I think the reason of this question is that i think according to the definition of inner product or outer product,they should be no relation with integral,but now these equations told me they do have the relation,so i can not understand this.
