It's not hard to prove that $$\int_S(\vec{dS}\times\vec\nabla)\times\vec P = \oint_C\vec{dl}\times \vec P$$

Is an alternative way to write Stoke's Theorem.

Now, from this alternative Theorem you can come to the following conclusion: $$\oint_c \vec r \times \vec{dr} = - \oint_c \vec{dr} \times \vec r = -\int_S (\vec{dS} \times \vec\nabla) \times \vec r$$

$$=\int_S[\vec{dS}(\vec r . \vec\nabla)-\vec\nabla(\vec{dS}.\vec r)] $$

$$=\int_S[3\vec{dS}-dS_j \partial_i x_j]$$

How do I get to the last line? I think that the 3 comes from the fact that we're working in 3 dimensions but other than that I'm a bit lost.

I'm not sure if $\vec r$ has any special properties or if it's just any vector. The task was simply: "This is the alternative Stokes Theorem. Use it to calculate $\oint_c \vec r \times \vec{dr}$.



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