Integral/Vector calculus $\oint_{\partial S} u \vec \nabla v \cdot d \vec \lambda=\int_S (\vec \nabla u)\times (\vec \nabla v)\cdot d\vec S.$

I am trying to show that $$\oint_{\partial S} u \vec \nabla v \cdot d \vec \lambda=\int_S (\vec \nabla u)\times (\vec \nabla v)\cdot d\vec S$$ using Levi Cevita notation methods only. The Levi Cevita tensor is given by $\epsilon_{ijk}$ and is a totally anti symmetric tensor. The functions u and v are dependent on the radius vector in 3 dimensions ($\vec r$). I am stuck on this I know the general idea of using this notation is to write things like $$(\vec \nabla \times \vec A)_i= \epsilon_{ijk} \partial_j A_k$$ but I still am stuck. Thank you

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We can prove the identity by using Stoke's theorem and tensor notation $$\oint_{\partial S}u \vec{\nabla} v \cdot d\vec{\lambda}=\int_S \big(\vec{\nabla} u\big)\times \big( \vec{\nabla}v \big)\cdot d\vec{S}.$$ We now use Stoke's theorem to relate line integral to surface integral by $$\oint_{\partial S}u \vec{\nabla} v \cdot d\vec{\lambda}=\int_{S}\vec{\nabla}\times (u \vec{\nabla} v) \cdot \hat{n} dS.$$ Now I will use your desired $\epsilon_{ijk}$ notation to show that $$\int_{S}\vec{\nabla}\times (u \vec{\nabla} v) \cdot \hat{n} dS=\int_S \big(\vec{\nabla} u\big)\times \big( \vec{\nabla}v \big)\cdot d\vec{S}.$$
We obtain $$\big( \vec{\nabla}\times (u \vec{\nabla}v) \big)_i=\epsilon_{ijk}\partial_j (u\partial_k v)=\epsilon_{ijk}( \partial_j u \partial_k v+u\partial_j \partial_k v)$$ where I used the product rule. However the term $\partial_j \partial_k v$ is symmetric, so we know this will vanish! We can see this by using $$\partial_j \partial_k v=\frac{1}{2} (\partial_j \partial_k +\partial_k \partial_j)v$$ since it is symmetric w.r.t j and k. Thus we can see by swapping the indices j and k we obtain $$\epsilon_{ijk}\partial_j\partial_k v=-\epsilon_{ijk}\partial_k \partial_jv$$
which is only true if this term is zero. Also you know that a total antisymmetric tensor times a symmetric quantity is zero. We are left with $$\big( \vec{\nabla}\times (u \vec{\nabla}v) \big)_i=\epsilon_{ijk}\partial_j u \partial_k v=(\vec{\nabla}u \times \vec{\nabla}v)_i$$ which concludes the desired proof.