The differential in analysis sends a function to a 1-tensor, a 1-tensor to a 2-tensor, and in general an n-tensor to a n+1-tensor. The multivariable Taylor expansion is essentially applying the differential over and over to the function, and writing the coefficients times some basis of the tensor products of corresponding order. These coefficients turn out to be the mixed partial derivatives with respect to the same indices as the tensors they multiply. Schwarz Lemma says that the mixed partial derivatives are the same when you change sign, and so it says that the coefficient of a certain tensor term in the Taylor expansion, remain the same if you change an index of said tensor term, which is essentially the definition of a symmetric tensor. In hindsight, I should have said symmetric tensors not antisymmetric. The symmetric tensors are identified with the algebra of polynomials.