"Definition" of Curl. Can anyone derive the gradient operator? Can anyone prove why this equality is true? http://en.wikipedia.org/wiki/Curl_(mathematics)#Definition Wikipedia says it is defined, however that's BS since the gradient operator was already defined so this needs to be proven. Either you take this for a definition and prove that the little "inverted triangle" is a derivative operator, or you prove the equality and don't call it a definition. I can't tell how to go about proving that differentiating a vector field with a weird determinant is EQUAL to the loop integral of F*dr divided by the area enclosed (as the are goes to zero). Its probably not hard, the cross product comes out of the "moment" of the field about a point, however I don't quite see how the derivative comes in.