I will use an example: -The surface is given by the intersection of the plane: y+z=2 -And the infinite cilinder: x2+y2<=1 We want to parametrize this surface, it could be done easily with: x=r cosθ y=r sin θ z=2 - r cos θ Then this surface could be written using vector notation: S= r cosθ i + r sin θ j + (2 - r sin θ)k I understand that these are a set of vectors starting from the origin and ending at the surface, so nothing has changed here, it is just a compact form of the same surface. Then the normal vector is calculated first finding the tangent vectors on r and θ and then taking the cross product and multiplying this by dr dθ. What I don't understand is, if I simply differenciate S, Why don't I get the same dS? I will show what I mean. 1.- Using the first method: Sr =(cos θ, sin θ,-sin θ) Sθ =(-sinθ r, r cosθ, -r cos θ) After doing the cross product I get: Sr x Sθ = r j + r k This vector is normal to the surface, and it must be multiplied by dr dθ. 2.- WHat if I simply differenciate S? Shouldn't I get the same result as in part 1? The differential of the vectors i, j, k is zero because they don't change. We had S= r cosθ i + r sin θ j + (2 - r sin θ)k, so: dS= (dr cos θ - sin θ dθ r) i + ... I don't need to continue since the i part is not zero, but in the previous calculation it was zero. What is wrong with the normal process of differenciation? Shouldn't it give us a differential of surface? I thought the differential of a parametrized surface is just the differential of its components but I don't know what to think now.