You compute r_u × r_v for a parametric surface and get a nonzero vector. You want to use it as the surface normal in a flux integral. What is wrong with using this vector directly without modification?
ANothing — r_u × r_v is already a unit normal vector suitable for any calculation
BIt points inward; you must always reverse it to point outward
CIts magnitude is not 1; it encodes area scaling and must not be normalized before multiplying by dA
DIt only gives the normal at the origin, not at arbitrary surface points
r_u × r_v is not generally a unit vector — its magnitude |r_u × r_v| is the area scaling factor that converts parameter-space area du dv into actual surface area. In a surface integral, dS = |r_u × r_v| du dv: you keep the full vector (or its magnitude) precisely because it carries this geometric information. Normalizing it to a unit vector and then multiplying by du dv would give the wrong element of surface area.
Question 2 Multiple Choice
A surface is parametrized with r(u, v). You accidentally compute r_v × r_u instead of r_u × r_v. What is the effect on the resulting normal vector?
AThe normal vector is unchanged — cross product is commutative for perpendicular vectors
BThe magnitude changes but the direction stays the same
CThe direction reverses (inward vs outward flip) but the magnitude is the same
DThe result is no longer perpendicular to the surface
The cross product is anti-commutative: r_v × r_u = −(r_u × r_v). The magnitude |r_v × r_u| = |r_u × r_v| is unchanged (so the area element dS is unaffected), but the direction flips. This means swapping the order reverses the orientation of the surface — what was the outward normal becomes the inward normal. For flux integrals, orientation matters: reversing the normal negates the integral.
Question 3 True / False
For the parametrization r(x, y) = ⟨x, y, f(x,y)⟩ of a graph surface, the surface area element dS equals √(f_x² + f_y² + 1) dx dy.
TTrue
FFalse
Answer: True
For this parametrization, r_x = ⟨1, 0, f_x⟩ and r_y = ⟨0, 1, f_y⟩. Their cross product is r_x × r_y = ⟨−f_x, −f_y, 1⟩, and its magnitude is √(f_x² + f_y² + 1). So dS = |r_x × r_y| dx dy = √(f_x² + f_y² + 1) dx dy. This is exactly the graph surface area formula, confirming that the parametric framework recovers it as a special case.
Question 4 True / False
At a point where r_u × r_v = 0, the parametric surface has a well-defined tangent plane with a degenerate (zero-length) normal.
TTrue
FFalse
Answer: False
When r_u × r_v = 0, the two tangent vectors r_u and r_v are parallel (or one is zero), meaning they do not span a plane. The surface has a singularity at that parameter value — a point where the geometry breaks down and a tangent plane is not defined. A common example is the north pole of a sphere in standard spherical coordinates, where the parameter lines all collapse to a single point. The cross product being zero signals a failure of the parametrization, not just a degenerate normal.
Question 5 Short Answer
Explain why the magnitude |r_u × r_v| must appear in a surface integral, rather than simply integrating over the parameter domain with area element du dv.
Think about your answer, then reveal below.
Model answer: A small rectangle in the parameter domain with sides du and dv maps to a small parallelogram on the surface with sides r_u du and r_v dv. The area of a parallelogram equals the magnitude of the cross product of its edge vectors, so the actual surface area element is |r_u × r_v| du dv, not du dv. Without this factor, you would be integrating over parameter area, not surface area — the two differ whenever the surface stretches or compresses the parameter domain, which is almost always.
This is the surface-integral analogue of the Jacobian in change-of-variables for double integrals (where r dr dθ replaces dx dy in polar coordinates). The parameter domain and the surface are different geometric objects; |r_u × r_v| is the local distortion factor that converts between them. Omitting it produces integrals that depend on the choice of parametrization rather than on the intrinsic geometry of the surface.