What is the role of |r_u × r_v| in the surface area formula A = ∬_D |r_u × r_v| du dv?
AIt computes the normal vector at each point, which is needed to orient the surface
BIt measures how much the parametrization stretches a small parameter rectangle du dv into actual surface area, accounting for distortion
CIt equals the determinant of the Jacobian matrix and is needed to convert between coordinate systems
DIt ensures the integral is taken over the correct parameter domain D
The cross product r_u × r_v is normal to the surface, but its *magnitude* |r_u × r_v| is the key quantity for area. A tiny parameter rectangle of area du dv maps to a surface patch whose area is approximately |r_u × r_v| du dv. This is the surface analog of |r'(t)| dt in arc length — the derivative magnitude corrects for the stretching introduced by the parametrization. Without this factor, integrating du dv over D would give the area of the parameter domain, not the surface.
Question 2 Multiple Choice
A surface S is the graph z = g(x, y) over domain D. Which formula gives the correct surface area?
A∬_D dA, since x and y directly parametrize the surface
B∬_D √(g_x² + g_y²) dA, the magnitude of the gradient of g
C∬_D √(1 + g_x² + g_y²) dA, where g_x and g_y are partial derivatives of g
D∬_D (1 + g_x + g_y) dA, a linear correction for the tilt of the surface
For the graph parametrization r(x, y) = ⟨x, y, g(x, y)⟩, the tangent vectors are r_x = ⟨1, 0, g_x⟩ and r_y = ⟨0, 1, g_y⟩. Their cross product has magnitude √(1 + g_x² + g_y²). The '1' arises because x and y serve directly as parameters; the g_x², g_y² terms come from the tilting of the surface. A flat horizontal surface has g_x = g_y = 0, reducing the formula to ∬_D dA — the plain area of D, as expected.
Question 3 True / False
The scalar surface integral ∬_S f dS equals the double integral of f over the parameter domain D, weighted by |r_u × r_v|.
TTrue
FFalse
Answer: True
This is exactly how surface integrals are computed. The integral ∬_S f dS is pulled back to parameter space as ∬_D f(r(u,v)) |r_u × r_v| du dv. The function f is evaluated at the surface point r(u,v), and |r_u × r_v| provides the correct area weighting to account for how the parametrization distorts parameter space. When f = 1 this gives total surface area; when f represents surface mass density it gives total mass.
Question 4 True / False
A different parametrization of the same surface will give a different value for the surface area integral.
TTrue
FFalse
Answer: False
Surface area is a geometric property of the surface itself — independent of parametrization. A different valid parametrization produces a different |r_u × r_v| factor, but when integrated over the corresponding parameter domain, the result is the same area. This is the surface analog of arc length being independent of how you parametrize a curve; the change-of-variables theorem guarantees the area element transforms correctly.
Question 5 Short Answer
Explain why the formula for the surface area of a graph z = g(x, y) is ∬_D √(1 + g_x² + g_y²) dA rather than simply ∬_D dA, and what goes wrong if you omit the square root factor.
Think about your answer, then reveal below.
Model answer: The formula ∬_D dA computes the area of the flat projection (the parameter domain D), not the area of the tilted surface. When g is not constant, the surface is tilted — a patch that projects onto a small rectangle in D has a larger area in 3D space because it is inclined relative to the xy-plane. The factor √(1 + g_x² + g_y²) corrects for this tilt: it equals 1 when the surface is flat (g_x = g_y = 0), and increases with steeper slopes. Omitting it systematically underestimates the surface area whenever the surface is not horizontal.
Surface area measures the actual 2D extent of the surface in 3D, not its shadow on the xy-plane. The cross product |r_u × r_v| is the local correction factor for how much the parametrization stretches each infinitesimal patch. For graph surfaces this becomes √(1 + g_x² + g_y²), directly analogous to √(1 + (dy/dx)²) for arc length — the tilt of the surface adds length/area beyond the projection.