MTH2010 Chap.11 Parametric Surfaces and Surface Integrals
Parametric Surfaces and Surface Integrals
Week 11 develops surfaces the way Week 1 developed curves. A parametric surface is the image of a function r(u, v) of two parameters. Fixing one parameter and varying the other traces grid curves on the surface, and the partial derivatives r_u and r_v are tangent vectors that span the tangent plane.
Their cross product is a normal vector, which recovers the tangent-plane normal of a graph from Chapter 3 as a special case and extends it to spheres, cylinders and other surfaces that are not graphs. The area of a small piece of surface is the length of that cross product times du dv, which defines the surface element dS.
Integrating 1 gives surface area, and integrating a scalar function gives quantities such as the mass of a shell. For a graph z = g(x, y) the surface element simplifies to sqrt(1 + g_x² + g_y²) dA. Surface integrals of vector fields, also called flux integrals, measure how much of a field passes through a surface and require an orientation, a consistent choice of unit normal.
The flux is the integral of F dotted with the normal, computed as F(r(u, v)) dotted with r_u x r_v over the parameter region, with the sign of the cross product chosen to match the requested orientation. Closed surfaces are oriented outward unless stated otherwise. The weekly page lists five topics: parametric surfaces, tangent planes, surface integrals, surface integrals of vector fields and oriented surfaces.
What this chapter covers
- 01
Parametric surfaces and grid curves
- 02
Tangent vectors r_u and r_v and the tangent plane
- 03
Normal vector as the cross product r_u x r_v
- 04
Surface element dS and surface area
- 05
Graphs as surfaces: sqrt(1 + g_x² + g_y²) dA
- 06
Scalar surface integrals: mass of a shell
- 07
Orientation: upward, downward, outward, inward
- 08
Flux integrals and the sign of the normal
Worked example · free
Area of a plane piece in the first octant
- 1Write the plane as z = 4 - 2x - 2y over D, the triangle x, y ≥ 0 with x + y ≤ 2, which has area 2.
- 1Surface element: sqrt(1 + (-2)² + (-2)²) dA = sqrt(9) dA = 3 dA.
- 1Area: 3 times the area of D = 3 times 2 = 6.
Key terms
- Parametric surface
- The set of points r(u, v) traced as two parameters vary over a region.
- Grid curve
- A curve on a parametric surface obtained by holding one parameter fixed.
- Surface element
- The quantity |r_u x r_v| du dv that measures the area of a small piece of surface.
- Oriented surface
- A surface together with a continuous choice of unit normal at every point.
- Flux
- The surface integral of the normal component of a vector field through an oriented surface.
Parametric Surfaces and Surface Integrals FAQ
How do I find a normal vector to a parametric surface?
Differentiate r(u, v) with respect to each parameter to get two tangent vectors, then take their cross product. Reverse the order of the cross product if the requested orientation points the other way.
What is the difference between a surface integral and a flux integral?
A scalar surface integral weights each piece of area by a function and has no orientation. A flux integral measures the normal component of a vector field and changes sign if the orientation is reversed.
How do I parametrise a sphere or a cylinder?
Freeze the radius in spherical or cylindrical coordinates. A sphere of radius a uses the two spherical angles, and a cylinder of radius a uses the angle θ and the height z.
Why does a graph have the surface element sqrt(1 + g_x² + g_y²) dA?
Parametrising the graph as (x, y, g(x, y)) gives tangent vectors (1, 0, g_x) and (0, 1, g_y), whose cross product (-g_x, -g_y, 1) has exactly that length.
Exam move
Build a table of standard parametrisations with their cross products already computed: graphs, planes, spheres, cylinders and cones. For each new problem, compute r_u, r_v and their cross product slowly and check that the result is perpendicular to both tangent vectors. Before any flux integral, write the required orientation in words and confirm that your normal points that way at one easy point.
Then practise splitting a closed surface into pieces, such as a cylinder with its two discs, each with its own outward normal.
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