Topic 5.6: Surfaces and Surface Elements
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1 Math 275 Notes Topic 5.6: Surfaces and Surface Elements Textbook Section: 16.6 From the Toolbox (what you need from previous classes): Using vector valued functions to parametrize curves. Derivatives of vector valued functions (aka tangent vectors). Cylindrical and spherical coordinates. Learning Objectives (New Skills) & Important Concepts Learning Objectives (New Skills): Compute surface elements ds and ds, given a parametrization r(u, v) for the surface. Important Concepts: Surfaces are 2-dimensional objects in R 3. A surfaces can be parametrized using a vector valued function, r(u, v). Grid curves are the curves on the surface parameterized by r(u, v) that occur by holding one of the parameters u or v constant. The derivatives r u (u, v) and r v (u, v) are tangent vectors to the surface, so the differentials dr u = r u (u, v)du and dr v = r v (u, v)dv are infinitesimal tangent vectors. The surface element ds is a vector that is normal to a surface S, and whose magnitude ds = ds is the area of an infinitesimal parallelogram on the surface. If r(u, v) parameterizes S, then ds = dr u dr v.
2 The Big Picture Surfaces are 2-dimensional objects in R 3. Because they are 2-d, it makes sense that they can be described using two variables (parameters). This can be done using a vector valued function: r(u, v) = x(u, v) î + y(u, v) ĵ + z(u, v) ˆk. Holding one parameter constant and letting the other vary results in a family of curves on the surface, called grid curves (or traces). The two families of grid curves chop up the surface into parallelograms. The surface element ds is a vector that is normal to the surface, whose magnitude ds = ds is the area of an infinitesimal parallelogram on the surface. The surface element and its magnitude are used when integrating over surfaces. More Details A curve (1-dimensional) can be parametrized using a vector valued function of a single parameter: r(t) = x(t) î + y(t) ĵ + z(t) ˆk, while a surface (2-dimensional) requires two parameters: r(u, v) = x(u, v) î + y(u, v) ĵ + z(u, v) ˆk. Both of the vector valued functions play the same role: to indicate position in terms of parameter values. You can find a list of common parameterizations of important surfaces here. The general idea behind constructing a parametrization for a surface, is to start with the general position vector: r = x î + y ĵ + z ˆk. 2
3 If you want to use cylindrical or spherical coordinates, use the changeof-coordinate functions to change from x, y, z to r, θ, z or ρ, φ, θ. This gives you the position vector in cylindrical or spherical coordinates: cylindrical: r = r cos θ î + r sin θ ĵ + z ˆk spherical: r = ρ sin φ cos θ î + ρ sin φ sin θ ĵ + ρ cos φ ˆk. Finally, find a way to get rid of one parameter. This is usually accomplished either by setting one parameter equal to a constant, or by writing one parameter as a function of the other two. Surface Elements: ds (vector surface element) and ds (scalar surface element). The scalar surface element ds is the area of an infinitesimal rectangle on a surface S. ds is used to measure area on a surface, and is the differential used with scalar surface integrals. (Compare it to the scalar line element ds in scalar line integrals.) The vector surface element ds is a vector normal to a surface S, with magnitude ds = ds. ds is used in vector surface integrals to find the flow of a field through an infinitesimal area of the surface. (Compare it to the vector line element dr in vector line integrals.) Line Element (1-d) Surface Element (2-d) Vector dr ds Element tangent to curve normal to surface Scalar ds = dr ds = ds Element measures length measures area 3
4 Computing ds and ds Using a Parameterization The idea: Suppose r(u, v) parametrizes a surface S in R 3. There are two families of grid curves of r(u, v). These are curves on the surface corresponding to holding one of the parameters constant, and parametrized by r(u, v 0 ) (v is held constant) and r(u 0, v) (u is held constant). Computing partial derivates r u (u, v) and r v (u, v) of these curves gives vectors that are tangent to the surface, in the direction of the grid curves. So the differentials dr u = r u (u, v)du and dr v = r v (u, v)dv are infinitesimal tangent vectors to the surface in the direction of the grid curves, forming the edges of infinitesimal parallelograms tangent to the surface. The cross product of these differentials results in a vector that is normal to the surface, and whose magnitude is the area of an infinitesimal parallelogram on the surface. These are the two properties we need for the vector and scalar line elements, ds and ds = ds. Computing ds and ds: Begin with a parametrization for the surface: r(u, v) = x(u, v) î + y(u, v) ĵ + z(u, v) ˆk Compute the differentials dr u and dr v : ( x dr u = r u (u, v) du = u î + y u ĵ + z ) u ˆk du ( x dr v = r v (u, v) dv = v î + y v ĵ + z ) v ˆk dv The vector surface element ds is (up to sign) the cross product of the differentials: ) ( ) ds = ± (dr u dr v = ± r u (u, v) r v (u, v) du dv Choosing a sign + or determines the orientation of the surface. This will be discussed in more detail when we look at vector surface integrals. 4
5 The scalar surface element ds is the magnitude of the vector surface element, so: ds = ds = r u (u, v) r v (u, v) du dv Note that du dv = da is the area of an infinitesimal rectangle in the uvplane. So r u (u, v) r v (u, v) is factor by which the parameterization r(u, v) changes the area of an infinitesimal rectangle in the uv-plane as it is mapped onto the corresponding infinitesimal parallelogram on the surface. 5
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