The image of this parameterization is simply point ( 1, 2 ), ( 1, 2 ), which is not a curve. For example, consider curve parameterization r ( t ) = 〈 1, 2 〉, 0 ≤ t ≤ 5. For a curve, this condition ensures that the image of r really is a curve, and not just a point. Recall that curve parameterization r ( t ), a ≤ t ≤ b r ( t ), a ≤ t ≤ b is regular if r ′ ( t ) ≠ 0 r ′ ( t ) ≠ 0 for all t in. Let’s now generalize the notions of smoothness and regularity to a parametric surface. Therefore the surface traced out by the parameterization is cylinder x 2 + y 2 = 1 x 2 + y 2 = 1 ( Figure 6.57).įigure 6.62 The simplest parameterization of the graph of a function is r ( x, y ) = 〈 x, y, f ( x, y ) 〉. If u is held constant, then we get vertical lines if v is held constant, then we get circles of radius 1 centered around the vertical line that goes through the origin. Then the curve traced out by the parameterization is 〈 cos u, sin u, K 〉, 〈 cos u, sin u, K 〉, which gives a circle in plane z = K z = K with radius 1 and center (0, 0, K). Then the curve traced out by the parameterization is 〈 cos K, sin K, v 〉, 〈 cos K, sin K, v 〉, which gives a vertical line that goes through point ( cos K, sin K, v ) ( cos K, sin K, v ) in the xy-plane. This allows us to build a “skeleton” of the surface, thereby getting an idea of its shape.įirst, suppose that u is a constant K. In the first family of curves we hold u constant in the second family of curves we hold v constant. To visualize S, we visualize two families of curves that lie on S. Since it is time-consuming to plot dozens or hundreds of points, we use another strategy. Similarly, points r ( π, 2 ) = ( −1, 0, 2 ) r ( π, 2 ) = ( −1, 0, 2 ) and r ( π 2, 4 ) = ( 0, 1, 4 ) r ( π 2, 4 ) = ( 0, 1, 4 ) are on S.Īlthough plotting points may give us an idea of the shape of the surface, we usually need quite a few points to see the shape. Since the parameter domain is all of ℝ 2, ℝ 2, we can choose any value for u and v and plot the corresponding point. To get an idea of the shape of the surface, we first plot some points. That is, we need a working concept of a parameterized surface (or a parametric surface), in the same way that we already have a concept of a parameterized curve.Ī parameterized surface is given by a description of the form In a similar way, to calculate a surface integral over surface S, we need to parameterize S. Recall that to calculate a scalar or vector line integral over curve C, we first need to parameterize C. However, before we can integrate over a surface, we need to consider the surface itself. Just as with line integrals, there are two kinds of surface integrals: a surface integral of a scalar-valued function and a surface integral of a vector field. In this sense, surface integrals expand on our study of line integrals. Parametric SurfacesĪ surface integral is similar to a line integral, except the integration is done over a surface rather than a path. In particular, surface integrals allow us to generalize Green’s theorem to higher dimensions, and they appear in some important theorems we discuss in later sections. They have many applications to physics and engineering, and they allow us to develop higher dimensional versions of the Fundamental Theorem of Calculus. Surface integrals are important for the same reasons that line integrals are important. We can extend the concept of a line integral to a surface integral to allow us to perform this integration. However, if we wish to integrate over a surface (a two-dimensional object) rather than a path (a one-dimensional object) in space, then we need a new kind of integral that can handle integration over objects in higher dimensions. We have seen that a line integral is an integral over a path in a plane or in space. 6.6.6 Use surface integrals to solve applied problems.6.6.5 Describe the surface integral of a vector field.6.6.4 Explain the meaning of an oriented surface, giving an example.
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