![]() ![]() 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.
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