This is going to be too complicated to integrate symbolically, so we do a numerical integration, using newnumint2 from the numerical integration toolbox nit, which is a folder in the mfiles you downloaded on Assignment 1. Surffactor = simple(veclength(cross(diff(ellipsoid,t). Now we compute the surface area factor: realdot = u*transpose(v) To check that this really is a parametrization, we verify the original equation: simplify(subs((x^2/4)+(y^2/9)+z^2,ellipsoid))Īnd we can also draw a picture with ezsurf: ezsurf(ellipsoid(1),ellipsoid(2),ellipsoid(3),) We may parametrize this ellipsoid as we have done in the past, using modified spherical coordinates: syms x y z p tĮllipsoid= To see how this works, let us compute the surface area of the ellipsoid whose equation is Which enables us to compute the area of a parametrized surface, or to integrate any function along the surface with respect to surface area. Thus, taking lengths on both sides of the above formula above gives The key idea behind all the computations is summarized in the formulaĪre vectors, and their cross-product is a vector with two important properties: it is normal to the surface parametrized by r, and its length gives the scale factor between area in the parameter space and the corresponding area on the surface. In other words, the surface is given by a vector-valued function r (encoding the x, y, and z coordinates of points on the surface) depending on two parameters, say u and v. Recall that a surface is an object in 3-dimensional space that locally looks like a plane. We begin this lesson by studying integrals over parametrized surfaces.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |