Vector Forms of Green s Theorem 18 Vector Forms of Green s Theorem The curl and divergence operators allow us to rewrite Green s Theorem in versions that will be useful in our later work We suppose that the plane region D its boundary curve C and the functions P and Q satisfy the hypotheses of Green s Theorem Then we consider the vector field F P i Q j 19 Vector Forms of Green s Theorem Its line integral is and regarding F as a vector field on component 0 we have with third 20 Vector Forms of Green s Theorem Therefore and we can now rewrite the equation in Green s Theorem in the vector form 21 Vector Forms of Green s Theorem Equation 12 expresses the line integral of the tangential component of F along C as the double integral of the vertical component of curl F over the region D enclosed by C We now derive a similar formula involving the normal component of F If C is given by the vector equation r t x t i y t j a t b then the unit tangent vector is 22 Vector Forms of Green s Theorem The outward unit normal vector to C is given by See Figure 2 Figure 2 23 Vector Forms of Green s Theorem Then from the equation we have 24 Vector Forms of Green s Theorem by Green s Theorem But the integrand in this double integral is just the divergence of F 25 Vector Forms of Green s Theorem So we have a second vector form of Green s Theorem This version says that the line integral of the normal component of F along C is equal to the double integral of the divergence of F over the region D enclosed by C 26 16 6 Parametric Surfaces and Their Areas Copyright Cengage Learning All rights reserved 27 Parametric Surfaces and Their Areas Here we use vector functions to describe more general surfaces called parametric surfaces and compute their areas Then we take the general surface area formula and see how it applies to special surfaces 28 Parametric Surfaces 29 Parametric Surfaces In much the same way that we describe a space curve by a vector function r t of a single parameter t we can describe a surface by a vector function r u v of two parameters u and v We suppose that r u v x u v i y u v j z u v k is a vector valued function defined on a region D in the uv plane 30 Parametric Surfaces So x y and z the component functions of r are functions of the two variables u and v with domain D The set of all points x y z in x x u v such that y y u v z z u v and u v varies throughout D is called a parametric surface S and Equations 2 are called parametric equations of S 31 Parametric Surfaces Each choice of u and v gives a point on S by making all choices we get all of S In other words the surface is traced out by the tip of the position vector r u v as u v moves throughout the region D See Figure 1 A parametric surface Figure 1 32 Example 1 Identify and sketch the surface with vector equation r u v 2 cos u i v j 2 sin u k Solution The parametric equations for this surface are x 2 cos u y v z 2 sin u 33 Example 1 Solution cont d So for any point x y z on the surface we have x2 z2 4 cos2u 4 sin2u 4 This means that vertical cross sections parallel to the xz plane that is with y constant are all circles with radius 2 34 Example 1 Solution cont d Since y v and no restriction is placed on v the surface is a circular cylinder with radius 2 whose axis is the yaxis see Figure 2 Figure 2 35 Parametric Surfaces If a parametric surface S is given by a vector function r u v then there are two useful families of curves that lie on S one family with u constant and the other with v constant These families correspond to vertical and horizontal lines in the uv plane 36 Parametric Surfaces If we keep u constant by putting u u0 then r u0 v becomes a vector function of the single parameter v and defines a curve C1 lying on S See Figure 4 Figure 4 37 Parametric Surfaces Similarly if we keep v constant by putting v v0 we get a curve C2 given by r u v0 that lies on S We call these curves grid curves In Example 1 for instance the grid curves obtained by letting u be constant are horizontal lines whereas the grid curves with v constant are circles In fact when a computer graphs a parametric surface it usually depicts the surface by plotting these grid curves as we see in the next example 38 Example 2 Use a computer algebra system to graph the surface r u v 2 sin v cos u 2 sin v sin u u cos v Which grid curves have u constant Which have v constant 39 Example 2 Solution We graph the portion of the surface with parameter domain 0 u 4 0 v 2 in Figure 5 Figure 5 40 Example 2 Solution cont d It has the appearance of a spiral tube To identify the grid curves we write the corresponding parametric equations x 2 sin v cos u v y 2 sin v sin u z u cos If v is constant then sin v and cos v are constant so the parametric equations resemble those of the helix Thus the grid curves with v constant are the spiral curves 41 in Figure 5 Example 2 Solution cont d We deduce that the grid curves with u constant must be curves that look like circles in the figure Further evidence for this assertion is that if u is kept constant u u0 then the equation z u0 cos v shows that the z values vary from u0 1 to u0 1 42 Example 4 Find a parametric representation of the sphere x2 y2 z2 a2 Solution The sphere has a simple representation a in spherical coordinates so let s choose the angles and in spherical coordinates as the parameters 43 Example 4 Solution cont d Then putting a in the equations for conversion from spherical to rectangular coordinates we obtain x a sin cos y a sin sin z a cos as the parametric equations of the sphere The corresponding vector equation is r a sin cos i a sin sin j a cos k 44 Example 4 Solution cont d We have 0 and 0 2 so the parameter domain is the rectangle D 0 0 2 The grid curves with constant are the circles of constant latitude including the equator 45 Example 4 Solution cont d The grid curves with constant are …
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