By Murray H. Protter, Charles B. Jr. Morrey
Chapters 1-5 of this ebook include all of the fabric usually integrated in a 3rd semester multivariable calculus direction. Chapters 6-10 disguise such themes as Fourier sequence, Green's and Stokes's Theorems, and the implicit functionality theorem. The authors have made their remedy of the themes within the moment 1/2 the booklet as self sufficient of one another as attainable, giving the teacher a excessive measure of suppleness in structuring the path. This a part of the e-book presents the subjects for an intensive advent to complicated calculus. a quick bankruptcy on linear algebra is incorporated within the Appendix.
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1,3, I), (2, 1,2), (4,2, -I) In each of Problems 9 through 12, find the equation of the plane passing through PI and perpendicular to the line L I . 9. PI (2, -1,3); L[ : x = -I + 2/, y = I 10. P I (I,2, -3); L, : x = t, Y = - 2 - 2t, II. P[(2, -I, -2); L[ : x = 2 + 3t, y = 0, 12. p[ ( - I, 2, - 3); L, : x = - I + 5t, Y = + 3t, Z Z Z = I = = -4t I + 3t -1 - 2t + 2t, Z = -I + 3t In each of Problems 13 through 16, find the equations of the line through PI and perpendicular to the given plane MI' + 3y + Z - 3 = 0 +4=0 + 2z + 3 = 0 13.
32 1. Analytic Geometry in Three Dimensions z p(x,y,z) (r, 8, z) Fig. 1-18 x 7. Other Coordinate Systems In plane analytic geometry we employ a rectangular coordinate system for certain types of problems and a polar coordinate system for others. We know that there are circumstances in which one system is more convenient than the other. A similar situation prevails in three-dimensional geometry, and we now take up systems of coordinates other than the rectangular one which we have studied exclusively so far.
Let OA be a representative of u and Qjj a representative of v. Let C be the point on line AB which is t of the way ~m A to B (Fig. 2-17). Express in terms of u and v the vector w which has DC as representative. SOLUTION. Let z be the vector with AC as representative, and t the vector with AB as representative. We have w Also we know that t = v- = u + z = u + tt. u, and so w = u + t(v - u) = tu + tv. Let Ali be a directed line segment. We introduce a convenient symbol for the vector which has Ali as a representative.