By Murray H. Protter, Charles B. Jr. Morrey

Chapters 1-5 of this ebook comprise all of the fabric usually integrated in a 3rd semester multivariable calculus path. 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 ebook as self sustaining of one another as attainable, giving the teacher a excessive measure of suppleness in structuring the direction. This a part of the booklet offers the subjects for a radical creation to complex calculus. a short bankruptcy on linear algebra is incorporated within the Appendix.

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**Example text**

A directed line segment AB is defined as before, except that now the base A and the head B may be situated anywhere in three-space. The magnitude of a directed line segment is its length. Two directed line segments AD and CD are saId to have the same magnitude and direction if and only if either one of the following two conditions holds: are both on the same directed line i and their directed lengths are equal; or ii) the points A, C, D, and B are the vertices of a parallelogram as shown in Fig.

Denote by (x o ,Yo, zo) the point of intersection of L and the plane. Then d2 = (Xl - x o)2 + (YI - Yo)2 + (z I - ZO)2. (3) Also (xo, Yo, zo) is on both the line and the plane. Therefore, we have for some value to (4) and Ax o + Byo + Cz o + D = 0 = A(x l + At o) + B(YI + Bt o) + C(ZI + Ct o) + D. JA 2 + B 2 + C2llol, and now, inserting the relation lo = -(Ax l + BYI + CZ I + D) + B2 + C 2 A2 in the preceding expression for d, we obtain the desired formula. EXAMPLE 5. Find the distance from the point (2, -1,5) to the plane 3x + 2y - 2z - 7 = O.

Draw a figure and devise, if you can, a general proof. 28. The same as Problem 27 for part (ii) of Theorem I. 29. Write out a proof establishing the associative, commutative, and distributive laws for vectors. ) 3, Operations with Plane Vectors, Continued. The Scalar Product Two vectors v and ware said to be parallel or proportional when each is a scalar multiple of the other (and neither is zero). Parallel vectors have parallel directed line segments. By the angle between two vectors v and w (neither = 0), we mean the measure of the angle between two representatives of v and w having the same base (see Fig.