Advanced Engineering Mathematics

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CHAPTER 9 Vector Differential Calculus. Grad, Div, Curl This chapter is independent of the previous two chapters (7 and 8). Changes The differential–geometric theory of curves in space and in the plane, which in the previous edition was distributed over three consecutive sections, along with its application in mechanics, is now streamlined and shortened and presented in a single section, with a discussion of tangential and normal acceleration in a more concrete fashion. Formulas for grad, div, and curl in curvilinear coordinates are placed for reference in App. A3.4. SECTION 9.1. Vector in 2-Space and 3-Space, page 364 Purpose. We introduce vectors in 3-space given geometrically by (families of parallel) directed segments or algebraically by ordered triples of real numbers, and we define addition of vectors and scalar multiplication (multiplication of vectors by numbers). Main Content, Important Concepts Vector, norm (length), unit vector, components Addition of vectors, scalar multiplication Vector space R 3 , linear independence, basis Comments on Content Our discussions in the whole chapter will be independent of Chaps. 7 and 8, and there will be no more need for writing vectors as columns and for distinguishing between row and column vectors. Our notation a 5 [ a 1 , a 2 , a 3 ] is compatible with that in Chap. 7. Engineers seem to like both notations a 5 [ a 1 , a 2 , a 3 ] 5 a 1 i 1 a 2 j 1 a 3 k, preferring the first for “short” components and the second in the case of longer expressions. The student is supposed to understand that the whole vector algebra (and vector calculus) has resulted from applications, with concepts that are practical, that is, they are “made to measure” for standard needs and situations; thus, in this section, the two algebraic operations resulted from forces (forming resultants and changing magnitudes of forces); similarly in the next sections. The restriction to three dimensions (as opposed to n dimensions in the previous two chapters) allows us to “visualize” concepts, relations, and results and to give geometrical explanations and interpretations. On a higher level, the equivalence of the geometric and the algebraic approach (Theorem 1) would require a consideration of how the various triples of numbers for the various choices of coordinate systems must be related (in terms of coordinate transformations) for a vector to have a norm and direction independent of the choice of coordinate systems. Teaching experience makes it advisable to cover the material in this first section rather slowly and to assign relatively many problems, so that the student gets a feel for vectors in R 3 (and R 2 ) and the interrelation between algebraic and geometric aspects. 174 im09.qxd 9/21/05 12:15 PM Page 174
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SOLUTION TO PROBLEM SET 9.1, page 370 2. The components are 2 5, 2 5, 2 5. The length is 5 Ï 3 w . Hence the unit vector in the direction of v is [ 2 1/ Ï 3 w , 2 1/ Ï 3 w , 2 1/ Ï 3 w ].
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ch09 - im09.qxd 12:15 PM Page 174 CHAPTER 9 Vector...

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