*This preview shows
pages
1–3. Sign up to
view the full content.*

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

This ** preview**
has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*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
].

This is the end of the preview. Sign up
to
access the rest of the document.