1.1 Vector Algebra hggdhhdhhdhdhdhdhd.ppt

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1. 1. Vector Analysis
1.1 Vector Algebra

2. 1.1.1 Vector operations
A scalar has a magnitude (mass, time, temperature, charge).
A vector has a magnitude (its length) and a direction.
Examples: velocity, force, momentum, field strength.
Boldface letters denote vectors.
On the blackboard I use 
A .A anˆ,ˆ
Unit vectors are denoted by

3. Vectors have no location.
-AA
Vector field A(r)

4. addition of two vectors:A+B
multiplication by a scalar:aA

5. dot product (scalar, inner):cosABBA 0


BA
BA
ABBA
AB
if parallel
if perpendicular
Example work)(
12rrFW

6. Example 1.1

7. cross product (vector, outer):nBA ˆsinAB nˆ
is the unit vector perpendicular to the AB-plane. nBAˆ,, nBAˆ,,
form a right-handed system.BAAB  0AA BA
is the area of the
parallelogram.
Example: angular momentumprL

8. 1.1.2 Component FormzyxA ˆˆˆ
zyx
AAA  3,2,1ˆ iA
iin
1: x, 2: y, 3: z
components:basis:zyxˆ,ˆ,ˆ zyx
AAA,, An
iiA

9. common notation: 
zyx
AAA,,A 312231123ˆˆˆ
0,0
,ˆˆ
321
cyclic
jiifjiif
ij
ijji
nnn
nn





Kronecker
symbol
Properties of the basis


3
1i
iiBABA iii BA)(BA 


3
1
)(
i
iiaAa nA

10. else
if
ijkif
BACor
BBB
AAA
ijk
kj
kjijki
zyx
zyx
0
321,213,1321
312,231,1231
ˆˆˆ
3
1,


 



zyx
BAC Levi-Civita
symbol

11. Example 1.2

12. 1.1.3 Triple Products
scalar triple product:cyclic)()()( BACACBCBA 
volumezyx
zyx
zyx
CCC
BBB
AAA
 )(CBA

13. vector triple product:)()()( BACCABCBA 
bac -cab
rule
Higher order products by repeated bac-cab and symmetries
of the scalar triple product.

14. 1.1.4 Notation r
r
rr 


ˆ , : vectorseparation
ˆˆˆ :ntdisplaceme malinfinitesi
ˆˆˆ : vectorposition
rr
zyxl
zyxr
dzdydxd
zyx

15. 1.1.5 How Vectors Transform
Rotation about the x-axis:

























z
y
z
y
A
A
A
A


cossin
sincos 


3
1j
jiji ARA
In general


3
1
3
1 i
ii
i
ii
BABABA