1. Chapter 6 Vector Analysis
April 16 Triple product
6.3 Triple products
Triple scalar product: A B C + _

2. This is called the “bac-cab” rule.
Triple vector product: C B q
B×C A x y z
This is called the “bac-cab” rule.

3. Example p282.1.
Problems 3.12b.

4. Read: Chapter 6:3
Homework: 6.3.9,12,19.
Due: April 27

5. 6.4 Differentiation of vectors
April 20 Gradient
6.4 Differentiation of vectors
Derivative of a vector:
Derivative of vector products: x y
Derivative in polar coordinates:

6. 6.6 Directional derivative; Gradient
Directional derivative: The changing rate of a field along a certain direction. r0 r us
Gradient:
The gradient is in the direction along which the field increases the fastest.
The gradient is perpendicular to the equipotential surface f =constant.
Examples p292.1,3.

7. Read: Chapter 6:4-6
Homework:
6.4.2,8;
6.6.1,6,7,9.
Due: April 27

8. April 23 Divergence and curl 6.7 Some other expressions involving
The operator:
Vector function:
Divergence:
Curl:
Laplacian:
Vector identities involving : p339.
Examples p297.1,2.

9. Physical meaning of divergence:
Let V be the flux density (particles across a unit area in a unit time). x0
x0+dx
is the net outflow flux per unit volume.
Physical meaning of curl:
Set the coordinate system so that z is along direction at the point (x0, y0, z0 ).
is the total circulation of V per unit area.

10. Read: Chapter 6: 7
Homework: 6.7.6,7,9,13,18.
Due: May 4

11. April 25 Line integrals 6.8 Line integrals Line integral: B
Circulation: A B
Examples p300.1,2
Conservative field: A field is said to be conservative if does not depend on the path in the calculation.
Theorem: If F and its first partial derivatives are continuous in a simply connected region, then the following five statements are equivalent to each other. 1)
2) does not depend on the path.
3) is an exact differential. 5)

12. Examples p304.3,5.
Examples p306.6.

13. Read: Chapter 6: 8
Homework: 6.8.9,15,17.
Due: May 4

14. 6.9 Green’s theorem in the plane
April 27 Green’s theorem
6.9 Green’s theorem in the plane
Double integral in an area:
Green’s theorem in the plane:
A double integral over an area may be evaluated by a line integral along the boundary of the area, and vice versa.
Examples p311.1,2.

15. Examples p312.3: Divergence theorem in two dimension:
Examples p312.4: Stokes’ theorem in two dimension:

16. Read: Chapter 6:9
Homework: 6.9.2,3,8,10.
Due: May 4

17. 6.10 The divergence and the divergence theorem
April 30 Gauss’ theorem
6.10 The divergence and the divergence theorem
Let r be the density, v be the velocity of water. The water flow in a unit time through a unit area that is perpendicular to v is given by V= r v, which is called flux density. The water flow rate through a surface with unit normal n is given by V·n.
Physical meaning of divergence:
Let V be the flux density. x0
x0+dx
is the net rate of outflow flux per unit volume.

18. Time rate of the increase of mass per unit volume:
Equation of continuity (when there is no sources):
The divergence theorem (Gauss’ theorem):
(Over a simply connected region.)
A volume integral may be evaluated by a closed surface integral on its boundary, and vice versa.
Proof: For a differential cube,
Sum over all differential cubes, at all interior surfaces will cancel, only the contributions from the exterior surfaces remain.
Examples p319.

19. Gauss’ law of electric field:
For a single charge,
For any charge distribution,
From the divergence theorem,

20. Read: Chapter 6:10
Homework: ,6,7,9.
Due: May 9

21. 6.11 The curl and Stokes’ theorem
May 2 Stokes’ theorem
6.11 The curl and Stokes’ theorem
Physical meaning of curl:
Set the coordinate system so that z is along at the point (x0, y0, z0 ).
is the total circulation of V per unit area.

22. Stokes’ theorem:
(Over a simply connected region. The surface does not need to be flat.)
A surface integral may be evaluated by a closed line integral at its boundary, and vice versa.
Proof: Set the coordinate system so that x is along nds at an arbitrarily chosen point on the surface. Suppose the coordinates of that point is (x0, y0, z0 ).
Sum over all differential squares, at all interior lines will cancel, only the contributions from the exterior lines remain.
Corollary:
Examples p328.1.

23. Ampere’s law:

24. Read: Chapter 6:11
Homework: ,10,12,14.
Due: May 9