1 April 14 Triple product 6.3 Triple products Triple scalar product: Chapter 6 Vector Analysis A B C + _

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

3 Example p Problems 3.12.

4 Read: Chapter 6:3 Homework: 6.3.9,12,19. Due: April 25

5 April 18 Gradient 6.4 Differentiation of vectors Derivative of a vector: Derivative of vector products: Derivative in polar coordinates: x y

6 6.6 Directional derivative; Gradient Directional derivative: The changing rate of a field along a certain direction. r0r0 r usus Gradient: Examples p292.1,3. 1.The gradient is in the direction along which the field increases the fastest. 2.The gradient is perpendicular to the equipotential surface  =constant.

7 Read: Chapter 6:4-6 Homework: 6.4.2,8; 6.6.1,6,7,9. Due: April 25

8 April 21 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). is the net rate of outflow flux per unit volume. x0x0 x 0 +dx Physical meaning of curl: Set the coordinate system so that z is along direction at the point (x 0, y 0, z 0 ). is the total circulation of V per unit area.

10 Read: Chapter 6: 7 Homework: 6.7.6,7,9,13,18. Due: May 2

11 April 23 Line integrals 6.8 Line integrals Line integral: 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. 4) 5)

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

13 Read: Chapter 6: 8 Homework: 6.8.9,15,17. Due: May 2

14 April 25 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 2

17 April 28 Gauss’ theorem 6.10 The divergence and the divergence theorem Let  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=  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. is the net rate of outflow flux per unit volume. x0x0 x 0 +dx

Proof: For a differential cube, Sum over all differential cubes, at all interior surfaces will cancel, only the contributions from the exterior surfaces remain. 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.) Examples p319. A volume integral may be evaluated by a closed surface integral on its boundary, and vice versa.

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 8

21 April 30 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 (x 0, y 0, z 0 ). 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.) Proof: Set the coordinate system so that x is along nd  at an arbitrarily chosen point on the surface. Suppose the coordinates of that point is (x 0, y 0, z 0 ). Sum over all differential squares, at all interior lines will cancel, only the contributions from the exterior lines remain. A surface integral may be evaluated by a closed line integral at its boundary, and vice versa. Examples p Corollary:

23 Ampere’s law:

24 Read: Chapter 6:11 Homework: ,10,12,14. Due: May 8