2-7 Divergence of a Vector Field

1. 2-7 Divergence of a Vector Field • where v is the volume enclosed by the closed surface S in which P is located. • Physical meaning: we may regard the divergence of the vector field at a given point as a measure of how much the field diverges or emanates from that point. the divergence of at a given point P (2-98) Lecture 05

2. It is possible to show (pp. 47-48), that • Thus, (2-108) Cartesian Cylindrical Spherical (2-113) Lecture 05

3. Example 2-17. Find the divergence of the position vector to an arbitrary point. • Solution. • a). Carlesion coordinates. • b). Spherical coordinates. • By using Table 2-1 Lecture 05

4. Properties of the divergence of a vector field • (a) • (b) the divergence of a scalar makes no sense • (c) it produces a scalar field • If at the point P, it is called a source point. • at the point P, it is called a sink point. • 2-8 Divergence theorem Gauss-Ostrogradsky theorem (2-115) Lecture 05

5. “The total outward flux of a vector field through the closed surface S is the same as the volume integral of the divergence of .” • Proof: Subdivide volume v into a large number of small cells: vk , Sk • Example: - electric flux density Gauss’s Law Lecture 05

6. The total electric flux  through any closed surface is equal to the total charge enclosed by that surface. • The theorem applies to any volume v bounded by the closed surface S provided that and are continuous in the region. • Practice exercise: Determine at the specified point • Determine the flux of over the closed surface of the cylinder Lecture 05

7. Example 2-20. Given • Determine whether the divergence theorem holds for the spherical shall. • Solution. • Outer surface: Lecture 05

8. Inner surface: • Adding the two results Lecture 05

9. From Eq. (2-113) • Therefore Lecture 05

10. 2-9 Curl of a vector • 2-10Stokes’s theorem • It is an axial vector whose magnitude is the maximum circulation of per unit area as the area tends to zero and whose direction is the normal direction of the area when the area is oriented so as to make the circulation maximum. • curl  a measure of the circulation or how much the field curls around P. Lecture 05

11. Lecture 05

12. In order to attach some physical meaning to the curl of a vector, we will employ the small “paddlewheel”. Let the vector field be a fluid velocity field. Place the small paddlewheel in this velocity field. The paddlewheel axis should be oriented in all possible directions. The maximum angular velocity of the paddlewheel at a point is proportional to the curl, while the axis points in the direction of the curl according to the right-hand rule. If the paddlewheel does not rotate, the vector field is irrotational, or has zero curl. Lecture 05

13. or Cartesian coordinates Cylindrical coordinates Lecture 05

14. or • Properties of the curl • 1) The curl of a vector is another vector • 2) The curl of a scalar V, V, makes no sense • 3) • 4) • 5) • 6) Lecture 05

15. Stokes’s Theorem • Proof: The circulation of around a closed path L is equal to the surface integral of the curl of over the open surface S bounded by L, provided that and are continuous on S. Lecture 05

16. Practice exercise: For a scalar field V, show that Lecture 05