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) EEE 340 Lecture 05

It is possible to show (pp. 47-48), that Thus, (2-108) Cartesian Cylindrical Spherical (2-113) EEE 340 Lecture 05

a). Carlesion coordinates. 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 EEE 340 Lecture 05

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) EEE 340 Lecture 05

“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 EEE 340 Lecture 05

Practice exercise: Determine at the specified point 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 EEE 340 Lecture 05

Example 2-20. Given Determine whether the divergence theorem holds for the spherical shall. Solution. Outer surface: EEE 340 Lecture 05

Inner surface: Adding the two results EEE 340 Lecture 05

From Eq. (2-113) Therefore EEE 340 Lecture 05

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. EEE 340 Lecture 05

EEE 340 Lecture 05

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. EEE 340 Lecture 05

Cartesian coordinates Cylindrical coordinates EEE 340 Lecture 05

1) The curl of a vector is another vector 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) EEE 340 Lecture 05

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. EEE 340 Lecture 05

Practice exercise: For a scalar field V, show that EEE 340 Lecture 05