Gradient Del Operator Properties of Gradient Divergence Divergence Theorem Curl

·                    Gradient of a Scalar Field

The gradient of a scalar function at any given point is the maximum spatial rate of change of that function at that point. Gradient is thus naturally a vector quantity because the maximum rate of change distance must occur in a given direction.  The direction of the gradient vector is that in which the  scalar function changes most rapidly with distance. Consider the change in the value of an arbitrary scalar field "I" as we move from (x, y, z) to (x+Dx, y+Dy, z+Dz) and denote this as dl.

Del operator or gradient operator:

##### Properties of Gradient Operator

1.    Ñ (U + V) = ÑU + ÑV

2.    Ñ (U V) = UÑV + VÑU

3.    ÑVn = nVn – 1 ÑV

·                    Divergence of a Vector Field

From Coulomb’s Law:

As shown in the figure, the electric field vector is represented by flux lines. At a surface boundary, flux density is defined as the amount of outward flux crossing a unit surface.

Consider the case of a differential rectangular parallelepiped such as cube whose edge are lined up with the axes of a Cartesian coordinate as shown and define E = Exax + Eyay + Ezaz.

Divergence Theorem:

Vector Identities Involving Divergence:

1.    Ñ · (A + B) = Ñ · A + Ñ · B

2.    Ñ · (UA) = U Ñ · A + A · ÑU

3.    Ñ · ÑV = Ñ2V

·                    The Curl of A Vector Field

The curl of a vector field A is a vector whose magnitude is the maximum net circulation ot A per unit area as the area tends to zero and whose direction is the normal direction of the area when the area is oriented to make the circulation maximum.

(a) Uniform Field

(b)Azimuthal Field

For Cartesian Coordinates:

(h1, h2, h3) = (1, 1, 1)

(u1, u2, u3) = (x, y, z)

For Cylindrical Coordinates:

(h1, h2, h3) = (1, r, 1)

(u1, u2, u3) = (r, f, z)

For Spherical Coordinates:

(h1, h2, h3) = (1, R, Rsinq)

(u1, u2, u3) = (R, q, f)

##### Vector Identities involving the curl

1.    Ñ x (A + B) = Ñ x A + Ñ x B

2.    Ñ x (UA) = UÑ x A + A Ñ x U

3.    Ñ · (Ñ x A) = 0

4.    Ñ x (ÑV) = 0

5.    Ñ x Ñ x A = Ñ(Ñ · A) – Ñ2A