** **

Gradient |
Del Operator |
Properties of Gradient |
Divergence |

Divergence Theorem |
Curl |

**
**

**
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:**

**1.**
**Ñ
(U + V) = ÑU
+ ÑV**

**2.**
**Ñ
(U V) = UÑV
+ VÑU**

**3.**
**ÑV ^{n}
= nV^{n – 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 = E_{x}ax + E_{y}a_{y} + E_{z}a_{z}.

**
Vector Identities
Involving Divergence:**

**1.**
**Ñ
·
(A + B) = Ñ
·
A + Ñ
·
B**

**2.**
**Ñ
·
(UA) = U Ñ
·
A + A ·
ÑU**

**3.**
**Ñ
·
ÑV
= Ñ ^{2}V**

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:__

(h_{1}, h_{2}, h_{3}) = (1, 1, 1)

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

__
For Cylindrical
Coordinates:__

(h_{1}, h_{2}, h_{3}) = (1, r, 1)

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

__
For Spherical
Coordinates:__

**
(h _{1}, h_{2},
h_{3}) = (1, R, Rsinq)**

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

**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) – Ñ ^{2}A**