محاضرة 4-6

GRADIENT OF A SCALAR
The gradient of a scalar field V is a vccior thai represents both the magnitude and the
direction of the maximum space rale of increase of V.
A mathematical expression for the gradient can be obtained by evaluating the difference in
the field dV between points Pl and P2 of Figure 3.12 where V,, V2, and V3 are contours on
which V is constant. From calculus,


Also take note of the following fundamental properties of the gradient of a scalar
field V:
1. The magnitude of Vy equals the maximum rate of change in V per unit distance.
2. Vy points in the direction of the maximum rate of change in V.
3. Vy at any point is perpendicular to the constant V surface that passes through that
point (see points P and Q in Figure 3.12).

4. The projection (or component) of VV in the direction of a unit vector a is W • a
and is called the directional derivative of V along a. This is the rate of change of V
in the direction of a. For example, dV/dl in eq. (3.26) is the directional derivative of
V along PiP2 in Figure 3.12. Thus the gradient of a scalar function V provides us
with both the direction in which V changes most rapidly and the magnitude of the
maximum directional derivative of V.
5. If A = VV, V is said to be the scalar potential of A.