University Mathematics Handbook (2015)

IX. Vector Analysis

Chapter 7. Scalar Field Gradient

a.  The function given in every point of body , defined a scalar field.

b.  Through every point , just one level surface passes (see VII.2.2).

c.  The change rate of a differentiable scalar field at in the direction of vector is directional derivative (see VII.7).

d.  Vector is called gradient of function , and is denoted or when operator (Nabla) is

operating on function , following the rule

e.  A gradient of scalar field at is in the direction of the normal to the level surface passing through (or a level curve, if the field is planar).

f.  Maximum change rate of scalar fiend is in the direction of vector .

g.  The maximum value of the directional derivative of function on is