the greatest rate of change of φ. The length of the gradient vector is |grad φ|and
represents the magnitude of this greatest rate of change.
In other words, the gradient of a scalar field is a vector field which represents
the direction and magnitude of the greatest rate of change of the scalar field.
Example 7-19. Show the gradient of φis a normal vector to the surface
φ=φ(x,y,z ) = c= constant.
Solution: Let r (s), where sis arc length, represent any curve lying in the surface
φ(x, y, z) = c. Along this curve the scalar field has the value φ=φ(x(s), y (s), z(s)) = c
and the rate of change of φalong this curve is given by
dφ
ds=∂φ
∂xdx
ds+∂φ
∂ydy
ds+∂φ
∂zdz
ds=dc
ds= 0or dφ
ds= grad φ·dr
ds= grad φ·ˆet= 0.The resulting equation tells us that the vector grad φis perpendicular to the unit
tangent vector to the curve on the surface. But this unit tangent vector lies in the
tangent plane to the surface at the point of evaluation for the gradient. Thus, grad φ
is normal to the surface φ(x,y,z ) = c. The family of surfaces φ=φ(x, y, z ) = c, for
various values of c, are called level surfaces. In two-dimensions, the family of curves
φ=φ(x, y ) = c, for various values of c, are called level curves. The gradient of φis a
vector perpendicular to these level surfaces or level curves.
Example 7-20. Find the unit tangent vector at a point on the curve defined
by the intersection of the two surfaces
F(x, y, z ) = c 1 and G(x, y, z) = c 2 ,
where c 1 and c 2 are constants.
Solution: If two surfaces F =c 1 and G=c 2 intersect in a curve, then at a point
(x 0 , y 0 , z 0 )common to both surfaces and on the curve one can calculate the normal
vectors to both surfaces. These normal vectors are
∇F= grad F and ∇G= grad G
which are evaluated at the point (x 0 , y 0 , z 0 ) common to both surfaces and on the
curve of intersection of the surfaces. The cross product
(∇F)×(∇G)