Begin2.DVI

is a vector tangent to the curve of intersection and perpendicular to both of the

normal vectors ∇F and ∇G. A unit tangent vector to the curve of intersection is

constructed having the form form

eˆt=

∇F×∇ G |∇ F×∇ G|

.

Example 7-21. In two-dimensions a curve y=f(x)or r =xeˆ 1 +f(x)ˆe 2 can be

represented in the implicit form φ=φ(x, y ) = y−f(x) = 0 so that

grad φ=∂φ ∂x

ˆe 1 +∂φ ∂y

ê 2 =−f′(x)ê 1 +ê 2 =N

is a vector normal^6 to the curve at the point (x, f (x)).A unit normal to this curve is

given by

ˆen=−f

′(x)ˆe 1 +ˆe 2 √ 1 + [f′(x)]^2

The vector T = dr

dx

=ê 1 +f′(x)ê 2 and unit vector êt=

ˆe 1 +f′(x)ˆe 2 √ 1 + [f′(x)]^2

are tangent to

the curve and one can verify that ˆen·ˆet= 0 showing these vectors are orthogonal.

Applications for the Gradient

In two-dimensions, let êα= cos αê 1 + sin αê 2 denote a unit vector in an arbitrary,

but constant, direction αand let φ=φ(x, y )denote any scalar function of position.

At a point (x 0 , y 0 ),the directional derivative of φin the direction αbecomes

dφ ds = grad φ·ˆeα=

∂φ ∂x cos α+

∂φ ∂y sin α

and the magnitude of this directional derivative changes as the angle αchanges. As

the angle αvaries, the maximum and minimum directional derivatives, at the point

(x 0 , y 0 ), occur in those directions αwhich satisfy

d dα

[ dφ ds

] =−∂φ ∂x

sin α+∂φ ∂y

cos α= 0. (7 .72)

Note there exists two angles αlying in the region between 0 and 2 π radians which

satisfy the above equation. These directions must be tested to see which corresponds

to a maximum and which corresponds to a minimum directional derivative. These

(^6) Always remember that there are two normals to a curve, namely ˆenand −ˆen,