angles specify the directions one should travel in order to achieve the maximum (or
minimum) rate of change of the scalar φ.
A physical example illustrating this idea is heat flow. Heat always flows from
regions of higher temperature to regions of lower temperature. Let T(x, y )denote a
scalar field which represents the temperature T at any point (x, y )in some region R
within a material medium. The level curves T(x, y ) = T 0 are called isothermal curves
and represent the constant “levels”of temperature. The vector grad T, evaluated
at a point on an isothermal curve, points in the direction of greatest temperature
change. The vector is also normal to the isothermal curve. Fourier’s law of heat
conduction states that the heat flow q [joules/cm^2 sec] is in a direction opposite to
this greatest rate of change and
q =−kgrad T,
where k[joules/cm −sec −deg C ]is the thermal conductivity of the material in which
the heat is flowing.
Example 7-22. In two-dimensions a curve y=f(x)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
ˆe 2 =−f′(x)ˆe 1 +ˆe 2 =N
is a vector normal to the curve at the point (x, f (x)).A unit normal vector to the
curve is given by
ˆen=−f
′(x)ˆe 1 +ˆe 2
√
1 + [f′(x)]^2
Another way to construct this normal vector is as follows. The position vector r
describing the curve y=f(x)is given by r =xˆe 1 +f(x)ˆe 2 with tangent dr
dx
=ˆe 1 +f′(x)ˆe 2.
The unit tangent vector to the curve is given by ˆet= eˆ^1 +f
′(x)ˆe 2
√
1 + [f′(x)]^2
. The vector ˆe 3
is perpendicular to the planar surface containing the curve and consequently the
vector ˆe 3 ׈etis normal to the curve. This cross product is given by
ˆe 3 ׈et=−f
′(x)ˆe 1 +ˆe 2
√
1 + [f′(x)]^2
=eˆn
and produces a unit normal vector to the curve. Note that there are always two
normals to every curve or surface. It is important to observe that if N is normal to
a point on the surface, then the vector −N is also a normal to the same point on
