The Unit Vector

The partial derivatives ∂z/∂x and ∂z/∂y are the rates of change of z = f(x, y) as the point (x, y) moves in the direction of the x-axis and the y-axis. We now consider the rate of change of z as the point (x, y) moves in other directions. Let P(a, b) be a point in the (x, y) plane and let

U = cos αi + sin αj

be a unit vector, a is the angle from the x-axis to U (see Figure 13.1.1). The line through P with direction vector U has the vector equation

X = P + tU

or in parametric form,

(1)

x = a + t cos α,
y = b + t sin α.

Figure 13.1.1 The unit vector

At t = 0 we have x = a and y = b. If we intersect the surface z = f(x, y) with the vertical plane through the line (Equation 1), we obtain the curve

z = f(a + t cos α, b + t sin α) = F(t).

The slope dz/dt = F'(0) of this curve at f = 0 is called the slope or derivative of f in the U direction and is written f_U(a, b) (Figure 13.1.2).

Figure 13.1.2: The directional derivative

Here is the precise definition.

f_U(a, b) is called a directional derivative of f at (a, b).

The partial derivatives of f(x, y) are equal to the derivatives of f(x, y) in the i and j directions: