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Corollary 1

The following corollary is analogous to the theorem that a differentiable function of one variable is continuous.

COROLLARY 1

If a function z = f(x, y) is smooth at (a, b) then it is continuous at (a, b).

PROOF

Let (x, y) be infinitely close to (a, b) and let

Δx = x - a, Δy = y - b.

Then

11_partial_differentiation-227.gif

Since Δx and Δy are infinitesimal, Δz is infinitesimal, so f(x, y) ≈ f(a, b).

Some examples of what can happen when the function is not smooth are given in the problem set.

If a function z = f(x, y) is smooth at (a, b), the curve z = f(x, b) has a tangent line L, on the plane y = b, and the curve z = f(a, y) has a tangent line L2 on the plane x = a.

L1 has the equation

z - f(a, b) = fx(a, b)(x - a)

and L2 has the equation

z - f(a, b) = fy(a, b)(y - b).

The plane determined by the lines L1 and L2 is called the tangent plane. It has the equation

z - f(a, b) = fx(a, b)(x -a) + fy(a, b)(y - b),

because the graph p of this equation is a plane and intersects the plane y = b in Lx and the plane x = a in L2 (Figure 11.4.4).

11_partial_differentiation-228.gif

Figure 11.4.4


Last Update: 2010-11-25