Differentiability and the Tangent Plane

References are to Salas & Hille, Calculus, 7th Edition

Assume that f(x, y) is defined in some neighborhood of the point (x_{0}, y_{0}) and let the surface S be the graph z = f(x, y) of f. Alternatively, we can view S as the level surface g = 0 of the function g(x, y, z) = z − f(x, y). Let P_{0}(x_{0}, y_{0}, z_{0}) be the point of S above (x_{0}, y_{0}), where z_{0} = f(x_{0}, y_{0}). We interpret the differentiability of f analytically and geometrically.

Analytically Let h = (h_{1}, h_{2}) be a vector in R^{2}, small enough for f(x_{0}+h_{1}, y_{0}+h_{2}) to be defined.

Definition 1 (cf. Defn. 15.1.1) The function f is differentiable at (x_{0}, y_{0}) if it has a good linear approximation near (x_{0}, y_{0}), in the sense that

f(x_{0} + h_{1}, y_{0} + h_{2}) = z_{0} + Ah_{1} + Bh_{2} + o(h) as h → 0.  (2)

Let P be the nearby point on S

(x, y, z) = (x_{0} + h_{1}, y_{0} + h_{2}, f(x_{0} + h_{1}, y_{0} + h_{2})),

so that h_{1} = x − x_{0} and h_{2} = y − y_{0}. The linear approximation to f has the graph

z = z_{0} + A(x − x_{0}) + B(y − y_{0}),  (3)

which is the equation of the plane T through P_{0} that is normal to the vector

n = −A i − B j + k.  (4)

We observe that this vector is just ∇g(P_{0}).

There is only one candidate for the coefficients A and B in equation (2).

Lemma 5 If f is differentiable at (x_{0}, y_{0}), we must have A = \frac{∂f}{∂x}(x_{0}, y_{0}) and B =

\frac{∂f}{∂y} (x_{0}, y_{0}), and these partial derivatives do exist.

Proof We do A only. In equation (2) we take h_{1} = t and h_{2} = 0; then

f(x_{0} + t, y_{0}) = f(x_{0}, y_{0}) + At + o(t) as t → 0.

This shows that the limit \frac{∂f}{∂x}(x_{0}, y_{0}) exists and has the value A.

Geometrically We show that T is the tangent plane at P_{0} to the surface S.

Theorem 6 If f is differentiable at (x_{0}, y_{0}), the angle between the vector \overrightarrow{P_{0}P} and the plane T tends to 0 as h → 0.

Proof We show instead that the angle θ between \overrightarrow{P_{0}P} and n tends to π/2. It is given by

cos θ = \frac{\overrightarrow{P_{0}P} • n}{∥\overrightarrow{P_{0}P}∥∥n∥}.  (7)

We have

\overrightarrow{P_{0}P} = h_{1} i + h_{2} j + (z − z_{0})k.

From equation (4), we compute and observe from equation (2) that

\overrightarrow{P_{0}P} • n = −Ah_{1} − Bh_{2} + z − z_{0} = o(h).  (8)

All we need to know about ∥\overrightarrow{P_{0}P}∥ is that ∥h∥ ≤ ∥\overrightarrow{P_{0}P}∥. Substituting into equation (7), we find

cos θ ≤ \frac{o(h)}{∥h∥} \frac{1}{∥n∥} → 0 as h → 0.

We apply the continuous function cos^{−1}(−) to get θ → cos^{−1}(0) = π/2.

*(Optional) The converse is also true, but somewhat trickier.

Theorem 9 If there exists a plane T through P_{0} such that the angle between the vector \overrightarrow{P_{0}P} and the plane T tends to 0 as h → 0, then f is differentiable at (x_{0}, y_{0}) and T is the tangent plane to S at P_{0}.

Proof From the triangle inequality, we have

∥\overrightarrow{P_{0}P}∥ ≤ ∥h∥ + |z − z_{0}|.  (10)

It is first necessary to show that z −z_{0} does not get too large compared to h. This can be seen geometrically. Alternatively, it follows from equation (7) and the hypothesis that cos θ → 0, that for small enough h,

|z − z_{0} − Ah_{1} − Bh_{2}| ≤ \frac{1}{2}(∥h∥ + |z − z_{0}|).

We rearrange to get

\frac{1}{2}|z − z_{0}| ≤ |Ah_{1} + Bh_{2}| + \frac{1}{2}∥h∥

and hence |z − z_{0}| ≤ M∥h∥, where we can take M = 2|A| + 2|B| + 1. Then from equation (10), ∥\overrightarrow{P_{0}P}∥ ≤ (1 + M)∥h∥.

Now we can deduce with the help of equation (8) that

\frac{|z − z_{0} − Ah_{1} − Bh_{2}|}{∥h∥} ≤ (1 + M)\frac{|z − z_{0} − Ah_{1} − Bh_{2}|}{∥\overrightarrow{P_{0}P}∥} = ∥n∥ cos θ → 0,

as required for Definition 1.

Remark The surface S is not as special as might appear. If the function g(x, y, z) is continuously differentiable in some neighborhood of P_{0} and ∇g(P_{0}) ≠ 0, we can

permute the coordinates to arrange \frac{∂g}{∂z} (P_{0}) ≠ 0. A major theorem shows that near

P_{0}, the level surface g(x, y, z) = c (where c = g(P_{0})) is the graph z = f(x_{0}, y_{0}) of a certain continuously differentiable function f.