Implicit Surface

Let us next consider the case where w depends on x, y, and z, while z depends on x and y,

w = F(x, y, z), z = g(x, y).

Figure 11.6.8 shows which variables depend on which.

Figure 11.6.8

If F(x, y, z) is smooth and ∂z/∂x, ∂z/∂y exist, the Chain Rule gives

or

Similarly,

We used the fact that for the independent variables x and y,

Notice that in this case ∂w/∂x alone is ambiguous so we had to use the more complete notation

for F_x(x, y, z),

for f_x(x, y), where f(x, y) = F(x, y, g(x, y)).

Example 6

The equation for the tangent plane is obtained as follows.

and finally

F_x(a, b, c)(x - a) + F_y(a, b, c)(y - b) + F_z(a, b, c)(z - c) = 0.

Example 7