Example 4: Tangent Plane to Sphere

Find the tangent plane to the sphere

x² + y² + z² = 14

at the point (1, 2, 3) (Figure 11.4.7).

Figure 11.4.7

The top hemisphere has the equation

Then

At (1,2), z = 3,

,

Then the tangent plane has the equation

or

x + 2y + 3z = 14.

The total differential of a function w = f(x, y, z) of three variables is the dependent variable dw given by

dw = f_x(x, y, z) dx + f_y(x, y, z) dy + f_z(x, y, z) dz,

or equivalently