Differentials — Question 6

Question 6 – Solution

Strategy. Differentiate the constant constraint; its total differential must vanish along allowed infinitesimal changes.

Step 1: Constraint $$2x\,dx+2y\,dy+2z\,dz=0,$$ so, because $z=6\ne 0$, $$\boxed{dz=-\frac{x\,dx+y\,dy}{z}}.$$

Step 2: Estimate $$dz=-\frac{2(0.03)+3(-0.02)}6 =-\frac{0.06-0.06}{6}=\boxed 0.$$

Step 3: Geometry The radius vector $\langle x,y,z\rangle$ is normal to the sphere. The relation $$\langle x,y,z\rangle\cdot\langle dx,dy,dz\rangle=0$$ says the first-order displacement is perpendicular to that normal, hence tangent to the sphere. Here the chosen horizontal change is already orthogonal to the radius projection, so no first-order vertical correction is needed.