#math/calculus The chain rule becomes geometric when a moving point is constrained to a surface: its velocity cannot point away from the surface. Suppose a differentiable curve $ \mathbf r(t)=\langle x(t),y(t),z(t)\rangle $ lies on the graph $z=f(x,y)$. Then $ z(t)=f(x(t),y(t)). $ Differentiating gives $ z'(t)=f_x(x(t),y(t))\,x'(t)+f_y(x(t),y(t))\,y'(t). $ Therefore its velocity $ \mathbf r'(t)=\langle x',y',z'\rangle $ satisfies $ \langle-f_x,-f_y,1\rangle\cdot\mathbf r'(t)=0. $ So every tangent vector to every curve on the surface through a point lies in the same [[15B2 - Tangent Planes and Normal Vectors|tangent plane]]. This is the geometric content of the chain rule. For the unit sphere $x^2+y^2+z^2=1$, differentiating along any surface curve yields $ xx'+yy'+zz'=0. $ Thus the position vector $\langle x,y,z\rangle$ is perpendicular to every surface velocity, so it is normal to the sphere. [[16B1 - Gradients and Level Sets|Gradients]] express this argument for any implicit level surface in one line.