Lecture 56: Geometric Applications of Multivariable Differentials

56.1 Tangent Plane and Normal of Surfaces

Concept One: Surface $F(x,y,z)=0$, normal vector: $n=({F_x,F_y,F_z})$

Concept 2: Surface $z=F(x,y)$, normal vector: $n=({F_x,F_y,-1})$

56.2 Tangents to Curves and Normal Planes

Concept

• Formula: $$1)\text{ Curve }\begin{cases}x=x(t)\\y=y(t)\\z=z(t)\end{cases}\quad\text{ Tangent vector: }\quad\tau=\{x'(t_0),y'(t_0),z'(t_0)\}$$
• Formula: $$2)\text{ Curve }\begin{cases}F(x,y,z)=0\\G(x,y,z)=0&\end{cases}\text{ Tangent vector: }\quad\mathbf{\tau}=\mathbf{n}_1\times\mathbf{n}_2$$