Surface defined explicitly

Surface defined implicitly

$z\=f\left(x\,y\right)$ 

$f\left(x\,y\,z\right)\=c$ 

$\mathbf{N}\=\left[\begin{array}{c}-f_x\\ -f_y\\ 1\end{array}\right]$

$\mathbf{N}\= \nabla f\=\left[\begin{array}{c}{f}_x\\ f_y\\ f_z\end{array}\right]$

Table 4.6.1   Surface normals for surfaces defined explicitly and implicitly

Example 4.6.1

At P:$\left(2\,-3\right)$ on the surface defined by $z\=f\left(x\,y\right)\equiv 5-x^2\/3-y^2\/2$, obtain and draw both the normal and tangent plane.

Example 4.6.2

At P:$\left(a\,b\right)$ on the surface defined by $z\=f\left(x\,y\right)$, obtain an equation for the tangent plane in the form $z\=\dots$.

Example 4.6.3

Derive the form of N for the surface given explicitly by $z\=f\left(x\,y\right)$. (See Table 4.6.1.)

Example 4.6.4

At P:$\left(1\,1\,1\right)$ on the surface defined by $f\left(x\,y\,z\right)\equiv x^2\+2 y^2\+3 z^2\=6$, obtain and draw both the normal and tangent plane.

Example 4.6.5

At P:$\left(a\,b\,c\right)$ on the surface defined implicitly by $f\left(x\,y\,z\right)\=0$, obtain an equation for the tangent plane in the form $z\=\dots$.

Example 4.6.6

Derive the form of N for the surface given implicitly by $f\left(x\,y\,z\right)\=c$, where $c$ is a real constant. (See Table 4.6.1.)