91Ó°ÊÓ

Let $w=f(x,y,z)$is a function of two variables defined on an open set containing the point $(a,b,c)$and

Let $\mathrm{Δ}w=f(a+\mathrm{Δ}x,b+\mathrm{Δ}y,c+\mathrm{Δ}z)-f(a,b,c)$

If $f_x(a,b,c),f_y(a,b,c)$and $f_z(a,b,c)$exist, the function$f$ is said to be differentiable and

$\mathrm{Δ}w=f_x(a,b,c)\mathrm{Δ}x+f_y(a,b,c)\mathrm{Δ}y+f_z(a,b,c)\mathrm{Δ}z+{\mathrm{Î\mu }}_1\mathrm{Δ}x+{\mathrm{Î\mu }}_2\mathrm{Δ}y+{\mathrm{Î\mu }}_3\mathrm{Δ}z……\text{(1)}$

where ${\mathrm{Î\mu }}_1,{\mathrm{Î\mu }}_2$and ${\mathrm{Î\mu }}_3$are functions of $\mathrm{Δ}x,\mathrm{Δ}y$and $\mathrm{Δ}z$and are zero when

$(\mathrm{Δ}x,\mathrm{Δ}y)→(0,0)$

Substitute $\mathrm{Δ}x=x-a,\mathrm{Δ}y=y-b,\mathrm{Δ}z=z-c$and $\mathrm{Δ}w=f(x,y,z)-f(a,b,c)$

in $\left(1\right)$and eliminate ${\mathrm{Î\mu }}_1\mathrm{Δ}x,{\mathrm{Î\mu }}_2\mathrm{Δ}y$and ${\mathrm{Î\mu }}_3\mathrm{Δ}z$terms.

$$\begin{gathered} ∆w=f_x(a,b,c)∆x+f_y(a,b,c)∆y+f_z(a,b,c)∆z+{\mathrm{Î\mu }}_1∆x+{\mathrm{Î\mu }}_2∆y+{\mathrm{Î\mu }}_3∆z \\ f(x,y,z)-f(a,b,c)=f_x(a,b,c)(x-a)+f_y(a,b,c)(y-b)+f_z(a,b,c)(z-c) \\ w-f(a,b,c)=\left(f_x\right(a,b,c),f_y(a,b,c),f_z(a,b,c\left)\right)·\left(\right(x-a),(x-b),(x-c\left)\right) \\ =∇f(a,b,c)·\left(\right(x,y,z)-(a,b,c\left)\right) \end{gathered}$$

Hence, the equation of tangent plane to the surface $w=f(x,y,z)$at $(a,b,c)$is

$w-f(a,b,c)=∇f(a,b,c)·⟨(x,y,z)-(a,b,c)âŸ\copyright$