# tangent plane to surface

I was wondering if somebody could check this question from schuam's outline for "Vector analysis" and tell me if this is a valid solution or has a typo...

Execise 3.27

Find an equation of the tangent plane to the surface $$x^2+2xy^2-3z^3 = 6$$ at the point P(1,2,1).

Solution:

We have $$F_x = 2x + 2y^2$$, $$F_y = 2x$$, $$F_z = 3z^2.$$

(At this point i'm wondering: is this Schuam's outline problem correct?)

Why isn't $$\frac{\partial F}{\partial y}=F_y = 4xy,\,\frac{\partial F}{\partial z}=F_z = -9z^2$$

Is there's something i'm missing or misunderstanding?

Here's the rest of the solution for reference.

Thus, at the point, the normal to the surface (And the tangent plane) is $$N(P)=[10,2,3]$$

The tangent plane E at P has the form $$10x+2y+3z=b$$. Substituting $$P$$ in the equation gives $$b=10+4+3=17$$. Thus $$10x_+2y+3z=17$$ is an equation for the tangent plane at $$P$$.

Yes, it is wrong. Note that $$F_x=2x+2y^2,\\ F_y=4xy\neq 2x,\\ F_z=-9z^2\neq 3z^2.$$ Then, at the point $$P=(1,2,1)$$, you get the normal of the plane $$(10,8,-9)$$.