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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 $P$.

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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 $(10,8,-9)$.

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