Finding the unit vector perpedicular to a surface

Question

The question asks to find a unit vector perpendicular to the surface z^2+2xy=y+x^4 at point (x0,y0)=(1, 1). How should I do this?

Answer 1

The function is F(x,y,z) = z²+2xy-y-x⁴ = 0 The gradient, del F, will point in the direction of the normal.

del F is <dF/dx, dF/dy, dF/dz>

<2y-4x³, 2x-1, 2z>

Now insert the point (1,1,0) z₀ obtained by substitution of the x and y values into main equation.

<-2,1,0> normalized to 1/sqrt(5) <2,1,0>

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