Problem on finding a normal vector to a surface
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Find a normal vector to the surface $x^3 + y^3 z = 3$ at the point $(1,1,2)$.
Solution
Recall that
Normal vector to an implicitly defined surface
Gradient is perpendicular to level curves
Level curves and surfaces
To find a normal vector to a surface, view that surface as a level set of some function $g(x,y,z)$.
A normal vector to the implicitly defined surface $g(x,y,z) = c$ is $\nabla g(x,y,z)$.
Level curves and surfaces
We identify the surface as the level curve of the value $c=3$ for $g(x,y,z) = x^3 + y^3 z$.
Definition of the gradient
The gradient of $g(x,y,z)$ is $$ \nabla g(x,y,z) = 3x^2 \ \mathbf{i} + 3 y^2 z \ \mathbf{j} + y^3 \ \mathbf{k}.$$
Evaluating at $x=1, y=1, z=2$, we get $$ \nabla g(1,1,2) = 3 \ \mathbf{i} + 6 \ \mathbf{j} + \mathbf{k}.$$
Hence a normal vector to the surface at $(1,1,2)$ is: $$3 \ \mathbf{i} + 6 \ \mathbf{j} + \mathbf{k}$$
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