Study guide and
5 practice problems
on:
Normal vector to an implicitly defined surface
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To find a normal vector to an implicitly defined surface, view that surface as a level set of some function $g(x,y,z)$.
A normal vector at $(x,y,z)$ is given by $\nabla g(x,y,z)$.
Related topics
Normal vectors to surfaces
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Surfaces in 3d
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Functions of several variables
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Multivariable calculus
(147 problems)
Practice problems
Find a normal vector to the surface $x^3 + y^3 z = 3$ at the point $(1,1,2)$.
Solution
Find the tangent plane at $(1,1,1)$ to the surface $$x^2 + y^2 + z^2 + xy + xz = 5.$$
Solution
Give the equation for the tangent plane to the surface $z x^2 + x y^2 + y z^2 = 5$ at the point $(-1,1,2)$.
Solution
Use 3d level surfaces to show that a normal vector to $z=f(x,y)$ is given by $\langle \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, -1 \rangle$.
Solution
Find a normal vector to the curve $\sqrt{x} + \sqrt{y} = 2$ at $(x,y) = (1,2)$. Use it to find the tangent line at $(1,1)$ expressed in the form $\bfn \cdot \bfx = b$.
Solution