Problem On Finding A Normal Vector To A Surface - Leading Lesson

Problem on finding a normal vector to a surface

$\newcommand{\bfA}{\mathbf{A}}$ $\newcommand{\bfB}{\mathbf{B}}$ $\newcommand{\bfC}{\mathbf{C}}$ $\newcommand{\bfF}{\mathbf{F}}$ $\newcommand{\bfI}{\mathbf{I}}$ $\newcommand{\bfa}{\mathbf{a}}$ $\newcommand{\bfb}{\mathbf{b}}$ $\newcommand{\bfc}{\mathbf{c}}$ $\newcommand{\bfd}{\mathbf{d}}$ $\newcommand{\bfe}{\mathbf{e}}$ $\newcommand{\bfi}{\mathbf{i}}$ $\newcommand{\bfj}{\mathbf{j}}$ $\newcommand{\bfk}{\mathbf{k}}$ $\newcommand{\bfn}{\mathbf{n}}$ $\newcommand{\bfr}{\mathbf{r}}$ $\newcommand{\bfu}{\mathbf{u}}$ $\newcommand{\bfv}{\mathbf{v}}$ $\newcommand{\bfw}{\mathbf{w}}$ $\newcommand{\bfx}{\mathbf{x}}$ $\newcommand{\bfy}{\mathbf{y}}$ $\newcommand{\bfz}{\mathbf{z}}$ 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: $$3 \ \mathbf{i} + 6 \ \mathbf{j} + \mathbf{k}$$

Related topics

• Multivariable calculus (147 problems)
  • Functions of several variables (36 problems)
    • The level curves of $f(x,y)$ are curves in the $xy$-plane along which $f$ has a constant value. (13 problems)
    • Gradient (18 problems)
      • $\nabla f(x,y) = \partial_x f(x,y) \mathbf{i} + \partial_y f(x,y) \mathbf{j}$ (15 problems)
      • The gradient $\nabla f(x,y)$ is perpendicular to the level curve of $f$ that contains $(x,y)$. (7 problems)
    • Surfaces in 3d (10 problems)
      • Normal vectors to surfaces (5 problems)
        • A normal vector to the surface $g(x,y,z)=0$ at $(x,y,z)$ is given by $\nabla g(x,y,z)$. (5 problems)

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