Calculus Small Changes, Big Ideas

Section 14.3 Geometry of the gradient

• Recall that the directional derivative can be written as

  \begin{equation*} \pdv{f}{\uv{u}} = \Grad f \cdot \uv{u}. \end{equation*}

  If this directional derivative is zero, then \(\uv{u}\) must be perpendicular to \(\Grad f\text{.}\)

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  Thus any direction in which the function does not change must be orthogonal to the gradient.

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• A level curve of \(f\) is defined by

  \begin{equation*} f(x,y)=k. \end{equation*}

  If a point moves along this curve, the value of \(f\) does not change.

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  Let \(\bv{r}(t)\) parametrize the curve. Then

  \begin{equation*} f(\bv{r}(t))=k. \end{equation*}

  Differentiating with respect to \(t\text{:}\)

  \begin{equation*} \dv{f}{t} = \Grad f(\bv{r}(t)) \cdot \bv{r}'(t) = 0. \end{equation*}

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  Since \(\bv{r}'(t)\) is tangent to the level curve, we conclude that \(\Grad f\) is perpendicular to the level curve.

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• In three dimensions, a level surface is given by

  \begin{equation*} f(x,y,z)=k. \end{equation*}

  The same reasoning shows that

  \begin{equation*} \Grad f \end{equation*}

  is perpendicular to the surface.

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  Thus the gradient provides a natural normal vector to any level surface.

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• This allows us to write the equation of the tangent plane to the surface \(z=f(x,y)\) at the point \((x_0,y_0)\) in vector form:

  \begin{equation*} z = f(\bv{r}_0) + \Grad f(\bv{r}_0)\cdot(\bv{r}-\bv{r}_0). \end{equation*}

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  Rearranging,

  \begin{equation*} \pdv{f}{x}(x-x_0)+\pdv{f}{y}(y-y_0)-1(z-z_0)=0. \end{equation*}

  The vector

  \begin{equation*} \qty(\pdv{f}{x},\pdv{f}{y},-1) \end{equation*}

  is normal to the tangent plane.

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• More generally, if a surface is defined implicitly by

  \begin{equation*} F(x,y,z)=0, \end{equation*}

  then \(\Grad F\) is normal to the surface. This gives us the tangent plane equation:

  \begin{equation*} \Grad F(x_0,y_0,z_0)\cdot\qty(x-x_0,y-y_0,z-z_0)=0. \end{equation*}

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• Why does the normal direction matter? In physics and computer graphics, surface normals determine how light reflects.

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