This page is a sub-page of our page on Calculus of Several Real Variables.

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The interactive simulations on this page can be navigated with the Free Viewer
of the Graphing Calculator.

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Anchors into the text below:

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A function $f$ from $\, \mathbb{R}^2 \,$ to $\, \mathbb{R} \,$ can be described by:

${{\mathbb{R}^2 \, \stackrel {f} {\longrightarrow} \, \mathbb{R} \:}\atop {\: (x,y) \, \longmapsto \, f(x,y) } } {\,}.$

The differential $df$ of the function $f$ at the point $(a,b) \in \mathbb{R}^2$ is given by:

$df_{(a,b)} = \frac{\partial f}{\partial x}_{(a,b)} dx + \frac{\partial f}{\partial y}_{(a,b)} dy.$

The equation of the level curve ($\, f = c_{onstant} \,$) of the function $f$ at the point $(a,b)$
is given by:

$f(x,y) = f(a,b)$.

The equation of the tangent to the level curve of the function $f$ at the point $(a,b)$
is given by:

$\frac{\partial f}{\partial x}_{(a,b)} (x-a) + \frac{\partial f}{\partial y}_{(a,b)} (y-b) = 0.$

The gradient of the function $f$ is the function ${\nabla f}$ defined by:

${{\mathbb{R}^2 \, \stackrel {\nabla f} {\longrightarrow} \, \mathbb{R}^2 \:}\atop {\: (x,y) \, \longmapsto \, (\frac{\partial f}{\partial x} , \frac{\partial f}{\partial y}) } } {\,}.$

Hence the value of ${\nabla f}$ at the point $(x,y)$ is:

${\nabla f}_{(x,y)} = (\frac{\partial f}{\partial x} , \frac{\partial f}{\partial y}).$

The value of ${\nabla f}_{(x,y)}$ at the point $x=a,y=b$ is obtained
by evaluating the function ${\nabla f}_{(x,y)}$ at the point $(a,b)$:

${\nabla f}_{(a,b)} = (\frac{\partial f}{\partial x} , \frac{\partial f}{\partial y})_{(a,b)}.$

Hence we see that the gradient ${\nabla f}_{(a,b)}$ of the function $f$ at the point $(a,b)$
is perpendicular to the tangent of the level curve of the function $f$ at the point $(a,b).$

Flying carpets and level surfaces

The “flying carpet” style equation for the graph of the function $f$ can be expressed as:

$z = f(x, y).$

The level surface style equation for the graph of the function $f$ can be expressed as:

$\mathrm g(x,y,z) \, \stackrel {\mathrm{def}}{=} f(x,y)-z = 0 \, .$

In the animation below, the “input” function $f$ is given by:

$f(x, y) = \frac{1}{4} (x^2 + 4 y^2)$.

The 3D-gradients of the level surfaces $\, g(x, y, z) = f(x, y) - z = c_{onstant} \,$
project (along the $\, z$-direction)
onto the 2D-gradients of the level curves
$\, f(x, y) = c_{onstant} \,$:

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$\, g(x, y, z) = \frac{1}{3} (x^2 + 4 y^2 + 9 z^2) \,$
$\, g(x, y, z) = \dfrac{x^2}{A} + \dfrac{y^2}{B} + \dfrac{z^2}{C} \; , \; C < B < A \,$