Gradients

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

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Related KMR-pages:

Directional Derivative
Vector Analysis

In Swedish:

Vektoranalys

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Other relevant sources of information:

Gradient
Contour line

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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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http://kmr.csc.kth.se/wp/research/math-rehab/learning-object-repository/calculus/calculus-of-several-real-variables/gradients/Gradients/

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Anchors into the text below:

http://kmr.csc.kth.se/wp/research/math-rehab/learning-object-repository/calculus/calculus-of-several-real-variables/gradients/Gradients/

Gradients
Flying carpets and level surfaces

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Gradients

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

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} \, :

The interactive simulation that created this movie (Explained in English).
The interactive simulation that created this movie (Explained in Swedish).

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Level Surface 1 – Gradient:

\, g(x, y, z) = \frac{1}{3} (x^2 + 4 y^2 + 9 z^2) \,

The interactive simulation that created this movie (Explained in English).
The interactive simulation that created this movie (Explained in Swedish).

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Level Surface 2 – Gradient:

\, g(x, y, z) = \dfrac{x^2}{A} + \dfrac{y^2}{B} + \dfrac{z^2}{C} \; , \; C < B < A \,

The interactive simulation that created this movie (Explained in English).
The interactive simulation that created this movie (Explained in Swedish).

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