Are visualisations of magnetic fields contour maps?

No, visualizations of magnetic fields are vector slope diagrams.

I've heard to contour integrals, what are they?

Contour integrals are line integrals over the complex plane.

Main content

Course: Multivariable calculus > Unit 1

Lesson 6: Visualizing multivariable functions (articles)

Contour maps

When drawing in three dimensions is inconvenient, a contour map is a useful alternative for representing functions with a two-dimensional input and a one-dimensional output.

Background

The process

Contour maps are a way to depict functions with a two-dimensional input and a one-dimensional output. For example, consider this function:

$f (x, y) = x^{4} - x^{2} + y^{2}$ ‍.

With graphs, the way to associate the input

(x, y)

with the output

f (x, y)

is to combine both into a triplet

(x, y, f (x, y))

, and plot that triplet as a point in three-dimensional space. The graph itself consists of all possible three-dimensional points of the form

(x, y, f (x, y))

, which collectively form a surface of some kind.

But sometimes rendering a three-dimensional image can be clunky, or difficult to do by hand on the fly. Contour maps give a way to represent the function while only drawing on the two-dimensional input space.

Here's how it's done:

Step 1: Start with the graph of the function.

Step 2: Slice the graph with a few evenly-spaced level planes, each of which should be parallel to the $x y$ ‍-plane. You can think of these planes as the spots where $z$ ‍ equals some given output, like $z = 2$ ‍.

Step 3: Mark the graph where the planes cut into it.

Step 4: Project these lines onto the $x y$ ‍-plane, and label the heights they correspond to.

In other words, you choose a set of output values to represent, and for each one of these output values you draw a line which passes through all the input values $(x, y)$ ‍ for which $f (x, y)$ ‍ equals that value. To keep track of which lines correspond to which values, people commonly write down the appropriate number somewhere along each line.

Note: The choice of outputs you want to represent, such as

{- 2, - 1, 0, 1, 2}

in this example, should almost always be evenly spaced. This makes it much easier to understand the "shape" of the function just by looking at the contour map.

Example 1: Paraboloid

Consider the function

f (x, y) = x^{2} + y^{2}

. The shape of its graph is what's known as a "paraboloid", the three-dimensional equivalent of a parabola.

Here's what its contour map looks like:

Notice that the circles are not evenly spaced. This is because the height of the graph increases more quickly as you get farther away from the origin. Therefore, increasing the height by a given amount requires a smaller step away from the origin in the input space.

Example 2: Waves

How about the function

f (x, y) = \cos (x) \cdot \sin (y)

? Its graph looks super wavy:

And here is its contour map:

One feature worth pointing out here is that peaks and valleys can easily look very similar on a contour map, and can only be distinguished by reading the labels.

Example 3: Linear function

Next, let's look at

f (x, y) = x + 2 y

. Its graph is a slanted plane.

This corresponds to a contour map with evenly spaced straight lines:

Example 4: Literal map

Contour maps are often used in actual maps to portray altitude in hilly terrains. The image on the right, for example, is a depiction of a certain crater on the moon.

Imagine walking around this crater. Where the contour lines are close together, the slope is rather steep. For instance, you descend from

7700

meters to

7650

meters over a very short distance. At the bottom, where lines are sparse, things are more flat, varying between

7650

meters and

7628

meters over larger distances.

Iso-stuffs

The lines on a contour map have various names:

Contour lines.
Level sets, so named because they represent values of $(x, y)$ ‍ where the height of the graph remains unchanged, hence level.
Isolines, where "iso" is a greek prefix meaning "same".

Depending on what the contour map represents, this iso prefix might come attached to a number of things. Here are two common examples from weather maps.

An isotherm is a line on a contour map for a function representing temperature.
An isobar is a line on a contour map representing pressure.

Gaining intuition from a contour map

You can tell how steep a portion of your graph is by how close the contour lines are to one another. When they are far apart, it takes a lot of lateral distance to increase altitude, but when they are close, altitude increases quickly for small lateral increments.

The level sets associated with heights that approach a peak of the graph will look like smaller and smaller closed loops, each one encompassing the next. Likewise for an inverted peak of the graph. This means you can spot the maximum or minimum of a function using its contour map by looking for sets of closed loops enveloping one another, like distorted concentric circles.

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sauj123
Posted 8 years ago. Direct link to sauj123's post “Are visualisations of mag...”
Are visualisations of magnetic fields contour maps?
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(11 votes)
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- Tejas
  Posted 8 years ago. Direct link to Tejas's post “No, visualizations of mag...”
  No, visualizations of magnetic fields are vector slope diagrams.
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samestimable2016
Posted 8 years ago. Direct link to samestimable2016's post “I've heard to contour int...”
I've heard to contour integrals, what are they?
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(6 votes)
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- Boltzmann human
  Posted 3 years ago. Direct link to Boltzmann human's post “Contour integrals are lin...”
  Contour integrals are line integrals over the complex plane.
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dena escot
Posted 2 years ago. Direct link to dena escot's post “could you explain more th...”
could you explain more this statement "Likewise for an inverted peak of the graph. This means you can spot the maximum or minimum of a function using its contour map by looking for sets of closed loops enveloping one another, like distorted concentric circles."
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- Jerry Nilsson
  Posted 2 years ago. Direct link to Jerry Nilsson's post “For example, if 𝑧 = 𝑓(�...”
  For example, if 𝑧 = 𝑓(𝑥, 𝑦) has a minimum at (𝑥, 𝑦) = (1, 2),
  then on the contour map of 𝑓 there will be these loops (varying in size and shape) that encompass the point (1, 2).
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alliesteele
Posted a year ago. Direct link to alliesteele's post “Can we expect exercises t...”
Can we expect exercises to be added to the multivariate calculus unit like the math units preceding it? It would be very helpful!
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jdh831
Posted 10 months ago. Direct link to jdh831's post “Are equipotential lines a...”
Are equipotential lines an application of contour lines or is that something different?
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(1 vote)
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- Venkata
  Posted 10 months ago. Direct link to Venkata's post “They're actually very sim...”
  They're actually very similar ideas. In a way, contour lines show some change in gravitational potential (elevation) and show how the elevation is constant along that line. Similarly, equipotential lines show some change in electric potential and how the potential stays the same along the line.
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reele91
Posted 7 years ago. Direct link to reele91's post “With Example 2: Waves, wh...”
With Example 2: Waves, what about z = sin (y). There is no x value. How would this work?
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(1 vote)
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- Rae
  Posted 7 years ago. Direct link to Rae's post “If you only have z and y ...”
  If you only have z and y that is a function with one input and one output. When you graph it in three dimensions, x can be any real number. You can slide the graph of z=sin y along the x-axis and make a wavy sheet.
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boharesolomon2006
Posted a year ago. Direct link to boharesolomon2006's post “find the approximate r.f ...”
find the approximate r.f of the map?
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Lottie Tomlinson
Posted 7 years ago. Direct link to Lottie Tomlinson's post “how do you read a contour...”
how do you read a contour map? Btw i thought contour map is a science thing but not math cuz we learned about contour interval stuff in science but not math
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- Still No Sheep
  Posted 7 years ago. Direct link to Still No Sheep's post “Lottie, try googling your...”
  Lottie, try googling your question, and checking out the explanantions e.g. http://sectionhiker.com/how-to-read-a-topographic-map/. A lot of science is mathematical, and contours can represent heights and depths, like on a geographical map, or temperatures or pressure, like on a weather chart, or a lot of other stuff, both in science and in maths. Like carpenters and geographers might both use rulers.
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chris wilson
Posted 7 years ago. Direct link to chris wilson's post “In the contour map for th...”
In the contour map for the cos(x)sin(y) function, how is the value of 0.7 arrived at?
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(1 vote)
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John Ortiz Ordoñez
Posted 8 years ago. Direct link to John Ortiz Ordoñez's post “How to graph the contour ...”
How to graph the contour map of f(x,y)=cos(x)sin(y)
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(1 vote)
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- Still No Sheep
  Posted 7 years ago. Direct link to Still No Sheep's post “Try plugging in cos(x)*si...”
  Try plugging in cos(x)*sin(y) to the z= at https://www.math.uri.edu/~bkaskosz/flashmo/graph3d/ and play with the max and min x and y ranges (tip: set grid = 22 and click Enter New Grid for a finer grained graph), then rotate the graph to help you visualise it more clearly.
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