This is a file from the Wikimedia Commons
From Wikipedia, the free encyclopedia

Line_integral_of_scalar_field.gif(400 × 300 pixels, file size: 580 KB, MIME type: image/gif, looped, 61 frames, 39 s)


English: Line integral of a scalar field, f. The area under the curve C, traced on the surface defined by z = f(x,y), is the value of the integral. See full description.
فارسی: انتگرال خطی یک میدان اسکالر f. مقدار انتگرال مساحت زیر منحنی C تعریف شده توسط سطح (z = f(x,y است.
Français : L′ intégrale curviligne d′un champ scalaire, f. L′aire sous la courbe C, tracée sur la surface définie par z = f(x,y), est la valeur de l'intégrale.
Italiano: Integrale di linea di un campo scalare, f. Il valore dell'integrale è pari all'area sotto la curva C, tracciata sulla superficie definita da z = f(x,y).
Русский: Иллюстрация криволинейного интеграла первого рода на скалярном поле.
Source Own work
Author Lucas Vieira
( Reusing this file)
Public domain I, the copyright holder of this work, release this work into the public domain. This applies worldwide.
In some countries this may not be legally possible; if so:
I grant anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law.
Other versions


Featured picture

Wikimedia CommonsWikipedia

This file was a finalist in Picture of the Year 2012.
This is a featured picture on Wikimedia Commons ( Featured pictures) and is considered one of the finest images. See its nomination here.

Cscr-featured.svg This is a featured picture on the English language Wikipedia ( Featured pictures) and is considered one of the finest images. See its nomination here.
Cscr-featured.svg This is a featured picture on the Persian language Wikipedia ( نگاره‌های برگزیده) and is considered one of the finest images. See its nomination here.

If you have an image of similar quality that can be published under a suitable copyright license, be sure to upload it, tag it, and nominate it.

Full description (English)

A scalar field has a value associated to each point in space. Examples of scalar fields are height, temperature or pressure maps. In a two-dimensional field, the value at each point can be thought of as a height of a surface embedded in three dimensions. The line integral of a curve along this scalar field is equivalent to the area under a curve traced over the surface defined by the field.

In this animation, all these processes are represented step-by-step, directly linking the concept of the line integral over a scalar field to the representation of integrals familiar to students, as the area under a simpler curve. A breakdown of the steps:

  1. The color-coded scalar field f and a curve C are shown. The curve C starts at a and ends at b
  2. The field is rotated in 3D to illustrate how the scalar field describes a surface. The curve C, in blue, is now shown along this surface. This shows how at each point in the curve, a scalar value (the height) can be associated.
  3. The curve is projected onto the plane XY (in gray), giving us the red curve, which is exactly the curve C as seen from above in the beginning. This is red curve is the curve in which the line integral is performed. The distances from the projected curve (red) to the curve along the surface (blue) describes a "curtain" surface (in blue).
  4. The graph is rotated to face the curve from a better angle
  5. The projected curve is rectified (made straight), and the same transformation follows on the blue curve, along the surface. This shows how the line integral is applied to the arc length of the given curve
  6. The graph is rotated so we view the blue surface defined by both curves face on
  7. This final view illustrates the line integral as the familiar integral of a function, whose value is the "signed area" between the X axis (the red curve, now a straight line) and the blue curve (which gives the value of the scalar field at each point). Thus, we conclude that the two integrals are the same, illustrating the concept of a line integral on a scalar field in an intuitive way.



Add a one-line explanation of what this file represents

Items portrayed in this file


24 July 2012


File history

Click on a date/time to view the file as it appeared at that time.

current 16:43, 14 August 2012 Thumbnail for version as of 16:43, 14 August 2012400 × 300 (580 KB)LucasVBUnoptimized. Sticking with local palettes for better color resolution per frame. Added bands of color to the field instead of a smooth gradient. Overall, it should look sharper, though the file will be bigger. Worth it, I say!
12:24, 25 July 2012 Thumbnail for version as of 12:24, 25 July 2012400 × 300 (328 KB)LucasVBAlternative illustration of the "straightening" of the curve. It should convey the concept better than the previous one, which may be interpreted as a mere projection. Also, changed to pattern dithering. Seems to look better, and file is smaller even t...
16:59, 24 July 2012 Thumbnail for version as of 16:59, 24 July 2012400 × 300 (337 KB)LucasVB{{Information |Description= |Source={{own}} |Date=2012-07-24 |Author= Kieff |Permission={{PD-self}} |other_versions= }}

Global file usage

The following other wikis use this file:

View more global usage of this file.