Math 252 Lecture 15: Friday , February 5th 1999.

When operating on a scalar field, the Laplacian produces a scalar field which is the sum of the second derivatives of the original scalar field. When operating on a vector field, the Laplacian produces a vector field, each component of which is the sum of the second derivatives of the corresponding component of the original vector field. [Transparancy 2]

We explored the meaning of the Laplacian [Transparancies 2 to 5], by starting with an analogy to the second derivative of a function of one variable. By comparing the value of a function "f" at a point to its average in the immediate vicinity of the point, one can see that "f" is greater than its average when the second derivative is negative, and conversely "f" is less than its average when the second derivative is positive. [Transparancy 3]

One can make this reasoning more exact by doing a Taylor expansion of the function about a point and using this to calculate the average of the function [Transparancy 4]. An expression for the second derivative in terms of the difference between a function and its average value was obtained.

Extending this reasoning to 3 dimensions, one can see that the Laplacian of of a field is a measure of the difference between the value of the field and its average in the immediate vicinity [Transparancy 5]. For a scalar field, the Laplacian is the divergence of the gradient. This has relevence to various transport processes which are governed by Fick's Law. If the Laplacian is negative (the concentration field of a solute is greater than its local average), there will be a net outflux per unit volume (positive divergence of the flow of solute). Conversely, if the Laplacian is positive (the concentration field of a solute is less than its local average), there will be a net influx per unit volume (negative divergence of the flow of solute). If a diffusive process is in equilibrium, the Laplacian of the concentration field will be zero (Laplace's equation). However, if sources of solute are present, the concentration field satisfies Poisson's equation.

Back in Lecture 11, various examples of scalar fields were given. We went back to these example fields and looked at the Laplacian of them [Transparancies 6 and 7]. We found the following:

For isotimic planes, the Laplacian is zero.

This makes sense because the flow lines are parallel.

For isotimic spheres, the Laplacian is 6.

For diffusive flow, this would indicate that there is a constant sink of solute throughout the space, and because the concentration is increasing radially outwards there is a net influx of solute per unit volume.

For isotimic ellipsoids, the Laplacian is a positive constant.

As for isotimic spheres, there is a constant sink of solute throughout the space.

For isotimic cones, the Laplacian is 1/r where "r" is the distance from the z-axis.

This indicates a source which becomes infinite along the z-axis. (Actually, it's a sink, since the Laplacian is positive. It's like a linear black hole sucking up all the solute!)

Here are the scanned-in transparancies, in full-colour JPEG format:

(You will find two versions -- a "screen" version, which is 400 pixels wide and a corresponding number of pixels long, at a resolution of 100 by 100 dots per inch, and a "print" version, which is generally between 8 and 8.5 inches wide and up to 11 inches long, also at a resolution of 100 by 100 dots per inch. The screen versions generally have a file size of between 60 and 75 K, which downloads reltively quickly. The print versions generally have a file size of between 170 and 225 K, which takes a bit longer to download.)

• Transparancy 1
  • Screen version
  • Print version

• Transparancy 2
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• Transparancy 3
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• Transparancy 4
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• Transparancy 5
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• Transparancy 6
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• Transparancy 7
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  • Print version