Double integrals and Jacobis

By Roy Ernest Ballard

1. Lengths, surfaces and volumes

Ordinary words like 'curve' or 'region' have to take on specific meanings if they are to be used mathematically. As a first step, empty space is endowed with imaginary points or locations in space defined by coordinates x_i. A curve is a set of points, usually an infinite set, given by some rule. One type of rule is...
Because this blogger does not support mathematical symbols please click the link to continue:

https://drive.google.com/open?id=0BxPw8lRke1HBZC1ySk9sYnFPQlk

STRESS

When a force is applied to an object the effect varies with direction. Take for instance a compressive force applied to an elastic cube like a rubber eraser. Between thumb and forefinger it can be squashed but then it bulges out sideways in directions perpendicular to the force. Not all substances bulge out sideways like this and  some even undergo a lateral contraction. . .
Because this blogger does not support mathematical symbols please click the link to continue:
https://drive.google.com/file/d/0BxPw8lRke1HBVDFFbE1Hc21EMlk/view?usp=sharing

Curvilinear coordinates

Roy Ernest Ballard

1 Position and coordinates

Coordinates are called curvilinear when coordinates are constant not over a plane but over some curved line or surface. For example any line drawn on the surface of a sphere is curved.
A one-dimensional creature living on a circle cannot measure distances other than  along the perimeter. If this creature were capable of rational thought it might want to calculate distances using axes at right angles but this requires an extra dimension, an abstract idea if you are thoroughly 1-D.  In higher dimensions the same puzzle is the subject of Riemannian geometry and the general theory of relativity.....
Because this blogger does not support mathematical symbols please click the link to continue.

https://drive.google.com/file/d/0BxPw8lRke1HBWmtubEhuQ1RuNkU/view?usp=sharing

Covariant and contravariant components, the metric tensor

This blog gives an elementary explanation of the terms covariant and contravariant as applied to vectors and describes how vectors transform under a change of coordinates. The notions of metric tensor and metric equation are explained.

For more information, open the link below.



https://drive.google.com/file/d/0BxPw8lRke1HBdzhGeHU1RkloSzQ/view?usp=sharing

The kinetics of three-stage, successive, first-order reactions.

By Roy Ernest Ballard

2 November 2014

This post is concerned with the kinetics of successive, irreversible, first order reactions. An earlier post on this subject  discussed the kinetics of a two-stage sequence. Here the complete solution is reported for the three-stage sequence:

The solution is:

For more details click the link below.


https://drive.google.com/file/d/0BxPw8lRke1HBUUVlT0RDT3RkQzA/view?usp=sharing

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Calculating matrix exponentials.

 I wanted to calculate by hand the exponential of a 3x3 matrix so I wrote out examples of two methods of doing this. To see the PDF click on the link below.

https://drive.google.com/file/d/0BxPw8lRke1HBQnd2RzFkNVYtQjA/view?usp=sharing

The kinetics of consecutive reactions.

This post presents the kinetic equation for consecutive chemical reactions of the type:

with irreversible, first order steps. They have rate constants k₁ and k₂ s^-1. Let the concentrations of the reactants be expressed as x_t,  y_t,  z_t  where x_t  is the concentration of x at time t.

The kinetics of the sequence are completely determined by the equation:

 This solution was derived using Williamson’s algorithm as described below
https://drive.google.com/file/d/0BxPw8lRke1HBdmpfQ2hNMFFfVVU/view?usp=sharing.