Friday, April 24, 2009

Practical Implications of Calculus

Calculus is widely used by engineers and scientists to analyse practical problems. One common approach is to analyse a particular system using fundamental physics for a particular geometry (e.g. a force balance around a spherical particle falling in a liquid) to form equations. These equations are than either integrated or differentiated to produce useful relationships for a given set of boundary conditions (e.g. settling time of a particle as a function of size and density for a given initial particle velocity). The success of this approach normally depends on the nature of the phenomena being studied (some very chaotic and/or highly non-linear phenomena are difficult to model), the assumptions made in setting up the problems and the difficulty in solving the equations formed. Often, numerical techniques are used to find solutions to these equations and any good engineering mathematics course teaches a range of relevant numerical techniques to differentiate and/or integrate equations that are either difficult or impossible to solve directly.

Another interesting application of calculus is to analyse data. Consider a set of data collected in an experiment ..... imagine we are measuring X and Y simultaneously. When we plot X against Y, the curve generated may clearly show a relationship exists but the relationship is not simple or immediately apparent. A very simple method to start analysing this mysterious relationship, is to differentiate the X Y plot numerically (i.e. calculate the slope at points along the curve) and form a new plot of dX/dY vrs X. Now remember that we differentiate particular functions, new very specific relationships are formed. For example, differentiating a trigonometric function will generate another trigonometric function, and in the case of simple trigonometric functions like sine and cosine, functions are formed that have very specific geometric relationships to the original functions (e.g. cosine has the same shape and periodic form of sine but is "out of phase" with that relationship). In the case of polynomials, differentiating produces a function of lower order; the slope of a cubic follows a parabolic relationship, the differential of a parabolic functions produces a linear function and so on. This means that by differentiating a curve (i.e. measuring the slope of the curve at each point) some of these underlying relationships in the data maybe revealed.

This approach can be extended to differentiating the dX/dY curve formed, as double differentiation also can unlock some underlying relationship For example, differentiating sinX will form cosX and differentiating that relationship will produce a negative version of the original relationship. Double differentiation of a cubic function will generate a linear function (try it !). Thus, the "slope of the slope" can potentially tell alot about the original relationship. This line of attack can be extended to integration, through measuring the area under the Y curve and plotting this relationship against X. Of course, both taking the slope and measuring the area can be used in combination to tackle the problem.

The beauty of this methodology is that the procedure is very simple (e.g. measuring a slope of a line) and easily automated. You can try it yourself .... I suggest asking a mathematically inclined friend to dream up a complex function that is the combination of well known simple functions (e.g. cosx + x^3 + exp(x)), get him or her to form an x y table of values from this relationship and than ask you to derive the underlying relationship from this data set. The detective job in front of you is made simple by modern graphical/CAS calculators that allow ready numerical differentiation and integration of curves. Sometimes, a combination of intuition, luck and insight is required to identify the underlying relationship but the journey is normally fun. Try it !!!

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