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Bioelectricity A Quantitative Approach
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INTRODUCTION
This text is directed to presenting the fundamentals of electrophysiology from a quantitative
standpoint. The treatment of a number of topics in this book is greatly facilitated using vectors
and vector calculus. This chapter reviews the concepts of vectors and scalars and the algebraic
operations of addition and multiplication as applied to vectors. The concepts of gradient and
divergence also are reviewed, since they will be encountered more frequently.1
VECTORS AND SCALARS
In any experiment or study of biophysical phenomena one identifies one or more variables
that arise in a consideration of the observed behavior. For physical observables, variables are
classified as either scalars or vectors, that is, the variable is defined by a simple value (e.g., temperature,
conductivity, voltage) or both a magnitude plus direction (e.g., current density, force,
electric field).
In a given preparation a scalar property might vary as a function of position (e.g., the conductivity
as a function of position in a body). The collection of such values at all positions is
referred to as a scalar field. A vector function of position (e.g., blood flow at different points
in a major artery) is similarly a vector field. We designate scalars by unmodified letters, while
vectors are designated with a bar over the letter. Thus T is for temperature, but J is for current
density.
Dot Product
The scalar product (or dot product) of two vectors is defined as the product of their magnitudes
times the cosine of the angle between the vectors (assumed drawn from a common origin). From
Figure 1.1 we note that the scalar product of A and B is the product of the magnitude of one of
them (say, |B|) times the projection of the other on the first (|A| cos θ).
Comments about the Gradient
One way to gain an intuitive concept2 of the gradient is as follows: If Φ(x, y) describes the
elevation of points on the surface of a hill [corresponding to each coordinate (x, y)], then the
height (Φ) will vary from place to place in the same way as in a conventional contour map. The
gradient of Φ evaluates the slope of the hill at each point. The slope is represented by a magnitude
and direction. The magnitude signifies how steep the slope is at a particular point. The direction
of the gradient points in the most uphill (steepest) direction. On most hills, both the magnitude
and direction of the slope will vary considerably from place to place.