William J Gibbs - Collected Quotes

"All field problems are really three-dimensional but a surprisinglylarge number can be treated as two-dimensional by assuming that the field distribution does not vary in the third dimension. We can put it another way by saying that the structure is treated as though it extends an infinite distance each way in the direction of the dimension not being used. Although this is never quite true it is often a reasonable idealisation of the problem for that part of the structure for which a solution is required." (William J Gibbs, "Conformal Transformations in Electrical Engineering", 1958)

"Conformal mapping as it is often called is the representation of a bounded area in the plane of a complex variable by an area in the plane of another complex variable. Thus the method is a branch of mathematics based on the theory of functions of a complex variable." (William J Gibbs, "Conformal Transformations in Electrical Engineering", 1958)

"The fields we are interested in are fields of force called inverse square fields. They are so named because the magnitude of the force at various points follows an inverse square law which is most easily conceived and explained in relation to one particular condition. The condition is that which obtains when the source of the field is concentrated at a point and is completelyisolated. The field does not contain the point but with this exception pervades all space. The reason for using this concept of a single point source is that it makes the best starting point from which to develop the equations." (William J Gibbs, "Conformal Transformations in Electrical Engineering", 1958)

"In considering the transformations it is helpful to imagine that the area enclosed by the boundaries contains an ideal elastic substance or membrane corresponding to the equipotentials and the lines of flow. The material is ideal in the sense that it is infinitely expansible and infinitely compressible without its changing character and without there being constraints of any kind. The spaces enclosed by the equipotentials and lines of flow are regarded as cells of this ideal material, which possesses the property that the boundaries can be distorted in any manner but that the lines forming the boundaries of the cells must always intersect at right angles." (William J Gibbs, "Conformal Transformations in Electrical Engineering", 1958)

"In order to describe the magnitude and direction of the force at any point in the field, a coordinate system is necessary not only to identify the position of the point in question but also to provide suitable components of the force at the point; these components considered together give both the magnitude and the direction of the force at the point selected. The coordinate system can be chosen to suit any particular problem and the form of the result." (William J Gibbs, "Conformal Transformations in Electrical Engineering", 1958)

"Laplace's equation does not yield easily to straightforward treatment: fortunately in the development of conformal transformations there is no need to seek a formal solution of the equation. It is only necessary to note first that all fields and functions to be considered in this book are those that satisfy the inverse square equation when emanating from a point source and therefore they also satisfy Laplace's equation. The second point is that the equation is necessary for the development of other important equations that govern these particular fields. Such fields are called Laplacian. When the field is not Laplacian, more recondite methods are necessary for determining its distribution." (William J Gibbs, "Conformal Transformations in Electrical Engineering", 1958)

"The concept of a vector field is difficult for some people. They have to make a mental effort, for instance, to develop the conception of the velocity field of a moving fluid. But provided there are no sources or sinks in the velocity field, the integration of the field produces a scalar field, viz. the field of flow. It is this scalar field associated with the potential that most people normally think of in connection with fluid motion. The scalar field is only a field in the mathematical sense ; it is really a map showing the potential at any point in a domain, and if lines are drawn through points of equal potential such lines are called equipotential lines. The difference of potential between two points in the field can be defined as the work done on a per unit test body when transferring it from one point in the field to another." (William J Gibbs, "Conformal Transformations in Electrical Engineering", 1958)

"With any point in the field there is associated a value of the force that would be exerted on any per unit test body placed at that point. The force on the test body must have direction as well as magnitude and hence such a field is called a vector field. The point source is the centre of the field and at every other point in it the force is directed either towards or away from this centre. Because it is an inverse square field the magnitude of the force is inversely proportional to the square of the distance from the centre." (William J Gibbs, "Conformal Transformations in Electrical Engineering", 1958)