8.1 Vectors and Operators as Shorthand for n-Tuples of Number and Functions


In secondary school the discussion of functions is mostly restricted to functions of one variable, i.e. to y = f(x) in Cartesian coordinates or r = g(φ) in polar coordinates. Therefore one gets used at school to the visualization of functional relationships in the xy-plane.

Real events cannot be described is this way, since they always take place in three-dimensional space with coordinates x,y,z or in a four-dimensional continuum, denoted by the space coordinates x,y,z and the time t. As an auxiliary workaround one uses only a restricted projection to a plane in space. That is possible if one assumes, that some variables are constant. On example would be y = f(t) for the movement along a straight path that is mapped to the y-axis and instead of the x-variable the parameter t is changing. One can possibly take into account a second quantity x that is changing in discrete steps by plotting a family of curves in a plane system of coordinates, for example y = f(t,xi),i = 1,2.

As soon as one wants to present events in space it becomes more complicated. The uniform movement of a point mass , i.e. without the influence of any force requires three ,, plane” parameter equations, for example x = at + a0;y = bt + b0;z = ct + c0. If one wants to describe its movement under the influence of a force that changes from point to point, one requires equations, which describe for every point in space both the absolute values as well as the direction of the force on the moving body. In coordinate notation this becomes easily messy and not vivid at all.

To come close to the vividness of two-dimensional presentations one instead uses a kind of shorthand, which combines the three space components in a vector and the functions connected to it or acting on it in a operator. If one combines the three coordinates in the vector X and three functions of time in the vector F, one can combine the above three equations as X = F(t), which is considerably clearer. If it makes sense, depends on the specific problem at hand, i.e. on whether the three components of F have a logical connection with each other. This is obviously the case for the simple movement considered above.

As soon as one starts to substitute numbers and to do calculations with them one does get around decomposing the relationship into its individual components and to formulate the corresponding algorithms. However, this process still often greatly benefits from the symbolic grouping of the individual relationships. Because of the repeated appearance of the always identical formalisms for physical problems the formulation often becomes routine.

This approach does not have to be restricted to three-dimensional descriptions, but can in principle be extended to an arbitrary number of dimensions. One can for example describe the position of two points in the three-dimensional space via two arrows or vectors starting at the origin (x1,y1,z1 and x2,y2,z2) in this space or via one vector in the six-dimensional space (x1,y1,z1,x2,y2,z2). In quantum mechanics one works with vectors in the infinitely-dimensional Hilbert-space. Plane problems can be described by two-dimensional vectors that can be considered to lie in the complex plane.

Vector algebra and vector analysis, in which partial differentiations take place, are an especially important mathematical tool of theoretical physics and therefore are often treated in depth in many textbooks for first year students.Their objects and operations are not easily accessible to the untrained imagination. Therefore the following sections concentrate only on the interactive visualization of fundamental aspects.