Any physical law should be expressible in a form that is invariant with respect to our choice of coordinate systems we certainly do not expect that the laws of physics change when we switch from spherical coordinates to cartesian coordinates! It follows that we should be able to express physical laws without making reference to any coordinate system. Coordinate systems are human inventions, and therefore are not part of physics, although they can be used in a discussion of physics. RandallĪ Quick Introduction to Vectors, Vector Calculus, and Coordinate Systemsįor the present discussion, we define a “coordinate system” as a system for describing positions in space. In atmospheric science, vectors are normally either Revised Janu6:39 PM 1 Quick Studies in Atmospheric Science Copyright 2011 David A. Vectors are “tensors of rank 1 ” a vector can be represented by a magnitude and one direction. ![]() For example, if someone tells you the temperature in Fort Collins, you don’t have to ask whether they are using spherical coordinates or some other coordinate system, because it makes no difference at all. A scalar is expressed in exactly the same way regardless of what coordinate system is used. An example of a (single) number that is not a scalar is the longitudinal component of the wind, which is defined with respect to a particular coordinate system, i.e., spherical coordinates. Not all quantities that are represented by a single number are scalars, because not all of them are defined without reference to any particular coordinate system. The simplest kind of tensor, called a “tensor of rank 0,” is a scalar, which is represented by a single number - essentially a magnitude with no direction. Tensors are, therefore, just what we need to formulate physical laws. A tensor is simply “out there,” and has a meaning that is the same whether we happen to be working in spherical coordinates, or Cartesian coordinates, or whatever. Scalars, vectors, and tensors A tensor is a quantity that is defined without reference to any particular coordinate system. ![]() ![]() ![]() Nevertheless, it is useful to understand how physical laws can be expressed in different coordinate systems, and in particular how various quantities “transform” as we change from one coordinate system to another. A Quick Introduction to Vectors, Vector Calculus, and Coordinate Systems David Randall Physical laws and coordinate systems For the present discussion, we define a “coordinate system” as a system for describing positions in space. Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space R 3.
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