4.1 Vector Fields

The state of a point (more precisely of an infinitely small volume) within a natural object is frequently characterized by a vector or a tensor. Hence inside a spatial object there arises what we call a vector or tensor field. As a rule these fields are governed by simultaneous partial differential equations. Such equations are usually derived using a Cartesian space frame. In this frame, the operations of calculus closely parallel those of one dimensional analysis: the differentiation of a vector function proceeds on a component by component basis, for example. However, it is often convenient to introduce so-called curvilinear coordinates in the body, in terms of which the problem formulation is simpler. In this way we get a frame that changes from point to point, and component-wise differentiation is not enough to characterize the change of the vector function. Thus we need to develop the apparatus of calculus for vector and tensor functions when the frame in the object is changeable. Another reason for introducing these tools is the objectivity of the laws of nature: we must be able to formulate frame independent statements of these laws. Finally, there is an aesthetic reason: in non-coordinate form many statements of mathematical physics look much less cumbersome than their counterparts stated in terms of coordinates. It is said that beauty governs the world; although this is not absolutely true, most students would prefer a short and beautiful statement to a nightmarish formula taking half a page.