Abstract
Mathematical ideas of the previous chapters, such as limits, derivatives and one-dimensional integrals, are the tools we developed for functions y = f(x) of a single variable. These functions, however, present only particular cases. In practical applications functions of more than one variable are frequently encountered. For instance, a temperature of an extended object is a function of all its coordinates (x, y, z), and may also be a function of time, t, i.e. it depends on four variables. Electric and magnetic fields, E and B, in general will also depend on all three coordinates and the time. Therefore, one has to generalise our definitions of the limits and derivatives for function of many variables. It is the purpose of this chapter to lie down the necessary principles of the theory of differentiation extending the foundations we developed so far to functions of many variables. In the next chapter the notion of integration will be extended to this case as well.
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Notes
- 1.
In fact, in the latter case we have three functions, not one, for either of the fields, as each component of the field is an independent function of all these variables.
- 2.
We shall define more accurately minimum (and maximum) points of functions of many variables later on as points where the function has the lowest (the highest) values within a finite vicinity of their arguments.
- 3.
Notation \(\partial \mathbf{r}/\partial \lambda\) corresponds to the vector \(\left (\partial x/\partial \lambda,\partial y/\partial \lambda,\partial z/\partial \lambda \right )\).
- 4.
We shall learn in Sect. 5.8 that these vectors are called gradients.
- 5.
Due to Sir William Rowan Hamilton.
- 6.
Named after Pierre-Simon, marquis de Laplace.
- 7.
The factor 1∕T, which turns an inexact differential into an exact one, is called the integrating factor. We shall come across these when considering differential equations.
- 8.
We are using here the product rule written directly for the differential of the function TS.
- 9.
- 10.
If a function is defined within a finite region, than a maximum or minimum may still exist at the region boundary.
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Kantorovich, L. (2016). Functions of Many Variables: Differentiation. In: Mathematics for Natural Scientists. Undergraduate Lecture Notes in Physics. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2785-2_5
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DOI: https://doi.org/10.1007/978-1-4939-2785-2_5
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