Abstract
We say that a function of several variables is homogeneous, when, by letting all of the variables grow or decline in a given ratio, we obtain for a result the original value of the function multiplied by a power of this ratio. The exponent of this power is the degree of the homogeneous function. By consequence, \(f(x, y, z, \dots )\) will be a homogeneous function of \( x, y, z, \dots \) and of degree a, if, t denoting a new variable, we have regardless of t, .
.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The rightmost ellipsis is omitted in equation (1) in the 1899 reprint but is present in the original 1823 edition. Except for the lack of specificity as to whether Cauchy has a finite or infinite number of variables in mind (the term “several” is used), Cauchy’s definition is very similar to a modern-day definition of a real homogeneous function.
- 2.
The original 1823 edition prints the y in the \({\varvec{L}}\Big (\frac{x}{y}\Big )\) expression of the second example in a slightly different font. However, this misprint is corrected in its ERRATA.
- 3.
There is a typographical error in the 1899 reprint that is not present in the original 1823 edition. The 1899 reprint reads, “...when we replace x by \(x=\alpha h.\)”
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Cates, D.M. (2019). THEOREM OF HOMOGENEOUS FUNCTIONS. MAXIMA AND MINIMA OF FUNCTIONS OF SEVERAL VARIABLES.. In: Cauchy's Calcul Infinitésimal. Springer, Cham. https://doi.org/10.1007/978-3-030-11036-9_10
Download citation
DOI: https://doi.org/10.1007/978-3-030-11036-9_10
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-11035-2
Online ISBN: 978-3-030-11036-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)