Editing Vector calculus (section)

{{short description|Calculus of vector-valued functions}}
{{distinguish|Geometric calculus|Matrix calculus}}
{{More footnotes|date=February 2016}}
{{Calculus}}

'''Vector calculus''' or '''vector analysis''' is a branch of mathematics concerned with the [[derivative|differentiation]] and [[integral|integration]] of [[vector field]]s, primarily in three-dimensional [[Euclidean space]], <math>\mathbb{R}^3.</math><ref>{{Cite book |last1=Kreyszig |first1=Erwin |title=Advanced Engineering Mathematics |last2=Kreyszig |first2=Herbert |last3=Norminton |first3=E. J. |date=2011 |publisher=John Wiley |isbn=978-0-470-45836-5 |edition=10th |location=Hoboken, NJ}}</ref> The term ''vector calculus'' is sometimes used as a synonym for the broader subject of [[multivariable calculus]], which spans vector calculus as well as [[partial derivative|partial differentiation]] and [[multiple integral|multiple integration]]. Vector calculus plays an important role in [[differential geometry]] and in the study of [[partial differential equation]]s. It is used extensively in physics and engineering, especially in the description of [[electromagnetic field]]s, [[gravitational field]]s, and [[fluid flow]].

Vector calculus was developed from the theory of [[quaternion]]s by [[J. Willard Gibbs]] and [[Oliver Heaviside]] near the end of the 19th century, and most of the notation and terminology was established by Gibbs and [[Edwin Bidwell Wilson]] in their 1901 book, ''[[Vector Analysis]]'', though earlier mathematicians such as [[Isaac Newton]] pioneered the field.<ref name=":17">{{Cite book |last=Rowlands |first=Peter |url=https://books.google.com/books?id=ipA4DwAAQBAJ&pg=PA26 |title=Newton and the Great World System |date=2017 |publisher=[[World Scientific Publishing]] |isbn=978-1-78634-372-7 |pages=26 |language=en |doi=10.1142/q0108}}</ref> In its standard form using the [[cross product]], vector calculus does not generalize to higher dimensions, but the alternative approach of [[geometric algebra]], which uses the [[Exterior algebra|exterior product]], does (see ''{{Section link|#Generalizations}}'' below for more).