In mathematics, calculus on Euclidean space is a generalization of calculus of functions in one or several variables to calculus of functions on Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ as well as a finite-dimensional real vector space. This calculus is also known as advanced calculus, especially in the United States. It is similar to multivariable calculus but is somewhat more sophisticated in that it uses linear algebra (or some functional analysis) more extensively and covers some concepts from differential geometry such as differential forms and Stokes' formula in terms of differential forms. This extensive use of linear algebra also allows a natural generalization of multivariable calculus to calculus on Banach spaces or topological vector spaces.

Calculus on Euclidean space is also a local model of calculus on manifolds, a theory of functions on manifolds.

This section is a brief review of function theory in one-variable calculus.

A real-valued function ${\displaystyle f:\mathbb {R} \to \mathbb {R} }$ is continuous at ${\displaystyle a}$ if it is approximately constant near ${\displaystyle a}$; i.e.,

In contrast, the function ${\displaystyle f}$ is differentiable at ${\displaystyle a}$ if it is approximately linear near ${\displaystyle a}$; i.e., there is some real number ${\displaystyle \lambda }$ such that

(For simplicity, suppose ${\displaystyle f(a)=0}$. Then the above means that ${\displaystyle f(a+h)=\lambda h+g(a,h)}$ where ${\displaystyle g(a,h)}$ goes to 0 faster than h going to 0 and, in that sense, ${\displaystyle f(a+h)}$ behaves like ${\displaystyle \lambda h}$.)

The number ${\displaystyle \lambda }$ depends on ${\displaystyle a}$ and thus is denoted as ${\displaystyle f'(a)}$. If ${\displaystyle f}$ is differentiable on an open interval ${\displaystyle U}$ and if ${\displaystyle f'}$ is a continuous function on ${\displaystyle U}$, then ${\displaystyle f}$ is called a C¹ function. More generally, ${\displaystyle f}$ is called a C^k function if its derivative ${\displaystyle f'}$ is C^k-1 function. Taylor's theorem states that a C^k function is precisely a function that can be approximated by a polynomial of degree k.

If ${\displaystyle f:\mathbb {R} \to \mathbb {R} }$ is a C¹ function and ${\displaystyle f'(a)\neq 0}$ for some ${\displaystyle a}$, then either ${\displaystyle f'(a)>0}$ or ${\displaystyle f'(a)<0}$; i.e., either ${\displaystyle f}$ is strictly increasing or strictly decreasing in some open interval containing a. In particular, ${\displaystyle f:f^{-1}(U)\to U}$ is bijective for some open interval ${\displaystyle U}$ containing ${\displaystyle f(a)}$. The inverse function theorem then says that the inverse function ${\displaystyle f^{-1}}$ is differentiable on U with the derivatives: for ${\displaystyle y\in U}$