Multivariable calculus

A study of limits and continuity in multivariable calculus yields many counterintuitive results not demonstrated by single-variable functions.[1]^: 19–22 For example, there are scalar functions of two variables with points in their domain which give different limits when approached along different paths. E.g., the function.

${\displaystyle f(x,y)={\frac {x^{2}y}{x^{4}+y^{2}}}}$

Plot of the function f(x, y) = (x²y)/(x⁴ + y²)

approaches zero whenever the point ${\displaystyle (0,0)}$ is approached along lines through the origin (${\displaystyle y=kx}$). However, when the origin is approached along a parabola ${\displaystyle y=\pm x^{2}}$, the function value has a limit of ${\displaystyle \pm 1/2}$. Since taking different paths toward the same point yields different limit values, a general limit does not exist there.

Continuity in each argument not being sufficient for multivariate continuity can also be seen from the following example.[1]^: 17–19 In particular, for a real-valued function with two real-valued parameters, ${\displaystyle f(x,y)}$, continuity of ${\displaystyle f}$ in ${\displaystyle x}$ for fixed ${\displaystyle y}$ and continuity of ${\displaystyle f}$ in ${\displaystyle y}$ for fixed ${\displaystyle x}$ does not imply continuity of ${\displaystyle f}$.

Consider

${\displaystyle f(x,y)={\begin{cases}{\frac {y}{x}}-y&{\text{if}}\quad 0\leq y<x\leq 1\\{\frac {x}{y}}-x&{\text{if}}\quad 0\leq x<y\leq 1\\1-x&{\text{if}}\quad 0<x=y\\0&{\text{everywhere else}}.\end{cases}}}$

It is easy to verify that this function is zero by definition on the boundary and outside of the quadrangle ${\displaystyle (0,1)\times (0,1)}$. Furthermore, the functions defined for constant ${\displaystyle x}$ and ${\displaystyle y}$ and ${\displaystyle 0\leq a\leq 1}$ by

${\displaystyle g_{a}(x)=f(x,a)\quad }$ and ${\displaystyle \quad h_{a}(y)=f(a,y)\quad }$

are continuous. Specifically,

${\displaystyle g_{0}(x)=f(x,0)=h_{0}(0,y)=f(0,y)=0}$ for all x and y.

However, the sequence ${\displaystyle f\left({\tfrac {1}{n}},{\tfrac {1}{n}}\right)}$ (for natural ${\displaystyle n}$) converges to ${\displaystyle \lim _{n\to \infty }f\left({\tfrac {1}{n}},{\tfrac {1}{n}}\right)=1}$, rendering the function as discontinuous at ${\displaystyle (0,0)}$. Approaching the origin not along parallels to the ${\displaystyle x}$- and ${\displaystyle y}$-axis reveals this discontinuity.

If ${\displaystyle f(x,y)}$ is continuous at ${\displaystyle (a,b),}$ and ${\displaystyle g}$ is a single variable function continuous at ${\displaystyle f(a,b),}$ then the composite function ${\displaystyle h=g\circ f}$ defined by ${\displaystyle h(x,y)=g(f(x,y))}$ is continuous at ${\displaystyle (a,b).}$

For examples, ${\displaystyle \exp(x-y)}$ and ${\displaystyle \ln(1+xy-4x+10y).}$

If ${\displaystyle f(x,y)}$ and ${\displaystyle g(x,y)}$ are both continuous at ${\displaystyle (a,b)}$ then

(i) ${\displaystyle f(x,y)\pm g(x,y)}$ are continuous at ${\displaystyle (a,b).}$

(ii) ${\displaystyle cf(x,y)}$ is continuous at ${\displaystyle (a,b)}$ for any constant c.

(iii) ${\displaystyle f(x,y)}$ ${\displaystyle .}$ ${\displaystyle g(x,y)}$ is continuous at point ${\displaystyle (a,b).}$

(iv) ${\displaystyle {\frac {f(x,y)}{g(x,y)}}}$ is continuous at ${\displaystyle (a,b),}$ if ${\displaystyle g(a,b)\neq 0.}$

(v) ${\displaystyle \mid f(x,y)\mid }$ is continuous at ${\displaystyle (a,b).}$

The partial derivative generalizes the notion of the derivative to higher dimensions. A partial derivative of a multivariable function is a derivative with respect to one variable with all other variables held constant.[1]^: 26ff

Partial derivatives may be combined in interesting ways to create more complicated expressions of the derivative. In vector calculus, the del operator (${\displaystyle \nabla }$) is used to define the concepts of gradient, divergence, and curl in terms of partial derivatives. A matrix of partial derivatives, the Jacobian matrix, may be used to represent the derivative of a function between two spaces of arbitrary dimension. The derivative can thus be understood as a linear transformation which directly varies from point to point in the domain of the function.

Differential equations containing partial derivatives are called partial differential equations or PDEs. These equations are generally more difficult to solve than ordinary differential equations, which contain derivatives with respect to only one variable.[1]^: 654ff

The multiple integral expands the concept of the integral to functions of any number of variables. Double and triple integrals may be used to calculate areas and volumes of regions in the plane and in space. Fubini's theorem guarantees that a multiple integral may be evaluated as a repeated integral or iterated integral as long as the integrand is continuous throughout the domain of integration.[1]^: 367ff

The surface integral and the line integral are used to integrate over curved manifolds such as surfaces and curves.

In single-variable calculus, the fundamental theorem of calculus establishes a link between the derivative and the integral. The link between the derivative and the integral in multivariable calculus is embodied by the integral theorems of vector calculus:[1]^: 543ff

• Gradient theorem
• Stokes' theorem
• Divergence theorem
• Green's theorem.

In a more advanced study of multivariable calculus, it is seen that these four theorems are specific incarnations of a more general theorem, the generalized Stokes' theorem, which applies to the integration of differential forms over manifolds.[2]