Function of several real variables in the context of Implicit surface

A contour line (also isoline, isopleth, isoquant or isarithm) of a function of two variables is a curve along which the function has a constant value, so that the curve joins points of equal value. It is a plane section of the three-dimensional graph of the function ${\displaystyle f(x,y)}$ parallel to the ${\displaystyle (x,y)}$-plane. More generally, a contour line for a function of two variables is a curve connecting points where the function has the same particular value.

In cartography, a contour line (often just called a "contour") joins points of equal elevation (height) above a given level, such as mean sea level. A contour map is a map illustrated with contour lines, for example a topographic map, which thus shows valleys and hills, and the steepness or gentleness of slopes. The contour interval of a contour map is the difference in elevation between successive contour lines.

In mathematics (specifically multivariable calculus), a multiple integral is a definite integral of a function of several real variables, for instance, f(x, y) or f(x, y, z).

Integrals of a function of two variables over a region in ${\displaystyle \mathbb {R} ^{2}}$ (the real-number plane) are called double integrals, and integrals of a function of three variables over a region in ${\displaystyle \mathbb {R} ^{3}}$ (real-number 3D space) are called triple integrals. For repeated antidifferentiation of a single-variable function, see the Cauchy formula for repeated integration.

In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives.

The function is often thought of as an "unknown" that solves the equation, similar to how x is thought of as an unknown number solving, e.g., an algebraic equation like x − 3x + 2 = 0. However, it is usually impossible to write down explicit formulae for solutions of partial differential equations. There is correspondingly a vast amount of modern mathematical and scientific research on methods to numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such as existence, uniqueness, regularity and stability. Among the many open questions are the existence and smoothness of solutions to the Navier–Stokes equations, named as one of the Millennium Prize Problems in 2000.

In mathematics, a level set of a real-valued function f of n real variables is a set where the function takes on a given constant value c, that is:

When the number of independent variables is two, a level set is called a level curve, also known as contour line or isoline; so a level curve is the set of all real-valued solutions of an equation in two variables x₁ and x₂. When n = 3, a level set is called a level surface (or isosurface); so a level surface is the set of all real-valued roots of an equation in three variables x₁, x₂ and x₃. For higher values of n, the level set is a level hypersurface, the set of all real-valued roots of an equation in n > 3 variables (a higher-dimensional hypersurface).

In mathematics (specifically multivariable calculus), a multiple integral is a definite integral of a function of several real variables, for instance, f(x, y) or f(x, y, z).

Integrals of a function of two variables over a region in ${\displaystyle \mathbb {R} ^{2}}$ (the real-number plane) are called double integrals, and integrals of a function of three variables over a region in ${\displaystyle \mathbb {R} ^{3}}$ (real-number 3D space) are called triple integrals.

In multivariable calculus, an iterated integral is the result of applying integrals to a function of more than one variable (for example ${\displaystyle f(x,y)}$ or ${\displaystyle f(x,y,z)}$) in such a way that each of the integrals considers some of the variables as given constants. For example, the function ${\displaystyle f(x,y)}$, if ${\displaystyle y}$ is considered a given parameter, can be integrated with respect to ${\displaystyle x}$, ${\textstyle \int f(x,y)\,dx}$. The result is a function of ${\displaystyle y}$ and therefore its integral can be considered. If this is done, the result is the iterated integral

It is key for the notion of iterated integrals that this is different, in principle, from the multiple integral

Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to functions of several variables: the differentiation and integration of functions involving multiple variables (multivariate), rather than just one.

Multivariable calculus may be thought of as an elementary part of calculus on Euclidean space. The special case of calculus in three dimensional space is often called vector calculus.

In mathematics, particularly in calculus, a stationary point of a differentiable function of one variable is a point on the graph of the function where the function's derivative is zero. Informally, it is a point where the function "stops" increasing or decreasing (hence the name).

For a differentiable function of several real variables, a stationary point is a point on the surface of the graph where all its partial derivatives are zero (equivalently, the gradient has zero norm).The notion of stationary points of a real-valued function is generalized as critical points for complex-valued functions.

In mathematics, and more specifically in geometry, parametrization (or parameterization; also parameterisation, parametrisation) is the process of finding parametric equations of a curve, a surface, or, more generally, a manifold or a variety, defined by an implicit equation. The inverse process is called implicitization. "To parameterize" by itself means "to express in terms of parameters".

Parametrization is a mathematical process consisting of expressing the state of a system, process or model as a function of some independent quantities called parameters. The state of the system is generally determined by a finite set of coordinates, and the parametrization thus consists of one function of several real variables for each coordinate. The number of parameters is the number of degrees of freedom of the system.

In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives.

The function is often thought of as an "unknown" that solves the equation. However, it is often impossible to write down explicit formulas for solutions of partial differential equations. Hence there is a vast amount of modern mathematical and scientific research on methods to numerically approximate solutions of partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematical research, where the focus is on the qualitative features of solutions of various partial differential equations, such as existence, uniqueness, regularity and stability. Among the many open questions are the existence and smoothness of solutions to the Navier–Stokes equations, named as one of the Millennium Prize Problems in 2000.