There are an infinitude of other curved shapes in two dimensions, notably including the conic sections: the ellipse, the parabola, and the hyperbola.

conic sections: the ellipse, the parabola, and the hyperbola.

In linear algebra

Another mathematical way of viewing two-dimensional space is found in linear algebra, where the idea of independence is crucial. The plane has two dimen

Another mathematical way of viewing two-dimensional space is found in linear algebra, where the idea of independence is crucial. The plane has two dimensions because the length of a rectangle is independent of its width. In the technical language of linear algebra, the plane is two-dimensional because every point in the plane can be described by a linear combination of two independent vectors.

Dot product, angle, and length B A vector can be pictured as an arrow. Its magnitude is its length, and its direction is the direction the arrow points. The magnitude of a vector A is denoted by ${\displaystyle \|\mathbf {A} \|}$. In this viewpoint, the dot product of two Euclidean vectors A and B is defined by^[5]

${\displaystyle \mathbf {A} \cdot \mathbf {B} =\|\mathbf {A} \|\,\|\mathbf {B} \|\cos \theta ,}$angle between A and B.

The dot product of a vector A by itself is

${\displaystyle \mathbf {A} \cdot \mathbf {A} =\|\mathbf {A} \|^{2},}$

which gives

A by itself is

which gives

${\displaystyle \|\mathbf {A} \|={\sqrt {\mathbf {A} \cdot \mathbf {A} }},}$Euclidean length of the vector.

In calculus

Gradient

In a rectangular coordinate system, the gradient is given by

${\displaystyle \mathrm{\nabla <!--\; \nabla \; -->}f=\frac{\mathrm{\partial <!--\; \partial \; -->}f}{\mathrm{\partial <!--\; \partial \; -->}}}$

In a rectangular coordinate system, the gradient is given by

${\displaystyle \mathrm{\nabla <!--\; \nabla \; -->}f=\frac{\mathrm{\partial <!--\; \partial \; -->}}{}}$

For some scalar field f : U ⊆ R² → R, the line integral along a piecewise smooth curve C ⊂ U is defined as

${\displaystyle \int \limits _{C}f\,ds=\int _{a}^{b}f(\mathbf {r} (t))|\mathbf {r} '(t)|\,dt.}$

where r: [a, b] → C is an arbitrary

where r: [a, b] → C is an arbitrary bijective parametrization of the curve C such that r(a) and r(b) give the endpoints of C and ${\displaystyle a<b}$.

For a vector field F : U ⊆ R² → R², the line integral along a piecewise smooth curve C ⊂ U, in the direction of r, is defined as

${\displaystyle \underset{C}{\int <!--\; \int \; -->}\mathbf{F}(\mathbf{r})\cdot <!--\; \cdot \; -->d\mathbf{r}={\int <!--\; \int \; -->}_a^b\mathbf{F}(\mathbf{r}(t))\cdot <!--\; \cdot \; -->{\mathbf{r}}^{\prime }}$

For a vector field F : U ⊆ R² → R², the line integral along a piecewise smooth curve C ⊂ U, in the direction of r, is defined as

where · is the dot product and r: [a, b] → C is a bijective parametrization of the curve C such that r(a) and r(b) give the endpoints of C.

A double integral refers to an integral within a region D in R² of a function ${\displaystyle f(x,y),}$ and is usually written as:

${\displaystyle \iint \limits _{D}f(x,y)\,dx\,dy.}$

Fundamental theorem of line integrals : U ⊆ R 2 → R {\displaystyle \varphi :U\subseteq \mathbb {R} ^{2}\to \mathbb {R} } . Then

${\displaystyle \varphi :U\subseteq \mathbb {R} ^{2}\to \mathbb {R} }$. Then

Let C be a positively oriented, piecewise smooth, simple closed curve in a plane, and let D be the region bounded by C. If L and M are functions of (x, y) defined on an open region containing D and have continuous partial derivatives there, then^[6][7]

${\displaystyle \oint _{C}(L\,dx+M\,dy)=\iint _{D}\left({\frac {\partial M}{\partial x}}-{\frac {\partial L}{\partial y}}\right)\,dx\,dy}$

where the path of integration along C is counterclockwise.

In topology

In topology, the plane is characterized as being the unique contractible 2-manifold.

Its dimension is characterized by the fact that removing a point from the plane leaves a space that is connected, but not simply connected.

In graph theory

In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other.^[8] Such a drawing is called a plane graph or planar embedding of the graph. A plane graph can be defined as a planar graph with a mapping from every node to a point on a plane, and from every edge to a plane curve on that plane, such that the extreme points of each curve are the points mapped from its end nodes, and all curves are disjoint except on their extreme points.

See also

• Picture function

References