Contents 1 History 2 In geometry 2.1 Coordinate systems 2.2 Polytopes 2.2.1 Convex 2.2.2 Degenerate (spherical) 2.2.3 Non-convex 2.3 Circle 2.4 Other shapes 3 In linear algebra 3.1 Dot product, angle, and length 4 In calculus 4.1 Gradient 4.2 Line integrals and double integrals 4.3 Fundamental theorem of line integrals 4.4 Green's theorem 5 In topology 6 In graph theory 7 References 8 See also History[edit]
Books I through IV and VI of
Cartesian coordinate system Polar coordinate system Polytopes[edit]
Main article: Polygon
In two dimensions, there are infinitely many polytopes: the polygons.
The first few regular ones are shown below:
Convex[edit]
The
Name Triangle (2-simplex) Square (2-orthoplex) (2-cube) Pentagon Hexagon Heptagon Octagon Schläfli 3 4 5 6 7 8 Image Name Nonagon Decagon Hendecagon Dodecagon Tridecagon Tetradecagon Schläfli 9 10 11 12 13 14 Image Name Pentadecagon Hexadecagon Heptadecagon Octadecagon Enneadecagon Icosagon ...n-gon Schläfli 15 16 17 18 19 20 n Image Degenerate (spherical)[edit] The regular henagon 1 and regular digon 2 can be considered degenerate regular polygons. They can exist nondegenerately in non-Euclidean spaces like on a 2-sphere or a 2-torus. Name Henagon Digon Schläfli 1 2 Image Non-convex[edit]
There exist infinitely many non-convex regular polytopes in two
dimensions, whose
Name Pentagram Heptagrams Octagram Enneagrams Decagram ...n-agrams Schläfli 5/2 7/2 7/3 8/3 9/2 9/4 10/3 n/m Image
Circle[edit] Main article: Circle The hypersphere in 2 dimensions is a circle, sometimes called a 1-sphere (S1) because it is a one-dimensional manifold. In a Euclidean plane, it has the length 2πr and the area of its interior is A = π r 2 displaystyle A=pi r^ 2 where r displaystyle r is the radius. Other shapes[edit] Main article: List of two-dimensional geometric shapes There are an infinitude of other curved shapes in two dimensions, notably including the conic sections: the ellipse, the parabola, and the hyperbola. In linear algebra[edit] 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[edit] Main article: Dot product The dot product of two vectors A = [A1, A2] and B = [B1, B2] is defined as:[4] A ⋅ B = A 1 B 1 + A 2 B 2 displaystyle mathbf A cdot mathbf B =A_ 1 B_ 1 +A_ 2 B_ 2 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 ‖ A ‖ displaystyle mathbf A . In this viewpoint, the dot product of two Euclidean vectors A and B is defined by[5] A ⋅ B = ‖ A ‖ ‖ B ‖ cos θ , displaystyle mathbf A cdot mathbf B =mathbf A ,mathbf B cos theta , where θ is the angle between A and B. The dot product of a vector A by itself is A ⋅ A = ‖ A ‖ 2 , displaystyle mathbf A cdot mathbf A =mathbf A ^ 2 , which gives ‖ A ‖ = A ⋅ A , displaystyle mathbf A = sqrt mathbf A cdot mathbf A , the formula for the
∇ f = ∂ f ∂ x i + ∂ f ∂ y j displaystyle nabla f= frac partial f partial x mathbf i + frac partial f partial y mathbf j Line integrals and double integrals[edit] For some scalar field f : U ⊆ R2 → R, the line integral along a piecewise smooth curve C ⊂ U is defined as ∫ C f d s = ∫ a b f ( r ( t ) )
r ′ ( t )
d t . 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 bijective parametrization of the curve C such that r(a) and r(b) give the endpoints of C and a < b 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 ∫ C F ( r ) ⋅ d r = ∫ a b F ( r ( t ) ) ⋅ r ′ ( t ) d t . displaystyle int limits _ C mathbf F (mathbf r )cdot ,dmathbf r =int _ a ^ b mathbf F (mathbf r (t))cdot mathbf r '(t),dt. 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 f ( x , y ) , displaystyle f(x,y), and is usually written as: ∬ D f ( x , y ) d x d y . displaystyle iint limits _ D f(x,y),dx,dy. Fundamental theorem of line integrals[edit] Main article: Fundamental theorem of line integrals The fundamental theorem of line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve. Let φ : U ⊆ R 2 → R displaystyle varphi :Usubseteq mathbb R ^ 2 to mathbb R . Then φ ( q ) − φ ( p ) = ∫ γ [ p , q ] ∇ φ ( r ) ⋅ d r . displaystyle varphi left(mathbf q right)-varphi left(mathbf p right)=int _ gamma [mathbf p ,,mathbf q ] nabla varphi (mathbf r )cdot dmathbf r . Green's theorem[edit] Main article: Green's theorem 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] ∮ C ( L d x + M d y ) = ∬ D ( ∂ M ∂ x − ∂ L ∂ y ) d x d y 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[edit] 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[edit] 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. References[edit] ^ "Analytic geometry". Encyclopædia Britannica (Encyclopædia Britannica Online ed.). 2008. access-date= requires url= (help) ^ Burton 2011, p. 374 ^ Wessel's memoir was presented to the Danish Academy in 1797; Argand's paper was published in 1806. (Whittaker & Watson, 1927, p. 9) ^ S. Lipschutz; M. Lipson (2009). Linear Algebra (Schaum’s Outlines) (4th ed.). McGraw Hill. ISBN 978-0-07-154352-1. ^ M.R. Spiegel; S. Lipschutz; D. Spellman (2009). Vector Analysis (Schaum’s Outlines) (2nd ed.). McGraw Hill. ISBN 978-0-07-161545-7. ^ Mathematical methods for physics and engineering, K.F. Riley, M.P. Hobson, S.J. Bence, Cambridge University Press, 2010, ISBN 978-0-521-86153-3 ^ Vector Analysis (2nd Edition), M.R. Spiegel, S. Lipschutz, D. Spellman, Schaum’s Outlines, McGraw Hill (USA), 2009, ISBN 978-0-07-161545-7 ^ Trudeau, Richard J. (1993). Introduction to Graph Theory (Corrected, enlarged republication. ed.). New York: Dover Pub. p. 64. ISBN 978-0-486-67870-2. Retrieved 8 August 2012. Thus a planar graph, when drawn on a flat surface, either has no edge-crossings or can be redrawn without them. See also[edit] Two-dimensional graph v t e Dimension Dimensional spaces Vector space Euclidean space Affine space Projective space Free module Manifold Algebraic variety Spacetime Other dimensions Krull Lebesgue covering Inductive Hausdorff Minkowski Fractal Degrees of freedom Polytopes and shapes Hyperplane Hypersurface Hypercube Hypersphere Hyperrectangle Demihypercube Cross-polytope Simplex Dimensions by number Zero One Two Three Four Five Six Seven Eight Nine n-dimensions Negative dimensions |