114 relations: Algebra over a field, Analytic geometry, Angle, Associative property, Ball (mathematics), Bijection, Binary operation, Cartesian coordinate system, Collinearity, Commutative property, Compact space, Cone, Conic section, Coordinate system, Coordinate vector, Coplanarity, Coxeter group, Cross product, Cube, Cuboid, Curl (mathematics), Curve, Curvilinear coordinates, Cylinder, Cylindrical coordinate system, Degeneracy (mathematics), Del in cylindrical and spherical coordinates, Differentiable function, Differential geometry of surfaces, Dimension, Dimensional analysis, Distance from a point to a plane, Divergence theorem, Dodecahedron, Dot product, Elevation, Ellipsoid, Engineering, Euclidean distance, Euclidean space, Euclidean vector, Function (mathematics), Geographic coordinate system, George B. Arfken, Gradient, Gradient theorem, Graph of a function, Great dodecahedron, Great icosahedron, Great stellated dodecahedron, ..., Height, Hyperboloid, Hyperplane, Icosahedral symmetry, Icosahedron, Integral, Kepler–Poinsot polyhedron, Knot theory, Length, Lie algebra, Line (geometry), Line integral, Linear algebra, Linear equation, Line–line intersection, Magnitude (mathematics), Mathematics, Matter, Multiple integral, Normal (geometry), Octahedral symmetry, Octahedron, Origin (mathematics), Paraboloid, Parallel (geometry), Parallelogram, Parameter, Parametric equation, Partial derivative, Perpendicular, Physics, Piecewise, Plane (geometry), Platonic solid, Point (geometry), Polyhedral group, Real number, Regular polyhedron, Regulus (geometry), Ruled surface, Scalar (mathematics), Scalar field, Seven-dimensional cross product, Skew lines, Small stellated dodecahedron, Sphere, Spherical coordinate system, Stokes' theorem, Surface (topology), Surface integral, Surface of revolution, Symmetry number, Tetrahedral symmetry, Tetrahedron, Two-dimensional space, Unit vector, Universe, University of Queensland, Varignon's theorem, Vector field, Vector space, Volume element, Volume integral, 3-manifold. Expand index (64 more) » « Shrink index
In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product.
In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system.
In plane geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle.
In mathematics, the associative property is a property of some binary operations.
In mathematics, a ball is the space bounded by a sphere.
In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set.
In mathematics, a binary operation on a set is a calculation that combines two elements of the set (called operands) to produce another element of the set.
A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length.
In geometry, collinearity of a set of points is the property of their lying on a single line.
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result.
In mathematics, and more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed (that is, containing all its limit points) and bounded (that is, having all its points lie within some fixed distance of each other).
A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex.
In mathematics, a conic section (or simply conic) is a curve obtained as the intersection of the surface of a cone with a plane.
In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space.
In linear algebra, a coordinate vector is a representation of a vector as an ordered list of numbers that describes the vector in terms of a particular ordered basis.
In geometry, a set of points in space are coplanar if there exists a geometric plane that contains them all.
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors).
In mathematics and vector algebra, the cross product or vector product (occasionally directed area product to emphasize the geometric significance) is a binary operation on two vectors in three-dimensional space \left(\mathbb^3\right) and is denoted by the symbol \times.
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.
In geometry, a cuboid is a convex polyhedron bounded by six quadrilateral faces, whose polyhedral graph is the same as that of a cube.
In vector calculus, the curl is a vector operator that describes the infinitesimal rotation of a vector field in three-dimensional Euclidean space.
In mathematics, a curve (also called a curved line in older texts) is, generally speaking, an object similar to a line but that need not be straight.
In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved.
A cylinder (from Greek κύλινδρος – kulindros, "roller, tumbler"), has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes.
A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis, the direction from the axis relative to a chosen reference direction, and the distance from a chosen reference plane perpendicular to the axis.
In mathematics, a degenerate case is a limiting case in which an element of a class of objects is qualitatively different from the rest of the class and hence belongs to another, usually simpler, class.
This is a list of some vector calculus formulae for working with common curvilinear coordinate systems.
In calculus (a branch of mathematics), a differentiable function of one real variable is a function whose derivative exists at each point in its domain.
In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric.
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it.
In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities (such as length, mass, time, and electric charge) and units of measure (such as miles vs. kilometers, or pounds vs. kilograms) and tracking these dimensions as calculations or comparisons are performed.
In Euclidean space, the point on a plane ax + by + cz.
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a result that relates the flow (that is, flux) of a vector field through a surface to the behavior of the vector field inside the surface.
In geometry, a dodecahedron (Greek δωδεκάεδρον, from δώδεκα dōdeka "twelve" + ἕδρα hédra "base", "seat" or "face") is any polyhedron with twelve flat faces.
In mathematics, the dot product or scalar productThe term scalar product is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space.
The elevation of a geographic location is its height above or below a fixed reference point, most commonly a reference geoid, a mathematical model of the Earth's sea level as an equipotential gravitational surface (see Geodetic datum § Vertical datum).
An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.
Engineering is the creative application of science, mathematical methods, and empirical evidence to the innovation, design, construction, operation and maintenance of structures, machines, materials, devices, systems, processes, and organizations.
In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" straight-line distance between two points in Euclidean space.
In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces.
In mathematics, physics, and engineering, a Euclidean vector (sometimes called a geometric or spatial vector, or—as here—simply a vector) is a geometric object that has magnitude (or length) and direction.
In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity.
A geographic coordinate system is a coordinate system used in geography that enables every location on Earth to be specified by a set of numbers, letters or symbols.
George Brown Arfken (born November 20, 1922) is an American theoretical physicist and the author of several mathematical physics texts.
In mathematics, the gradient is a multi-variable generalization of the derivative.
The gradient theorem, also known as the fundamental theorem of calculus for 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.
In mathematics, the graph of a function f is, formally, the set of all ordered pairs, and, in practice, the graphical representation of this set.
In geometry, the great dodecahedron is a Kepler–Poinsot polyhedron, with Schläfli symbol and Coxeter–Dynkin diagram of.
In geometry, the great icosahedron is one of four Kepler-Poinsot polyhedra (nonconvex regular polyhedra), with Schläfli symbol and Coxeter-Dynkin diagram of.
In geometry, the great stellated dodecahedron is a Kepler-Poinsot polyhedron, with Schläfli symbol.
Height is the measure of vertical distance, either how "tall" something or someone is, or how "high" the position is.
In geometry, a hyperboloid of revolution, sometimes called circular hyperboloid, is a surface that may be generated by rotating a hyperbola around one of its principal axes.
In geometry, a hyperplane is a subspace whose dimension is one less than that of its ambient space.
A regular icosahedron has 60 rotational (or orientation-preserving) symmetries, and a symmetry order of 120 including transformations that combine a reflection and a rotation.
In geometry, an icosahedron is a polyhedron with 20 faces.
In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data.
In geometry, a Kepler–Poinsot polyhedron is any of four regular star polyhedra.
In topology, knot theory is the study of mathematical knots.
In geometric measurements, length is the most extended dimension of an object.
In mathematics, a Lie algebra (pronounced "Lee") is a vector space \mathfrak g together with a non-associative, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g; (x, y) \mapsto, called the Lie bracket, satisfying the Jacobi identity.
The notion of line or straight line was introduced by ancient mathematicians to represent straight objects (i.e., having no curvature) with negligible width and depth.
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve.
Linear algebra is the branch of mathematics concerning linear equations such as linear functions such as and their representations through matrices and vector spaces.
In mathematics, a linear equation is an equation that may be put in the form where x_1, \ldots, x_n are the variables or unknowns, and c, a_1, \ldots, a_n are coefficients, which are often real numbers, but may be parameters, or even any expression that does not contain the unknowns.
In Euclidean geometry, the intersection of a line and a line can be the empty set, a point, or a line.
In mathematics, magnitude is the size of a mathematical object, a property which determines whether the object is larger or smaller than other objects of the same kind.
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
In the classical physics observed in everyday life, matter is any substance that has mass and takes up space by having volume.
The multiple integral is a definite integral of a function of more than one real variable, for example, or.
In geometry, a normal is an object such as a line or vector that is perpendicular to a given object.
A regular octahedron has 24 rotational (or orientation-preserving) symmetries, and a symmetry order of 48 including transformations that combine a reflection and a rotation.
In geometry, an octahedron (plural: octahedra) is a polyhedron with eight faces, twelve edges, and six vertices.
In mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space.
In geometry, a paraboloid is a quadric surface that has (exactly) one axis of symmetry and no center of symmetry.
In geometry, parallel lines are lines in a plane which do not meet; that is, two lines in a plane that do not intersect or touch each other at any point are said to be parallel.
In Euclidean geometry, a parallelogram is a simple (non-self-intersecting) quadrilateral with two pairs of parallel sides.
A parameter (from the Ancient Greek παρά, para: "beside", "subsidiary"; and μέτρον, metron: "measure"), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when identifying the system, or when evaluating its performance, status, condition, etc.
In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters.
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary).
In elementary geometry, the property of being perpendicular (perpendicularity) is the relationship between two lines which meet at a right angle (90 degrees).
Physics (from knowledge of nature, from φύσις phýsis "nature") is the natural science that studies matterAt the start of The Feynman Lectures on Physics, Richard Feynman offers the atomic hypothesis as the single most prolific scientific concept: "If, in some cataclysm, all scientific knowledge were to be destroyed one sentence what statement would contain the most information in the fewest words? I believe it is that all things are made up of atoms – little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another..." and its motion and behavior through space and time and that studies the related entities of energy and force."Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of events." Physics is one of the most fundamental scientific disciplines, and its main goal is to understand how the universe behaves."Physics is one of the most fundamental of the sciences. Scientists of all disciplines use the ideas of physics, including chemists who study the structure of molecules, paleontologists who try to reconstruct how dinosaurs walked, and climatologists who study how human activities affect the atmosphere and oceans. Physics is also the foundation of all engineering and technology. No engineer could design a flat-screen TV, an interplanetary spacecraft, or even a better mousetrap without first understanding the basic laws of physics. (...) You will come to see physics as a towering achievement of the human intellect in its quest to understand our world and ourselves."Physics is an experimental science. Physicists observe the phenomena of nature and try to find patterns that relate these phenomena.""Physics is the study of your world and the world and universe around you." Physics is one of the oldest academic disciplines and, through its inclusion of astronomy, perhaps the oldest. Over the last two millennia, physics, chemistry, biology, and certain branches of mathematics were a part of natural philosophy, but during the scientific revolution in the 17th century, these natural sciences emerged as unique research endeavors in their own right. Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry, and the boundaries of physics are not rigidly defined. New ideas in physics often explain the fundamental mechanisms studied by other sciences and suggest new avenues of research in academic disciplines such as mathematics and philosophy. Advances in physics often enable advances in new technologies. For example, advances in the understanding of electromagnetism and nuclear physics led directly to the development of new products that have dramatically transformed modern-day society, such as television, computers, domestic appliances, and nuclear weapons; advances in thermodynamics led to the development of industrialization; and advances in mechanics inspired the development of calculus.
In mathematics, a piecewise-defined function (also called a piecewise function or a hybrid function) is a function defined by multiple sub-functions, each sub-function applying to a certain interval of the main function's domain, a sub-domain.
In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far.
In three-dimensional space, a Platonic solid is a regular, convex polyhedron.
In modern mathematics, a point refers usually to an element of some set called a space.
In geometry, the polyhedral group is any of the symmetry groups of the Platonic solids.
In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.
A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags.
In three-dimensional space, a regulus R is a set of skew lines, every point of which is on a transversal which intersects an element of R only once, and such that every point on a transversal lies on a line of R The set of transversals of R forms an opposite regulus S. In ℝ3 the union R ∪ S is the ruled surface of a hyperboloid of one sheet.
In geometry, a surface S is ruled (also called a scroll) if through every point of S there is a straight line that lies on S. Examples include the plane, the curved surface of a cylinder or cone, a conical surface with elliptical directrix, the right conoid, the helicoid, and the tangent developable of a smooth curve in space.
A scalar is an element of a field which is used to define a vector space.
In mathematics and physics, a scalar field associates a scalar value to every point in a space – possibly physical space.
In mathematics, the seven-dimensional cross product is a bilinear operation on vectors in seven-dimensional Euclidean space.
In three-dimensional geometry, skew lines are two lines that do not intersect and are not parallel.
In geometry, the small stellated dodecahedron is a Kepler-Poinsot polyhedron, named by Arthur Cayley, and with Schläfli symbol.
A sphere (from Greek σφαῖρα — sphaira, "globe, ball") is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball (viz., analogous to the circular objects in two dimensions, where a "circle" circumscribes its "disk").
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its polar angle measured from a fixed zenith direction, and the azimuth angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane.
In vector calculus, and more generally differential geometry, Stokes' theorem (also called the generalized Stokes theorem or the Stokes–Cartan theorem) is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus.
In topology and differential geometry, a surface is a two-dimensional manifold, and, as such, may be an "abstract surface" not embedded in any Euclidean space.
In mathematics, a surface integral is a generalization of multiple integrals to integration over surfaces.
A surface of revolution is a surface in Euclidean space created by rotating a curve (the generatrix) around an axis of rotation.
The symmetry number or symmetry order of an object is the number of different but indistinguishable (or equivalent) arrangements (or views) of the object, i.e. the order of its symmetry group.
A regular tetrahedron, an example of a solid with full tetrahedral symmetry A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection and a rotation.
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners.
Two-dimensional space or bi-dimensional space is a geometric setting in which two values (called parameters) are required to determine the position of an element (i.e., point).
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1.
The Universe is all of space and time and their contents, including planets, stars, galaxies, and all other forms of matter and energy.
The University of Queensland (UQ) is a public research university primarily located in Queensland's capital city, Brisbane, Australia.
Varignon's theorem is a statement in Euclidean geometry, that deals with the construction of a particular parallelogram, the Varignon parallelogram, from an arbitrary quadrilateral (quadrangle).
In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space.
A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.
In mathematics, a volume element provides a means for integrating a function with respect to volume in various coordinate systems such as spherical coordinates and cylindrical coordinates.
In mathematics—in particular, in multivariable calculus—a volume integral refers to an integral over a 3-dimensional domain, that is, it is a special case of multiple integrals.
In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space.
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