To introduce the unit vectors of the local coordinate system, one approach is to begin in Cartesian coordinates and describe the local coordinates in terms of these Cartesian coordinates.

In this approach, a point is represented by its Cartesian ( x, y ) coordinates, a line is represented by its equation, and so on.

Critiquing the reductionistic Cartesian view that everything can be studied in parts to understand the whole, Capra encourages his readers to take a holistic approach.

In this approach, the state of the system can be described by any type of generalized coordinates q ; the laws of motion need not be expressed in a Cartesian coordinate system, as was customary in Newtonian mechanics.

Another approach to solving the grid spacing problem is to deform a Cartesian cube such that it covers the surface of a sphere.

This rise of vernacular philosophy, which quite predated the Cartesian approach, is a new field of research whose contours are only now beginning to be clarified.

Gilbert Ryle ( 19 August 1900, Brighton – 6 October 1976, Oxford ), was a British philosopher, a representative of the generation of British ordinary language philosophers who shared Wittgenstein's approach to philosophical problems, and is principally known for his critique of Cartesian dualism, for which he coined the phrase " the ghost in the machine.

The advantage of the level set method is that one can perform numerical computations involving curves and surfaces on a fixed Cartesian grid without having to parameterize these objects ( this is called the Eulerian approach ).

The Cartesian approach has the advantage that no prior knowledge of Legendre functions, spherical harmonics, etc., is required.

His axioms, however, do not guarantee that the circles actually intersect, because they do not assert the geometrical property of continuity, which in Cartesian terms is equivalent to the completeness property of the real numbers.

A uniform space ( X, Φ ) is a set X equipped with a nonempty family Φ of subsets of the Cartesian product X × X ( Φ is called the uniform structure or uniformity of X and its elements entourages ( French: neighborhoods or surroundings )) that satisfies the following axioms:

A binary relation R is usually defined as an ordered triple ( X, Y, G ) where X and Y are arbitrary sets ( or classes ), and G is a subset of the Cartesian product X × Y.

If ( x < sub > 1 </ sub >, x < sub > 2 </ sub >, x < sub > 3 </ sub >) are the Cartesian coordinates and ( u < sub > 1 </ sub >, u < sub > 2 </ sub >, u < sub > 3 </ sub >) are the orthogonal coordinates, then

The two integers a and b are coprime if and only if the point with coordinates ( a, b ) in a Cartesian coordinate system is " visible " from the origin ( 0, 0 ), in the sense that there is no point with integer coordinates between the origin and ( a, b ).

These polar unit vectors can be expressed in terms of Cartesian unit vectors in the x and y directions, denoted i and j respectively :< ref > Note: unlike the Cartesian unit vectors i and j, which are constant, in polar coordinates the direction of the unit vectors u < sub > r </ sub > and u < sub > θ </ sub > depend on θ, and so in general have non-zero time derivatives .</ ref >

A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length.

Cartesian coordinates are the foundation of analytic geometry, and provide enlightening geometric interpretations for many other branches of mathematics, such as linear algebra, complex analysis, differential geometry, multivariate calculus, group theory, and more.

Cartesian coordinates are also essential tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering, and many more.

One can generalize the concept of Cartesian coordinates to allow axes that are not perpendicular to each other, and / or different units along each axis.

Since Cartesian coordinates are unique and non-ambiguous, the points of a Cartesian plane can be identified with all possible pairs of real numbers ; that is with the Cartesian product, where is the set of all reals.

If ( x, y ) are the Cartesian coordinates of a point, then (− x, y ) are the coordinates of its reflection across the second coordinate axis ( the Y axis ), as if that line were a mirror.

To make a figure larger or smaller is equivalent to multiplying the Cartesian coordinates of every point by the same positive number m. If ( x, y ) are the coordinates of a point on the original figure, the corresponding point on the scaled figure has coordinates

Although four-and higher-dimensional spaces are difficult to visualize, the algebra of Cartesian coordinates can be extended relatively easily to four or more variables, so that certain calculations involving many variables can be done.

Still, in addition to already discussed theories of dualism ( particularly the Christian and Cartesian Models ) there are new theories in the defense of dualism.

Examples are the product of sets ( see Cartesian product ), groups ( described below ), the product of rings and of other algebraic structures.

Some philosophers, such as Dennett, reject both epiphenomenalism and the existence of qualia with the same charge that Gilbert Ryle leveled against a Cartesian " ghost in the machine ", that they too are category mistakes.

Consistent with the symmetry of the hyperbola, if the transverse axis is aligned with the x-axis of a Cartesian coordinate system, the slopes of the asymptotes are equal in magnitude but opposite in sign, ±, where b = a × tan ( θ ) and where θ is the angle between the transverse axis and either asymptote.

Convenient Cartesian coordinates for the vertices of an icosidodecahedron with unit edges are given by:

Contrary to pre-existing Cartesian philosophy, he maintained that we are born without innate ideas, and that knowledge is instead determined only by experience derived from sense perception.

In classical mathematics, analytic geometry, also known as coordinate geometry, or Cartesian geometry, is the study of geometry using a coordinate system and the principles of algebra and analysis.

The invention of Cartesian coordinates in the 17th century by René Descartes ( Latinized name: Cartesius ) revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra.

One of Descartes ' most enduring legacies was his development of Cartesian or analytic geometry, which uses algebra to describe geometry.

* Cylinder ( algebra ), the Cartesian product of a set with its superset

We could associate an algebra to our geometry using a Cartesian coordinate system made of two lines, and represent points of our plane by vectors.

In algebra, a ring is commonly defined as a 3-tuple, where is some set, and "", and "" are functions mapping the Cartesian product to with specific properties.

Five primitive operators of Codd's algebra are the selection, the projection, the Cartesian product ( also called the cross product or cross join ), the set union, and the set difference.

Descartes unified geometry and algebra ( see: Cartesian coordinate system ).

* A Heyting algebra is a Cartesian closed ( bounded ) lattice.

Hamilton ’ s discovery derived from his attempts to find an algebra of " triplets " or 3-tuples that he believed would reflect the three Cartesian axes.

In linear algebra terms, the theorem states that, in 3D space, any two Cartesian coordinate systems with a common origin are related by a rotation about some fixed axis.

In linear algebra, a squeeze mapping is a type of linear map that preserves Euclidean area of regions in the Cartesian plane, but is not a Euclidean motion.

This is the ordinary two-dimensional Cartesian graph studied in school algebra.