In elementary 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 both a magnitude (or length) and direction. A Euclidean vector is frequently represented by a line segment with a definite direction, or graphically as an arrow, connecting an initial point A with a terminal point B, and denoted by $\backslash overrightarrow\{AB\}.$

A vector is what is needed to "carry" the point A to the point B; the Latin word vector means "carrier". The magnitude of the vector is the distance between the two points and the direction refers to the direction of displacement from A to B. Many algebraic operations on real numbers such as addition, subtraction, multiplication, and negation have close analogues for vectors, operations which obey the familiar algebraic laws of commutativity, associativity, and distributivity. These operations and associated laws qualify Euclidean vectors as an example of the more generalized concept of vectors defined simply as elements of a vector space.

Vectors play an important role in physics: velocity and acceleration of a moving object and forces acting on it are all described by vectors. Many other physical quantities can be usefully thought of as vectors. Although most of them do not represent distances (such as position or displacement), their magnitude and direction can be still represented by the length and direction of an arrow. The mathematical representation of a physical vector depends on the coordinate system used to describe it. Other vector-like objects that describe physical quantities and transform in a similar way under changes of the coordinate system include pseudovectors and tensors.

Overview

A vector is a geometric entity characterized by a magnitude (in mathematics a number, in physics a number times a unit) and a direction. In rigorous mathematical treatments, a vector is defined as a directed line segment, or arrow, in a Euclidean space. When it becomes necessary to distinguish it from vectors as defined elsewhere, this is sometimes referred to as a geometric, spatial, or Euclidean vector.

As an arrow in Euclidean space, a vector possesses a definite initial point and terminal point. Such a vector called a bound vector. When only the magnitude and direction of the vector matter, then the particular initial point is of no importance, and the vector is called a free vector. Thus two arrows $\backslash overrightarrow\{AB\}$ and $\backslash overrightarrow\{A\text{'}B\text{'}\}$ in space represent the same free vector if they have the same magnitude and direction: that is, they are equivalent if the quadrilateral ABB′A′ is a parallelogram. If the Euclidean space is equipped with a choice of origin, then a free vector is equivalent to the bound vector of the same magnitude and direction whose initial point is the origin.

The term vector also has generalizations to higher dimensions and to more formal approaches with much wider applications.

Examples in one dimension

Since the physicist's concept of force has a direction and a magnitude, it may be seen as a vector. As an example, consider a rightward force F of 15 newtons. If the positive axis is also directed rightward, then F is represented by the vector 15 N, and if positive points leftward, then the vector for F is −15 N. In either case, the magnitude of the vector is 15 N. Likewise, the vector representation of a displacement Δs of 4 meters to the right would be 4 m or −4 m, and its magnitude would be 4 m regardless.

In physics and engineering

Vectors are fundamental in the physical sciences. They can be used to represent any quantity that has both a magnitude and direction, such as velocity, the magnitude of which is speed. For example, the velocity 5 meters per second upward could be represented by the vector (0,5) (in 2 dimensions with the positive y axis as 'up'). Another quantity represented by a vector is force, since it has a magnitude and direction. Vectors also describe many other physical quantities, such as displacement, acceleration, momentum, and angular momentum. Other physical vectors, such as the electric and magnetic field, are represented as a system of vectors at each point of a physical space; that is, a vector field.

In Cartesian space

In the Cartesian coordinate system, a vector can be represented by identifying the coordinates of its initial and terminal point. For instance, the points A = (1,0,0) and B = (0,1,0) in space determine the free vector $\backslash overrightarrow\{AB\}$ pointing from the point x=1 on the x-axis to the point y=1 on the y-axis.

Typically in Cartesian coordinates, one considers primarily bound vectors. A bound vector is determined by the coordinates of the terminal point, its initial point always having the coordinates of the origin O = (0,0,0). Thus the bound vector represented by (1,0,0) is a vector of unit length pointing from the origin up the positive x-axis.

The coordinate representation of vectors allows the algebraic features of vectors to be expressed in a convenient numerical fashion. For example, the sum of the vectors (1,2,3) and (−2,0,4) is the vector :(1, 2, 3) + (−2, 0, 4) = (1 − 2, 2 + 0, 3 + 4) = (−1, 2, 7).

Euclidean and affine vectors

In the geometrical and physical settings, sometimes it is possible to associate, in a natural way, a length or magnitude and a direction to vectors. In turn, the notion of direction is strictly associated with the notion of an angle between two vectors. When the length of vectors is defined, it is possible to also define a dot product — a scalar-valued product of two vectors — which gives a convenient algebraic characterization of both length (the square root of the dot product of a vector by itself) and angle (a function of the dot product between any two vectors). In three dimensions, it is further possible to define a cross product which supplies an algebraic characterization of the area and orientation in space of the parallelogram defined by two vectors (used as sides of the parallelogram).

However, it is not always possible or desirable to define the length of a vector in a natural way. This more general type of spatial vector is the subject of vector spaces (for bound vectors) and affine spaces (for free vectors). An important example is Minkowski space that is important to our understanding of special relativity, where there is a generalization of length that permits non-zero vectors to have zero length. Other physical examples come from thermodynamics, where many of the quantities of interest can be considered vectors in a space with no notion of length or angle.

Generalizations

In physics, as well as mathematics, a vector is often identified with a tuple, or list of numbers, which depend on some auxiliary coordinate system or reference frame. When the coordinates are transformed, for example by rotation or stretching, then the components of the vector also transform. The vector itself has not changed, but the reference frame has, so the components of the vector (or measurements taken with respect to the reference frame) must change to compensate. The vector is called covariant or contravariant depending on how the transformation of the vector's components is related to the transformation of coordinates. In general, contravariant vectors are "regular vectors" with units of distance (such as a displacement) or distance times some other unit (such as velocity or acceleration); covariant vectors, on the other hand, have units of one-over-distance such as gradient. If you change units (a special case of a change of coordinates) from meters to milimeters, a scale factor of 1/1000, a displacement of 1 m becomes 1000 mm–a contravariant change in numerical value. In contrast, a gradient of 1 K/m becomes 0.001 K/mm–a covariant change in value. See covariance and contravariance of vectors. Tensors are another type of quantity that behave in this way; in fact a vector is a special type of tensor.

In pure mathematics, a vector is any element of a vector space over some field and is often represented as a coordinate vector. The vectors described in this article are a very special case of this general definition because they are contravariant with respect to the ambient space. Contravariance captures the physical intuition behind the idea that a vector has "magnitude and direction".

History

The concept of vector, as we know it today, evolved gradually over a period of more than 200 years. About a dozen people made significant contributions.

Representations

Vectors are usually denoted in lowercase boldface, as a or lowercase italic boldface, as a. (Uppercase letters are typically used to represent matrices.) Other conventions include $\backslash vec\{a\}$ or a, especially in handwriting. Alternately, some use a tilde (~) or a wavy underline drawn beneath the symbol, which is a convention for indicating boldface type. If the vector represents a directed distance or displacement from a point A to a point B (see figure), it can also be denoted as $\backslash overrightarrow\{AB\}$ or AB.

Vectors are usually shown in graphs or other diagrams as arrows (directed line segments), as illustrated in the figure. Here the point A is called the origin, tail, base, or initial point; point B is called the head, tip, endpoint, terminal point or final point. The length of the arrow is proportional to the vector's magnitude, while the direction in which the arrow points indicates the vector's direction.

On a two-dimensional diagram, sometimes a vector perpendicular to the plane of the diagram is desired. These vectors are commonly shown as small circles. A circle with a dot at its centre (Unicode U+2299 ⊙) indicates a vector pointing out of the front of the diagram, toward the viewer. A circle with a cross inscribed in it (Unicode U+2297 ⊗) indicates a vector pointing into and behind the diagram. These can be thought of as viewing the tip of an arrow head on and viewing the vanes of an arrow from the back.

In order to calculate with vectors, the graphical representation may be too cumbersome. Vectors in an n-dimensional Euclidean space can be represented in a Cartesian coordinate system. The endpoint of a vector can be identified with an ordered list of n real numbers (n-tuple). As an example in two dimensions (see figure), the vector from the origin O = (0,0) to the point A = (2,3) is simply written as :$\backslash mathbf\{a\}\; =\; (2,3).$

The notion that the tail of the vector coincides with the origin is implicit and easily understood. Thus, the more explicit notation $\backslash overrightarrow\{OA\}$ is usually not deemed necessary and very rarely used.

In three dimensional Euclidean space (or $\backslash mathbb\{R\}^3$), vectors are identified with triples of numbers corresponding to the Cartesian coordinates of the endpoint (a,b,c): :$\backslash mathbf\{a\}\; =\; (a,\; b,\; c).$

These numbers are often arranged into a column vector or row vector, particularly when dealing with matrices, as follows: :$\backslash mathbf\{a\}\; =\; \backslash begin\{bmatrix\}\; a\backslash \backslash \; b\backslash \backslash \; c\backslash \backslash \; \backslash end\{bmatrix\}$ :$\backslash mathbf\{a\}\; =\; [\; a\backslash \; b\backslash \; c\; ].$

Another way to express a vector in three dimensions is to introduce the three standard basis vectors: :$\{\backslash mathbf\; e\}\_1\; =\; (1,0,0),\backslash \; \{\backslash mathbf\; e\}\_2\; =\; (0,1,0),\backslash \; \{\backslash mathbf\; e\}\_3\; =\; (0,0,1).$ These have the intuitive interpretation as vectors of unit length pointing up the x, y, and z axis of a Cartesian coordinate system, respectively, and they are sometimes referred to as versors of those axes. In terms of these, any vector in $\backslash mathbb\{R\}^3$ can be expressed in the form: :$(a,b,c)\; =\; a(1,0,0)\; +\; b(0,1,0)\; +\; c(0,0,1)\; =\; a\{\backslash mathbf\; e\}\_1\; +\; b\{\backslash mathbf\; e\}\_2\; +\; c\{\backslash mathbf\; e\}\_3.$

In introductory physics classes, these three special vectors are often instead denoted $\backslash mathbf\{\backslash hat\{i\}\},\backslash mathbf\{\backslash hat\{j\}\},\backslash mathbf\{\backslash hat\{k\}\}$ (or $\backslash mathbf\{\backslash hat\{x\}\},\; \backslash mathbf\{\backslash hat\{y\}\},\; \backslash mathbf\{\backslash hat\{z\}\}$), the versors of the three dimensional space, in which the hat symbol (^) typically denotes unit vectors (vectors with unit length). The notation e_i is compatible with the index notation and the summation convention commonly used in higher level mathematics, physics, and engineering.

The use of Cartesian versors such as $\backslash mathbf\{\backslash hat\{x\}\},\; \backslash mathbf\{\backslash hat\{y\}\},\; \backslash mathbf\{\backslash hat\{z\}\}$ as a basis in which to represent a vector is not mandated. Vectors can also be expressed in terms of cylindrical unit vectors $\backslash boldsymbol\{\backslash hat\{\backslash rho\}\},\; \backslash boldsymbol\{\backslash hat\{\backslash phi\}\},\; \backslash mathbf\{\backslash hat\{z\}\}$ or spherical unit vectors $\backslash mathbf\{\backslash hat\{r\}\},\; \backslash boldsymbol\{\backslash hat\{\backslash theta\}\},\; \backslash boldsymbol\{\backslash hat\{\backslash phi\}\}$. The latter two choices are more convenient for solving problems which possess cylindrical or spherical symmetry respectively.

Basic properties

The following section uses the Cartesian coordinate system with basis vectors :$\{\backslash mathbf\; e\}\_1\; =\; (1,0,0),\backslash \; \{\backslash mathbf\; e\}\_2\; =\; (0,1,0),\backslash \; \{\backslash mathbf\; e\}\_3\; =\; (0,0,1)$ and assume that all vectors have the origin as a common base point. A vector a will be written as :$\{\backslash mathbf\; a\}\; =\; a\_1\{\backslash mathbf\; e\}\_1\; +\; a\_2\{\backslash mathbf\; e\}\_2\; +\; a\_3\{\backslash mathbf\; e\}\_3.$

Equality

Two vectors are said to be equal if they have the same magnitude and direction. Equivalently they will be equal if their coordinates are equal. So two vectors :$\{\backslash mathbf\; a\}\; =\; a\_1\{\backslash mathbf\; e\}\_1\; +\; a\_2\{\backslash mathbf\; e\}\_2\; +\; a\_3\{\backslash mathbf\; e\}\_3$ and :$\{\backslash mathbf\; b\}\; =\; b\_1\{\backslash mathbf\; e\}\_1\; +\; b\_2\{\backslash mathbf\; e\}\_2\; +\; b\_3\{\backslash mathbf\; e\}\_3$ are equal if :$a\_1\; =\; b\_1,\backslash quad\; a\_2=b\_2,\backslash quad\; a\_3=b\_3.\backslash ,$

Addition and subtraction

Assume now that a and b are not necessarily equal vectors, but that they may have different magnitudes and directions. The sum of a and b is :$\backslash mathbf\{a\}+\backslash mathbf\{b\}\; =(a\_1+b\_1)\backslash mathbf\{e\_1\}\; +(a\_2+b\_2)\backslash mathbf\{e\_2\}\; +(a\_3+b\_3)\backslash mathbf\{e\_3\}.$

The addition may be represented graphically by placing the start of the arrow b at the tip of the arrow a, and then drawing an arrow from the start of a to the tip of b. The new arrow drawn represents the vector a + b, as illustrated below:

This addition method is sometimes called the parallelogram rule because a and b form the sides of a parallelogram and a + b is one of the diagonals. If a and b are bound vectors that have the same base point, it will also be the base point of a + b. One can check geometrically that a + b = b + a and (a + b) + c = a + (b + c).

The difference of a and b is

:$\backslash mathbf\{a\}-\backslash mathbf\{b\}\; =(a\_1-b\_1)\backslash mathbf\{e\_1\}\; +(a\_2-b\_2)\backslash mathbf\{e\_2\}\; +(a\_3-b\_3)\backslash mathbf\{e\_3\}.$

Subtraction of two vectors can be geometrically defined as follows: to subtract b from a, place the end points of a and b at the same point, and then draw an arrow from the tip of b to the tip of a. That arrow represents the vector a − b, as illustrated below:

Scalar multiplication

A vector may also be multiplied, or re-scaled, by a real number r. In the context of conventional vector algebra, these real numbers are often called scalars (from scale) to distinguish them from vectors. The operation of multiplying a vector by a scalar is called scalar multiplication. The resulting vector is

:$r\backslash mathbf\{a\}=(ra\_1)\backslash mathbf\{e\_1\}\; +(ra\_2)\backslash mathbf\{e\_2\}\; +(ra\_3)\backslash mathbf\{e\_3\}.$

Intuitively, multiplying by a scalar r stretches a vector out by a factor of r. Geometrically, this can be visualized (at least in the case when r is an integer) as placing r copies of the vector in a line where the endpoint of one vector is the initial point of the next vector.

If r is negative, then the vector changes direction: it flips around by an angle of 180°. Two examples (r = −1 and r = 2) are given below:

Scalar multiplication is distributive over vector addition in the following sense: r(a + b) = ra + rb for all vectors a and b and all scalars r. One can also show that a − b = a + (−1)b.

===Length=== The length or magnitude or norm of the vector a is denoted by ||a|| or, less commonly, |a|, which is not to be confused with the absolute value (a scalar "norm").

The length of the vector a can be computed with the Euclidean norm

:$\backslash left\backslash |\backslash mathbf\{a\}\backslash right\backslash |=\backslash sqrt.$

;Unit vector

A unit vector is any vector with a length of one; normally unit vectors are used simply to indicate direction. A vector of arbitrary length can be divided by its length to create a unit vector. This is known as normalizing a vector. A unit vector is often indicated with a hat as in â.

To normalize a vector a = [a₁, a₂, a₃], scale the vector by the reciprocal of its length ||a||. That is:

:$\backslash mathbf\{\backslash hat\{a\}\}\; =\; \backslash frac\{\backslash mathbf\{a\}\}\{\backslash left\backslash |\backslash mathbf\{a\}\backslash right\backslash |\}\; =\; \backslash frac\{a\_1\}\{\backslash left\backslash |\backslash mathbf\{a\}\backslash right\backslash |\}\backslash mathbf\{e\_1\}\; +\; \backslash frac\{a\_2\}\{\backslash left\backslash |\backslash mathbf\{a\}\backslash right\backslash |\}\backslash mathbf\{e\_2\}\; +\; \backslash frac\{a\_3\}\{\backslash left\backslash |\backslash mathbf\{a\}\backslash right\backslash |\}\backslash mathbf\{e\_3\}$

;Null vector

The null vector (or zero vector) is the vector with length zero. Written out in coordinates, the vector is (0,0,0), and it is commonly denoted $\backslash vec\{0\}$, or 0, or simply 0. Unlike any other vector, it does not have a direction, and cannot be normalized (that is, there is no unit vector which is a multiple of the null vector). The sum of the null vector with any vector a is a (that is, 0+a=a).

Dot product

The dot product of two vectors a and b (sometimes called the inner product, or, since its result is a scalar, the scalar product) is denoted by a ∙ b and is defined as:

:$\backslash mathbf\{a\}\backslash cdot\backslash mathbf\{b\}\; =\backslash left\backslash |\backslash mathbf\{a\}\backslash right\backslash |\backslash left\backslash |\backslash mathbf\{b\}\backslash right\backslash |\backslash cos\backslash theta$

where θ is the measure of the angle between a and b (see trigonometric function for an explanation of cosine). Geometrically, this means that a and b are drawn with a common start point and then the length of a is multiplied with the length of that component of b that points in the same direction as a.

The dot product can also be defined as the sum of the products of the components of each vector as

:$\backslash mathbf\{a\}\; \backslash cdot\; \backslash mathbf\{b\}\; =\; a\_1\; b\_1\; +\; a\_2\; b\_2\; +\; a\_3\; b\_3.$

Cross product

The cross product (also called the vector product or outer product) is only meaningful in three or seven dimensions. The cross product differs from the dot product primarily in that the result of the cross product of two vectors is a vector. The cross product, denoted a × b, is a vector perpendicular to both a and b and is defined as

:$\backslash mathbf\{a\}\backslash times\backslash mathbf\{b\}\; =\backslash left\backslash |\backslash mathbf\{a\}\backslash right\backslash |\backslash left\backslash |\backslash mathbf\{b\}\backslash right\backslash |\backslash sin(\backslash theta)\backslash ,\backslash mathbf\{n\}$

where θ is the measure of the angle between a and b, and n is a unit vector perpendicular to both a and b which completes a right-handed system. The right-handedness constraint is necessary because there exist two unit vectors that are perpendicular to both a and b, namely, n and (–n).

The cross product a × b is defined so that a, b, and a × b also becomes a right-handed system (but note that a and b are not necessarily orthogonal). This is the right-hand rule.

The length of a × b can be interpreted as the area of the parallelogram having a and b as sides.

The cross product can be written as :$\{\backslash mathbf\; a\}\backslash times\{\backslash mathbf\; b\}\; =\; (a\_2\; b\_3\; -\; a\_3\; b\_2)\; \{\backslash mathbf\; e\}\_1\; +\; (a\_3\; b\_1\; -\; a\_1\; b\_3)\; \{\backslash mathbf\; e\}\_2\; +\; (a\_1\; b\_2\; -\; a\_2\; b\_1)\; \{\backslash mathbf\; e\}\_3.$

For arbitrary choices of spatial orientation (that is, allowing for left-handed as well as right-handed coordinate systems) the cross product of two vectors is a pseudovector instead of a vector (see below).

Scalar triple product

The scalar triple product (also called the box product or mixed triple product) is not really a new operator, but a way of applying the other two multiplication operators to three vectors. The scalar triple product is sometimes denoted by (a b c) and defined as:

:$(\backslash mathbf\{a\}\backslash \; \backslash mathbf\{b\}\backslash \; \backslash mathbf\{c\})\; =\backslash mathbf\{a\}\backslash cdot(\backslash mathbf\{b\}\backslash times\backslash mathbf\{c\}).$

It has three primary uses. First, the absolute value of the box product is the volume of the parallelepiped which has edges that are defined by the three vectors. Second, the scalar triple product is zero if and only if the three vectors are linearly dependent, which can be easily proved by considering that in order for the three vectors to not make a volume, they must all lie in the same plane. Third, the box product is positive if and only if the three vectors a, b and c are right-handed.

In components (with respect to a right-handed orthonormal basis), if the three vectors are thought of as rows (or columns, but in the same order), the scalar triple product is simply the determinant of the 3-by-3 matrix having the three vectors as rows :

The scalar triple product is linear in all three entries and anti-symmetric in the following sense: :$(\backslash mathbf\{a\}\backslash \; \backslash mathbf\{b\}\backslash \; \backslash mathbf\{c\})\; =\; (\backslash mathbf\{c\}\backslash \; \backslash mathbf\{a\}\backslash \; \backslash mathbf\{b\})\; =\; (\backslash mathbf\{b\}\backslash \; \backslash mathbf\{c\}\backslash \; \backslash mathbf\{a\})=\; -(\backslash mathbf\{a\}\backslash \; \backslash mathbf\{c\}\backslash \; \backslash mathbf\{b\})\; =\; -(\backslash mathbf\{b\}\backslash \; \backslash mathbf\{a\}\backslash \; \backslash mathbf\{c\})\; =\; -(\backslash mathbf\{c\}\backslash \; \backslash mathbf\{b\}\backslash \; \backslash mathbf\{a\}).$

Multiple Cartesian bases

All examples thus far have dealt with vectors expressed in terms of the same basis, namely, e₁,e₂,e₃. However, a vector can be expressed in terms of any number of different bases that are not necessarily aligned with each other, and still remain the same vector. For example, using the vector a from above,

:$\backslash mathbf\{a\}\; =\; a\_1\backslash mathbf\{e\}\_1\; +\; a\_2\backslash mathbf\{e\}\_2\; +\; a\_3\backslash mathbf\{e\}\_3\; =\; u\backslash mathbf\{n\}\_1\; +\; v\backslash mathbf\{n\}\_2\; +\; w\backslash mathbf\{n\}\_3$

where n₁,n₂,n₃ form another orthonormal basis not aligned with e₁,e₂,e₃. The values of u, v, and w are such that the resulting vector sum is exactly a.

It is not uncommon to encounter vectors known in terms of different bases (for example, one basis fixed to the Earth and a second basis fixed to a moving vehicle). In order to perform many of the operations defined above, it is necessary to know the vectors in terms of the same basis. One simple way to express a vector known in one basis in terms of another uses column matrices that represent the vector in each basis along with a third matrix containing the information that relates the two bases. For example, in order to find the values of u, v, and w that define a in the n₁,n₂,n₃ basis, a matrix multiplication may be employed in the form

:

where each matrix element c_jk is the direction cosine relating n_j to e_k. The term direction cosine refers to the cosine of the angle between two unit vectors, which is also equal to their dot product. Both remain orthogonal to the axis of rotation at all times. (In two dimensions this requirement becomes redundant as the axis degenerates to a point of rotation.) The choice of a coordinate system doesn't affect properties of a vector or its behaviour under transformations.

Vectors as directional derivatives

A vector may also be defined as a directional derivative: consider a function $f(x^\backslash alpha)$ and a curve $x^\backslash alpha\; (\backslash tau)$. Then the directional derivative of $f$ is a scalar defined as

:$\backslash frac\{df\}\{d\backslash tau\}\; =\; \backslash sum\_\{\backslash alpha=1\}^n\; \backslash frac\{dx^\backslash alpha\}\{d\backslash tau\}\backslash frac\{\backslash partial\; f\}\{\backslash partial\; x^\backslash alpha\}.$

where the index $\backslash alpha$ is summed over the appropriate number of dimensions (for example, from 1 to 3 in 3-dimensional Euclidean space, from 0 to 3 in 4-dimensional spacetime, etc.). Then consider a vector tangent to $x^\backslash alpha\; (\backslash tau)$:

:$t^\backslash alpha\; =\; \backslash frac\{dx^\backslash alpha\}\{d\backslash tau\}.$

The directional derivative can be rewritten in differential form (without a given function $f$) as

:$\backslash frac\{d\}\{d\backslash tau\}\; =\; \backslash sum\_\backslash alpha\; t^\backslash alpha\backslash frac\{\backslash partial\}\{\backslash partial\; x^\backslash alpha\}.$

Therefore any directional derivative can be identified with a corresponding vector, and any vector can be identified with a corresponding directional derivative. A vector can therefore be defined precisely as

:$\backslash mathbf\{a\}\; \backslash equiv\; a^\backslash alpha\; \backslash frac\{\backslash partial\}\{\backslash partial\; x^\backslash alpha\}.$

Vectors, pseudovectors, and transformations

An alternative characterization of Euclidean vectors, especially in physics, describes them as lists of quantities which behave in a certain way under a coordinate transformation. A contravariant vector is required to have components that "transform like the coordinates" under changes of coordinates such as rotation and dilation. The vector itself does not change under these operations; instead, the components of the vector make a change that cancels the change in the spatial axes, in the same way that co-ordinates change. In other words, if the reference axes were rotated in one direction, the component representation of the vector would rotate in exactly the opposite way. Similarly, if the reference axes were stretched in one direction, the components of the vector, like the co-ordinates, would reduce in an exactly compensating way. Mathematically, if the coordinate system undergoes a transformation described by an invertible matrix M, so that a coordinate vector x is transformed to x′ = Mx, then a contravariant vector v must be similarly transformed via v′ = Mv. This important requirement is what distinguishes a contravariant vector from any other triple of physically meaningful quantities. For example, if v consists of the x, y, and z-components of velocity, then v is a contravariant vector: if the coordinates of space are stretched, rotated, or twisted, then the components of the velocity transform in the same way. On the other hand, for instance, a triple consisting of the length, width, and height of a rectangular box could make up the three components of an abstract vector, but this vector would not be contravariant, since rotating the box does not change the box's length, width, and height. Examples of contravariant vectors include displacement, velocity, electric field, momentum, force, and acceleration.

In the language of differential geometry, the requirement that the components of a vector transform according to the same matrix of the coordinate transition is equivalent to defining a contravariant vector to be a tensor of contravariant rank one. Alternatively, a contravariant vector is defined to be a tangent vector, and the rules for transforming a contravariant vector follow from the chain rule.

Some vectors transform like contravariant vectors, except that when they are reflected through a mirror, they flip and gain a minus sign. A transformation that switches right-handedness to left-handedness and vice versa like a mirror does is said to change the orientation of space. A vector which gains a minus sign when the orientation of space changes is called a pseudovector or an axial vector. Ordinary vectors are sometimes called true vectors or polar vectors to distinguish them from pseudovectors. Pseudovectors occur most frequently as the cross product of two ordinary vectors.

One example of a pseudovector is angular velocity. Driving in a car, and looking forward, each of the wheels has an angular velocity vector pointing to the left. If the world is reflected in a mirror which switches the left and right side of the car, the reflection of this angular velocity vector points to the right, but the actual angular velocity vector of the wheel still points to the left, corresponding to the minus sign. Other examples of pseudovectors include magnetic field, torque, or more generally any cross product of two (true) vectors.

This distinction between vectors and pseudovectors is often ignored, but it becomes important in studying symmetry properties. See parity (physics).

See also

Affine space, which distinguishes between vectors and points

Four-vector, a non-Euclidean vector in Minkowski space (i.e. four-dimensional spacetime), important in relativity

Normal vector

Null vector

Pseudovector

Tangential and normal components (of a vector)

Unit vector

Vector calculus

Vector-valued function

Vector bundle

Vector notation

Function space

Banach space

Hilbert space

Coordinate system

Complex number

Quaternion

Grassmann's Ausdehnungslehre

Clifford algebra

Tensor

Covariance and contravariance of vectors

Notes

References

Mathematical treatments .

Physical treatments

External links

Online vector identities (PDF)

Introducing Vectors A conceptual introduction (applied mathematics)

Addition of forces (vectors) Java Applet

French tutorials on vectors and their application to video games

Category:Abstract algebra Category:Vector calculus Category:Linear algebra Category:Introductory physics Category:Fundamental physics concepts

Name	LOG Records
Founded	2008
Founder	Larissa Lam
Label	LOG Records
Genre	Hip Hop
Country	United States
Location	Los Angeles
Url	LOG Records Website

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Category:Living people Category:1974 births Category:American rappers Category:American actors

Name	David Hilbert
Image width	200px
Caption	David Hilbert (1912)
Birth date	January 23, 1862
Birth place	Königsberg or Wehlau (today Znamensk, Kaliningrad Oblast), Province of Prussia
Death date	February 14, 1943
Death place	Göttingen, Germany
Residence	Germany
Nationality	German
Field	Mathematician and Philosopher
Work institutions	University of KönigsbergGöttingen University
Alma mater	University of Königsberg
Doctoral advisor	Ferdinand von Lindemann
Doctoral students	Wilhelm AckermannOtto BlumenthalWerner BoyRichard CourantHaskell CurryMax DehnErich HeckeHellmuth KneserRobert KönigEmanuel LaskerErhard SchmidtHugo SteinhausTeiji TakagiHermann WeylErnst Zermelo
Known for	Hilbert's basis theoremHilbert's axiomsHilbert's problemsHilbert's programEinstein–Hilbert actionHilbert space
Prizes

David Hilbert (; January 23, 1862 – February 14, 1943) was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of geometry. He also formulated the theory of Hilbert spaces, one of the foundations of functional analysis.

Hilbert adopted and warmly defended Georg Cantor's set theory and transfinite numbers. A famous example of his leadership in mathematics is his 1900 presentation of a collection of problems that set the course for much of the mathematical research of the 20th century.

Hilbert and his students contributed significantly to establishing rigor and developed important tools used in modern mathematical physics. Hilbert is known as one of the founders of proof theory and mathematical logic, as well as for being among the first to distinguish between mathematics and metamathematics.

Life

Hilbert, the first of two children and only son of Otto and Maria Therese (Erdtmann) Hilbert, was born in the Province of Prussia - either in Königsberg (according to Hilbert's own statement) or in Wehlau (known since 1946 as Znamensk) near Königsberg where his father worked at the time of his birth. In the fall of 1872, he entered the Friedrichskolleg Gymnasium (Collegium fridericianum, the same school that Immanuel Kant had attended 140 years before), but after an unhappy duration he transferred (fall 1879) to and graduated from (spring 1880) the more science-oriented Wilhelm Gymnasium. Upon graduation he enrolled (autumn 1880) at the University of Königsberg, the "Albertina". In the spring of 1882, Hermann Minkowski (two years younger than Hilbert and also a native of Königsberg but so talented he had graduated early from his gymnasium and gone to Berlin for three semesters), returned to Königsberg and entered the university. "Hilbert knew his luck when he saw it. In spite of his father's disapproval, he soon became friends with the shy, gifted Minkowski." In 1884, Adolf Hurwitz arrived from Göttingen as an Extraordinarius, i.e., an associate professor. An intense and fruitful scientific exchange between the three began and especially Minkowski and Hilbert would exercise a reciprocal influence over each other at various times in their scientific careers. Hilbert obtained his doctorate in 1885, with a dissertation, written under Ferdinand von Lindemann, titled Über invariante Eigenschaften spezieller binärer Formen, insbesondere der Kugelfunktionen ("On the invariant properties of special binary forms, in particular the spherical harmonic functions").

Hilbert remained at the University of Königsberg as a professor from 1886 to 1895. In 1892, Hilbert married Käthe Jerosch (1864–1945), "the daughter of a Konigsberg merchant, an outspoken young lady with an independence of mind that matched his own". While at Königsberg they had their one child Franz Hilbert (1893–1969). In 1895, as a result of intervention on his behalf by Felix Klein he obtained the position of Chairman of Mathematics at the University of Göttingen, at that time the best research center for mathematics in the world and where he remained for the rest of his life.

His son Franz would suffer his entire life from an (undiagnosed) mental illness, his inferior intellect a terrible disappointment to his father and this tragedy a matter of distress to the mathematicians and students at Göttingen. Sadly, Minkowski — Hilbert's "best and truest friend" — would die prematurely of a ruptured appendix in 1909.

, was opened by Hilbert and Courant in 1930.]]

The Göttingen school

Among the students of Hilbert were: Hermann Weyl, chess champion Emanuel Lasker, Ernst Zermelo, and Carl Gustav Hempel. John von Neumann was his assistant. At the University of Göttingen, Hilbert was surrounded by a social circle of some of the most important mathematicians of the 20th century, such as Emmy Noether and Alonzo Church.

Among his 69 Ph.D. students in Göttingen were many who later became famous mathematicians, including (with date of thesis): Otto Blumenthal (1898), Felix Bernstein (1901), Hermann Weyl (1908), Richard Courant (1910), Erich Hecke (1910), Hugo Steinhaus (1911), Wilhelm Ackermann (1925). Between 1902 and 1939 Hilbert was editor of the Mathematische Annalen, the leading mathematical journal of the time.

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Later years

Hilbert lived to see the Nazis purge many of the prominent faculty members at University of Göttingen in 1933. Those forced out included Hermann Weyl (who had taken Hilbert's chair when he retired in 1930), Emmy Noether and Edmund Landau. One of those who had to leave Germany was Paul Bernays, Hilbert's collaborator in mathematical logic, and co-author with him of the important book Grundlagen der Mathematik (which eventually appeared in two volumes, in 1934 and 1939). This was a sequel to the Hilbert-Ackermann book Principles of Mathematical Logic from 1928.

About a year later, he attended a banquet and was seated next to the new Minister of Education, Bernhard Rust. Rust asked, "How is mathematics in Göttingen now that it has been freed of the Jewish influence?" Hilbert replied, "Mathematics in Göttingen? There is really none any more." By the time Hilbert died in 1943, the Nazis had nearly completely restaffed the university, inasmuch as many of the former faculty had either been Jewish or married to Jews. Hilbert's funeral was attended by fewer than a dozen people, only two of whom were fellow academics, among them Arnold Sommerfeld, a theoretical physicist and also a native of Königsberg. News of his death only became known to the wider world six months after he had died.

The epitaph on his tombstone in Göttingen is the famous lines he had spoken at the conclusion of his retirement address to the Society of German Scientists and Physicians in the fall of 1930:

:Wir müssen wissen. :Wir werden wissen.

In English: : We must know. : We will know.

The day before Hilbert pronounced these phrases at the 1930 annual meeting of the Society of German Scientists and Physicians, Kurt Gödel—in a roundtable discussion during the Conference on Epistemology held jointly with the Society meetings—tentatively announced the first expression of his incompleteness theorem.

The finiteness theorem

Hilbert's first work on invariant functions led him to the demonstration in 1888 of his famous finiteness theorem. Twenty years earlier, Paul Gordan had demonstrated the theorem of the finiteness of generators for binary forms using a complex computational approach. Attempts to generalize his method to functions with more than two variables failed because of the enormous difficulty of the calculations involved. Hilbert realized that it was necessary to take a completely different path. As a result, he demonstrated Hilbert's basis theorem: showing the existence of a finite set of generators, for the invariants of quantics in any number of variables, but in an abstract form. That is, while demonstrating the existence of such a set, it was not a constructive proof — it did not display "an object" — but rather, it was an existence proof and relied on use of the Law of Excluded Middle in an infinite extension.

Hilbert sent his results to the Mathematische Annalen. Gordan, the house expert on the theory of invariants for the Mathematische Annalen, was not able to appreciate the revolutionary nature of Hilbert's theorem and rejected the article, criticizing the exposition because it was insufficiently comprehensive. His comment was:

:Das ist nicht Mathematik. Das ist Theologie. ::(This is not Mathematics. This is Theology.)

Klein, on the other hand, recognized the importance of the work, and guaranteed that it would be published without any alterations. Encouraged by Klein and by the comments of Gordan, Hilbert in a second article extended his method, providing estimations on the maximum degree of the minimum set of generators, and he sent it once more to the Annalen. After having read the manuscript, Klein wrote to him, saying:

:Without doubt this is the most important work on general algebra that the Annalen has ever published.

Later, after the usefulness of Hilbert's method was universally recognized, Gordan himself would say:

:I have convinced myself that even theology has its merits.

For all his successes, the nature of his proof stirred up more trouble than Hilbert could have imagined at the time. Although Kronecker had conceded, Hilbert would later respond to others' similar criticisms that "many different constructions are subsumed under one fundamental idea" — in other words (to quote Reid): "Through a proof of existence, Hilbert had been able to obtain a construction"; "the proof" (i.e. the symbols on the page) was "the object". Indeed Hilbert would lose his "gifted pupil" Weyl to intuitionism — "Hilbert was disturbed by his former student's fascination with the ideas of Brouwer, which aroused in Hilbert the memory of Kronecker". Brouwer the intuitionist in particular opposed the use of the Law of Excluded Middle over infinite sets (as Hilbert had used it). Hilbert would respond: :Taking the Principle of the Excluded Middle from the mathematician ... is the same as ... prohibiting the boxer the use of his fists.

Axiomatization of geometry

The text Grundlagen der Geometrie (tr.: Foundations of Geometry) published by Hilbert in 1899 proposes a formal set, the Hilbert's axioms, substituting the traditional axioms of Euclid. They avoid weaknesses identified in those of Euclid, whose works at the time were still used textbook-fashion. Independently and contemporaneously, a 19-year-old American student named Robert Lee Moore published an equivalent set of axioms. Some of the axioms coincide, while some of the axioms in Moore's system are theorems in Hilbert's and vice-versa.

Hilbert's approach signaled the shift to the modern axiomatic method. In this, Hilbert was anticipated by Peano's work from 1889. Axioms are not taken as self-evident truths. Geometry may treat things, about which we have powerful intuitions, but it is not necessary to assign any explicit meaning to the undefined concepts. The elements, such as point, line, plane, and others, could be substituted, as Hilbert says, by tables, chairs, glasses of beer and other such objects. It is their defined relationships that are discussed.

Hilbert first enumerates the undefined concepts: point, line, plane, lying on (a relation between points and planes), betweenness, congruence of pairs of points, and congruence of angles. The axioms unify both the plane geometry and solid geometry of Euclid in a single system.

The 23 Problems

Hilbert put forth a most influential list of 23 unsolved problems at the International Congress of Mathematicians in Paris in 1900. This is generally reckoned the most successful and deeply considered compilation of open problems ever to be produced by an individual mathematician.

After re-working the foundations of classical geometry, Hilbert could have extrapolated to the rest of mathematics. His approach differed, however, from the later 'foundationalist' Russell-Whitehead or 'encyclopedist' Nicolas Bourbaki, and from his contemporary Giuseppe Peano. The mathematical community as a whole could enlist in problems, which he had identified as crucial aspects of the areas of mathematics he took to be key.

The problem set was launched as a talk "The Problems of Mathematics" presented during the course of the Second International Congress of Mathematicians held in Paris. Here is the introduction of the speech that Hilbert gave:

:Who among us would not be happy to lift the veil behind which is hidden the future; to gaze at the coming developments of our science and at the secrets of its development in the centuries to come? What will be the ends toward which the spirit of future generations of mathematicians will tend? What methods, what new facts will the new century reveal in the vast and rich field of mathematical thought?

He presented fewer than half the problems at the Congress, which were published in the acts of the Congress. In a subsequent publication, he extended the panorama, and arrived at the formulation of the now-canonical 23 Problems of Hilbert. The full text is important, since the exegesis of the questions still can be a matter of inevitable debate, whenever it is asked how many have been solved.

Some of these were solved within a short time. Others have been discussed throughout the 20th century, with a few now taken to be unsuitably open-ended to come to closure. Some even continue to this day to remain a challenge for mathematicians.

Formalism

In an account that had become standard by the mid-century, Hilbert's problem set was also a kind of manifesto, that opened the way for the development of the formalist school, one of three major schools of mathematics of the 20th century. According to the formalist, mathematics is manipulation of symbols according to agreed upon formal rules. It is therefore an autonomous activity of thought. There is, however, room to doubt whether Hilbert's own views were simplistically formalist in this sense.

Hilbert's program

In 1920 he proposed explicitly a research project (in metamathematics, as it was then termed) that became known as Hilbert's program. He wanted mathematics to be formulated on a solid and complete logical foundation. He believed that in principle this could be done, by showing that:

#all of mathematics follows from a correctly chosen finite system of axioms; and #that some such axiom system is provably consistent through some means such as the epsilon calculus.

He seems to have had both technical and philosophical reasons for formulating this proposal. It affirmed his dislike of what had become known as the ignorabimus, still an active issue in his time in German thought, and traced back in that formulation to Emil du Bois-Reymond.

This program is still recognizable in the most popular philosophy of mathematics, where it is usually called formalism. For example, the Bourbaki group adopted a watered-down and selective version of it as adequate to the requirements of their twin projects of (a) writing encyclopedic foundational works, and (b) supporting the axiomatic method as a research tool. This approach has been successful and influential in relation with Hilbert's work in algebra and functional analysis, but has failed to engage in the same way with his interests in physics and logic.

Hilbert wrote in 1919:

:We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules. Rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise.

Gödel's work

Hilbert and the mathematicians who worked with him in his enterprise were committed to the project. His attempt to support axiomatized mathematics with definitive principles, which could banish theoretical uncertainties, was however to end in failure.

Gödel demonstrated that any non-contradictory formal system, which was comprehensive enough to include at least arithmetic, cannot demonstrate its completeness by way of its own axioms. In 1931 his incompleteness theorem showed that Hilbert's grand plan was impossible as stated. The second point cannot in any reasonable way be combined with the first point, as long as the axiom system is genuinely finitary.

Nevertheless, the subsequent achievements of proof theory at the very least clarified consistency as it relates to theories of central concern to mathematicians. Hilbert's work had started logic on this course of clarification; the need to understand Gödel's work then led to the development of recursion theory and then mathematical logic as an autonomous discipline in the 1930s. The basis for later theoretical computer science, in Alonzo Church and Alan Turing also grew directly out of this 'debate'.

Functional analysis

Around 1909, Hilbert dedicated himself to the study of differential and integral equations; his work had direct consequences for important parts of modern functional analysis. In order to carry out these studies, Hilbert introduced the concept of an infinite dimensional Euclidean space, later called Hilbert space. His work in this part of analysis provided the basis for important contributions to the mathematics of physics in the next two decades, though from an unanticipated direction. Later on, Stefan Banach amplified the concept, defining Banach spaces. Hilbert space is the most important single idea in the area of functional analysis, particularly of the spectral theory of self-adjoint linear operators, that grew up around it during the 20th century.

Physics

Until 1912, Hilbert was almost exclusively a "pure" mathematician. When planning a visit from Bonn, where he was immersed in studying physics, his fellow mathematician and friend Hermann Minkowski joked he had to spend 10 days in quarantine before being able to visit Hilbert. In fact, Minkowski seems responsible for most of Hilbert's physics investigations prior to 1912, including their joint seminar in the subject in 1905.

In 1912, three years after his friend's death, Hilbert turned his focus to the subject almost exclusively. He arranged to have a "physics tutor" for himself. He started studying kinetic gas theory and moved on to elementary radiation theory and the molecular theory of matter. Even after the war started in 1914, he continued seminars and classes where the works of Albert Einstein and others were followed closely.

By 1907 Einstein had framed the fundamentals of the theory of gravity, but then struggled for nearly 8 years with a confounding problem of putting the theory into final form. By early summer 1915, Hilbert's interest in physics had focused him on general relativity, and he invited Einstein to Göttingen to deliver a week of lectures on the subject. Einstein received an enthusiastic reception at Göttingen. Over the summer Einstein learned that Hilbert was also working on the field equations and redoubled his own efforts. During November 1915 Einstein published several papers culminating in "The Field Equations of Gravitation" (see Einstein field equations). Nearly simultaneously David Hilbert published "The Foundations of Physics", an axiomatic derivation of the field equations (see Einstein–Hilbert action). Hilbert fully credited Einstein as the originator of the theory, and no public priority dispute concerning the field equations ever arose between the two men during their lives (see more at priority).

Additionally, Hilbert's work anticipated and assisted several advances in the mathematical formulation of quantum mechanics. His work was a key aspect of Hermann Weyl and John von Neumann's work on the mathematical equivalence of Werner Heisenberg's matrix mechanics and Erwin Schrödinger's wave equation and his namesake Hilbert space plays an important part in quantum theory. In 1926 von Neuman showed that if atomic states were understood as vectors in Hilbert space, then they would correspond with both Schrödinger's wave function theory and Heisenberg's matrices.

Throughout this immersion in physics, Hilbert worked on putting rigor into the mathematics of physics. While highly dependent on higher math, physicists tended to be "sloppy" with it. To a "pure" mathematician like Hilbert, this was both "ugly" and difficult to understand. As he began to understand physics and how physicists were using mathematics, he developed a coherent mathematical theory for what he found, most importantly in the area of integral equations. When his colleague Richard Courant wrote the now classic Methods of Mathematical Physics including some of Hilbert's ideas, he added Hilbert's name as author even though Hilbert had not directly contributed to the writing. Hilbert said "Physics is too hard for physicists", implying that the necessary mathematics was generally beyond them; the Courant-Hilbert book made it easier for them.

Number theory

Hilbert unified the field of algebraic number theory with his 1897 treatise Zahlbericht (literally "report on numbers"). He also resolved a significant number theory problem formulated by Waring in 1770. As with the finiteness theorem, he used an existence proof that shows there must be solutions for the problem rather than providing a mechanism to produce the answers. He then had little more to publish on the subject; but the emergence of Hilbert modular forms in the dissertation of a student means his name is further attached to a major area.

He made a series of conjectures on class field theory. The concepts were highly influential, and his own contribution is seen in the names of the Hilbert class field and the Hilbert symbol of local class field theory. Results on them were mostly proved by 1930, after work by Teiji Takagi that established Tagaki as Japan's first mathematician of international stature.

Hilbert did not work in the central areas of analytic number theory, but his name has become known for the Hilbert–Pólya conjecture, for reasons that are anecdotal.

Miscellaneous talks, essays, and contributions

Hilbert's paradox of the Grand Hotel, a meditation on strange properties of the infinite, is often used in popular accounts of infinite cardinal numbers.

His Erdős number is (at most) 4.

He was a Foreign member of the Royal Society

He received the second Bolyai Prize in 1910.

His collected works (Gesammelte Abhandlungen) have been published several times. The original versions of his papers contained errors; when the collection was first published, the errors were corrected and it was found that this could be done without major changes in the statements of the theorems, with one exception—a claimed proof of the Continuum hypothesis. The errors were nonetheless so numerous and significant that it took Olga Taussky-Todd three years to make the corrections.

Hilbert's axioms

Hilbert class field

Hilbert C*-module

Hilbert cube

Hilbert curve

Hilbert function

Hilbert inequality

Hilbert matrix

Hilbert metric

Hilbert modular form

Hilbert number

Hilbert polynomial

Hilbert's problems

Hilbert's program

Hilbert–Poincaré series

Hilbert space

Hilbert spectrum

Hilbert symbol

Hilbert transform

Hilbert's Arithmetic of Ends

Hilbert's basis theorem

Hilbert's constants

Hilbert's irreducibility theorem

Hilbert's Nullstellensatz

Hilbert's paradox of the Grand Hotel

Hilbert's theorem (differential geometry)

Hilbert's Theorem 90

Hilbert's syzygy theorem

Hilbert-style deduction system

Hilbert–Pólya conjecture

Hilbert–Schmidt operator

Hilbert–Smith conjecture

Hilbert–Speiser theorem

Principles of Mathematical Logic

Relativity priority dispute

Notes

References

Primary literature in English translation

Ewald, William B., ed., 1996. From Kant to Hilbert: A Source Book in the Foundations of Mathematics, 2 vols. Oxford Uni. Press.

*1918. "Axiomatic thought," 1115–14.

*1922. "The new grounding of mathematics: First report," 1115–33.

*1923. "The logical foundations of mathematics," 1134–47.

*1930. "Logic and the knowledge of nature," 1157–65.

*1931. "The grounding of elementary number theory," 1148–56.

*1904. "On the foundations of logic and arithmetic," 129–38.

*1925. "On the infinite," 367–92.

*1927. "The foundations of mathematics," with comment by Weyl and Appendix by Bernays, 464–89.

Jean van Heijenoort, 1967. From Frege to Godel: A Source Book in Mathematical Logic, 1879–1931. Harvard Univ. Press.

- an accessible set of lectures originally for the citizens of Göttingen.

Secondary literature

Bottazzini Umberto, 2003. Il flauto di Hilbert. Storia della matematica. UTET, ISBN 88-7750-852-3

Corry, L., Renn, J., and Stachel, J., 1997, "Belated Decision in the Hilbert-Einstein Priority Dispute," Science 278: nn-nn.

Dawson, John W. Jr 1997. Logical Dilemmas: The Life and Work of Kurt Gödel. Wellesley MA: A. K. Peters. ISBN 1-56881-256-6.

Folsing, Albrecht, 1998. Albert Einstein. Penguin.

Grattan-Guinness, Ivor, 2000. The Search for Mathematical Roots 1870-1940. Princeton Univ. Press.

Gray, Jeremy, 2000. The Hilbert Challenge. ISBN 0-19-850651-1

Mehra, Jagdish, 1974. Einstein, Hilbert, and the Theory of Gravitation. Reidel.

Piergiorgio Odifreddi, 2003. Divertimento Geometrico - Da Euclide ad Hilbert. Bollati Boringhieri, ISBN 88-339-5714-4. A clear exposition of the "errors" of Euclid and of the solutions presented in the Grundlagen der Geometrie, with reference to non-Euclidean geometry.

Reid, Constance, 1996. Hilbert, Springer, ISBN 0-387-94674-8. The biography in English.

Sauer, Tilman, 1999, "The relativity of discovery: Hilbert's first note on the foundations of physics," Arch. Hist. Exact Sci. 53: 529-75.

Sieg, Wilfried, and Ravaglia, Mark, 2005, "Grundlagen der Mathematik" in Grattan-Guinness, I., ed., Landmark Writings in Western Mathematics. Elsevier: 981-99. (in English)

Thorne, Kip, 1995. Black Holes and Time Warps: Einstein's Outrageous Legacy, W. W. Norton & Company; Reprint edition. ISBN 0-393-31276-3.

External links

Hilbert Bernays Project

Hilbert's 23 Problems Address

Hilbert's Program

Hilbert's radio speech recorded in Königsberg 1930 (in German), with English translation

'From Hilbert's Problems to the Future', lecture by Professor Robin Wilson, Gresham College, 27 February 2008 (available in text, audio and video formats).

Category:German mathematicians Category:19th-century mathematicians Category:20th-century mathematicians Category:University of Königsberg alumni Category:University of Königsberg faculty Category:University of Göttingen faculty Category:Relativists Category:Geometers Category:Operator theorists Category:Mathematical analysts Category:Number theorists Category:Foreign Members of the Royal Society Category:German Lutherans Category:People from Königsberg Category:People from the Province of Prussia Category:1862 births Category:1943 deaths Category:Burials in Germany Category:Formalism (deductive)