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67-443: Del is a vector differential operator represented by the symbol ∇ (nabla). Del or DEL can also refer to: Del Del is a very convenient mathematical notation for those three operations (gradient, divergence, and curl) that makes many equations easier to write and remember. The del symbol (or nabla) can be formally defined as a vector operator whose components are the corresponding partial derivative operators. As

134-414: A = 0 {\displaystyle a=0} . While not explicitly studied by Hamilton, this indirectly introduced notions of basis, here given by the quaternion elements i , j , k {\displaystyle i,j,k} , as well as the dot product and cross product , which correspond to (the negative of) the scalar part and the vector part of the product of two vector quaternions. It

201-434: A n -dimensional Euclidean space and a Cartesian coordinate system . When n = 3 , this space is called the three-dimensional Euclidean space (or simply "Euclidean space" when the context is clear). In classical physics , it serves as a model of the physical universe , in which all known matter exists. When relativity theory is considered, it can be considered a local subspace of space-time . While this space remains

268-487: A parallelogram , and hence are coplanar. A sphere in 3-space (also called a 2-sphere because it is a 2-dimensional object) consists of the set of all points in 3-space at a fixed distance r from a central point P . The solid enclosed by the sphere is called a ball (or, more precisely a 3-ball ). The volume of the ball is given by V = 4 3 π r 3 , {\displaystyle V={\frac {4}{3}}\pi r^{3},} and

335-404: A scalar field f {\displaystyle f} is called the gradient , and it can be represented as: It always points in the direction of greatest increase of f {\displaystyle f} , and it has a magnitude equal to the maximum rate of increase at the point—just like a standard derivative. In particular, if a hill is defined as a height function over

402-517: A three-dimensional space ( 3D space , 3-space or, rarely, tri-dimensional space ) is a mathematical space in which three values ( coordinates ) are required to determine the position of a point . Most commonly, it is the three-dimensional Euclidean space , that is, the Euclidean space of dimension three, which models physical space . More general three-dimensional spaces are called 3-manifolds . The term may also refer colloquially to

469-522: A choice of basis, corresponding to a set of axes. But in rotational symmetry, there is no reason why one set of axes is preferred to say, the same set of axes which has been rotated arbitrarily. Stated another way, a preferred choice of axes breaks the rotational symmetry of physical space. Computationally, it is necessary to work with the more concrete description R 3 {\displaystyle \mathbb {R} ^{3}} in order to do concrete computations. A more abstract description still

536-528: A field , which is not commutative nor associative , but is a Lie algebra with the cross product being the Lie bracket. Specifically, the space together with the product, ( R 3 , × ) {\displaystyle (\mathbb {R} ^{3},\times )} is isomorphic to the Lie algebra of three-dimensional rotations, denoted s o ( 3 ) {\displaystyle {\mathfrak {so}}(3)} . In order to satisfy

603-709: A formal cross product —to give a vector field called the curl. These formal products do not necessarily commute with other operators or products. These three uses, detailed below, are summarized as: In the Cartesian coordinate system R n {\displaystyle \mathbb {R} ^{n}} with coordinates ( x 1 , … , x n ) {\displaystyle (x_{1},\dots ,x_{n})} and standard basis { e 1 , … , e n } {\displaystyle \{\mathbf {e} _{1},\dots ,\mathbf {e} _{n}\}} , del

670-412: A given plane, intersect that plane in a unique point, or be parallel to the plane. In the last case, there will be lines in the plane that are parallel to the given line. A hyperplane is a subspace of one dimension less than the dimension of the full space. The hyperplanes of a three-dimensional space are the two-dimensional subspaces, that is, the planes. In terms of Cartesian coordinates, the points of

737-400: A hyperplane satisfy a single linear equation , so planes in this 3-space are described by linear equations. A line can be described by a pair of independent linear equations—each representing a plane having this line as a common intersection. Varignon's theorem states that the midpoints of any quadrilateral in R 3 {\displaystyle \mathbb {R} ^{3}} form

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804-426: A plane h ( x , y ) {\displaystyle h(x,y)} , the gradient at a given location will be a vector in the xy-plane (visualizable as an arrow on a map) pointing along the steepest direction. The magnitude of the gradient is the value of this steepest slope. In particular, this notation is powerful because the gradient product rule looks very similar to the 1d-derivative case: However,

871-419: A plane curve about a fixed line in its plane as an axis is called a surface of revolution . The plane curve is called the generatrix of the surface. A section of the surface, made by intersecting the surface with a plane that is perpendicular (orthogonal) to the axis, is a circle. Simple examples occur when the generatrix is a line. If the generatrix line intersects the axis line, the surface of revolution

938-465: A scalar field f ( x , y , z ) {\displaystyle f(x,y,z)} in the direction a ( x , y , z ) = a x x ^ + a y y ^ + a z z ^ {\displaystyle \mathbf {a} (x,y,z)=a_{x}{\hat {\mathbf {x} }}+a_{y}{\hat {\mathbf {y} }}+a_{z}{\hat {\mathbf {z} }}}

1005-467: A scalar field f or a vector field v ; the use of the scalar Laplacian and vector Laplacian gives two more: These are of interest principally because they are not always unique or independent of each other. As long as the functions are well-behaved ( C ∞ {\displaystyle C^{\infty }} in most cases), two of them are always zero: Two of them are always equal: The 3 remaining vector derivatives are related by

1072-445: A single coherent unit; fluid dynamics uses this convention extensively, terming it the convective derivative —the "moving" derivative of the fluid. Note that ( a ⋅ ∇ ) {\displaystyle (\mathbf {a} \cdot \nabla )} is an operator that takes scalar to a scalar. It can be extended to operate on a vector, by separately operating on each of its components. The Laplace operator

1139-409: A subset of space, a three-dimensional region (or 3D domain ), a solid figure . Technically, a tuple of n numbers can be understood as the Cartesian coordinates of a location in a n -dimensional Euclidean space. The set of these n -tuples is commonly denoted R n , {\displaystyle \mathbb {R} ^{n},} and can be identified to the pair formed by

1206-442: A subtle way. By definition, there exists a basis B = { e 1 , e 2 , e 3 } {\displaystyle {\mathcal {B}}=\{e_{1},e_{2},e_{3}\}} for V {\displaystyle V} . This corresponds to an isomorphism between V {\displaystyle V} and R 3 {\displaystyle \mathbb {R} ^{3}} :

1273-445: A unique plane, so skew lines are lines that do not meet and do not lie in a common plane. Two distinct planes can either meet in a common line or are parallel (i.e., do not meet). Three distinct planes, no pair of which are parallel, can either meet in a common line, meet in a unique common point, or have no point in common. In the last case, the three lines of intersection of each pair of planes are mutually parallel. A line can lie in

1340-421: A vector A is denoted by || A || . The dot product of a vector A = [ A 1 , A 2 , A 3 ] with itself is which gives the formula for the Euclidean length of the vector. Without reference to the components of the vectors, the dot product of two non-zero Euclidean vectors A and B is given by where θ is the angle between A and B . The cross product or vector product

1407-402: A vector operator, it can act on scalar and vector fields in three different ways, giving rise to three different differential operations: first, it can act on scalar fields by a formal scalar multiplication—to give a vector field called the gradient; second, it can act on vector fields by a formal dot product —to give a scalar field called the divergence; and lastly, it can act on vector fields by

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1474-609: A vector. Though one can often replace del with a vector and obtain a vector identity, making those identities mnemonic, the reverse is not necessarily reliable, because del does not commute in general. A counterexample that demonstrates the divergence ( ∇ ⋅ v {\displaystyle \nabla \cdot \mathbf {v} } ) and the advection operator ( v ⋅ ∇ {\displaystyle \mathbf {v} \cdot \nabla } ) are not commutative: A counterexample that relies on del's differential properties: Central to these distinctions

1541-627: Is a binary operation on two vectors in three-dimensional space and is denoted by the symbol ×. The cross product A × B of the vectors A and B is a vector that is perpendicular to both and therefore normal to the plane containing them. It has many applications in mathematics, physics , and engineering . In function language, the cross product is a function × : R 3 × R 3 → R 3 {\displaystyle \times :\mathbb {R} ^{3}\times \mathbb {R} ^{3}\rightarrow \mathbb {R} ^{3}} . The components of

1608-408: Is a scalar field that can be represented as: The divergence is roughly a measure of a vector field's increase in the direction it points; but more accurately, it is a measure of that field's tendency to converge toward or diverge from a point. The power of the del notation is shown by the following product rule: The formula for the vector product is slightly less intuitive, because this product

1675-428: Is a vector function that can be represented as: The curl at a point is proportional to the on-axis torque that a tiny pinwheel would be subjected to if it were centered at that point. The vector product operation can be visualized as a pseudo- determinant : Again the power of the notation is shown by the product rule: The rule for the vector product does not turn out to be simple: The directional derivative of

1742-474: Is a 9-term second-rank tensor – that is, a 3×3 matrix – but can be denoted simply as ∇ ⊗ v {\displaystyle \nabla \otimes \mathbf {v} } , where ⊗ {\displaystyle \otimes } represents the dyadic product . This quantity is equivalent to the transpose of the Jacobian matrix of the vector field with respect to space. The divergence of

1809-758: Is a right circular cone with vertex (apex) the point of intersection. However, if the generatrix and axis are parallel, then the surface of revolution is a circular cylinder . In analogy with the conic sections , the set of points whose Cartesian coordinates satisfy the general equation of the second degree, namely, A x 2 + B y 2 + C z 2 + F x y + G y z + H x z + J x + K y + L z + M = 0 , {\displaystyle Ax^{2}+By^{2}+Cz^{2}+Fxy+Gyz+Hxz+Jx+Ky+Lz+M=0,} where A , B , C , F , G , H , J , K , L and M are real numbers and not all of A , B , C , F , G and H are zero,

1876-456: Is a scalar operator that can be applied to either vector or scalar fields; for cartesian coordinate systems it is defined as: and the definition for more general coordinate systems is given in vector Laplacian . The Laplacian is ubiquitous throughout modern mathematical physics , appearing for example in Laplace's equation , Poisson's equation , the heat equation , the wave equation , and

1943-458: Is a vector operator whose x 1 , … , x n {\displaystyle x_{1},\dots ,x_{n}} components are the partial derivative operators ∂ ∂ x 1 , … , ∂ ∂ x n {\displaystyle {\partial \over \partial x_{1}},\dots ,{\partial \over \partial x_{n}}} ; that is, Where

2010-466: Is called a quadric surface . There are six types of non-degenerate quadric surfaces: The degenerate quadric surfaces are the empty set, a single point, a single line, a single plane, a pair of planes or a quadratic cylinder (a surface consisting of a non-degenerate conic section in a plane π and all the lines of R through that conic that are normal to π ). Elliptic cones are sometimes considered to be degenerate quadric surfaces as well. Both

2077-400: Is defined as: Which is equal to the following when the gradient exists This gives the rate of change of a field f {\displaystyle f} in the direction of a {\displaystyle \mathbf {a} } , scaled by the magnitude of a {\displaystyle \mathbf {a} } . In operator notation, the element in parentheses can be considered

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2144-406: Is found in linear algebra , where the idea of independence is crucial. Space has three dimensions because the length of a box is independent of its width or breadth. In the technical language of linear algebra, space is three-dimensional because every point in space can be described by a linear combination of three independent vectors . A vector can be pictured as an arrow. The vector's magnitude

2211-425: Is its length, and its direction is the direction the arrow points. A vector in R 3 {\displaystyle \mathbb {R} ^{3}} can be represented by an ordered triple of real numbers. These numbers are called the components of the vector. The dot product of two vectors A = [ A 1 , A 2 , A 3 ] and B = [ B 1 , B 2 , B 3 ] is defined as: The magnitude of

2278-397: Is not commutative: The curl of a vector field v ( x , y , z ) = v x x ^ + v y y ^ + v z z ^ {\displaystyle \mathbf {v} (x,y,z)=v_{x}{\hat {\mathbf {x} }}+v_{y}{\hat {\mathbf {y} }}+v_{z}{\hat {\mathbf {z} }}}

2345-669: Is the Kronecker delta . Written out in full, the standard basis is E 1 = ( 1 0 0 ) , E 2 = ( 0 1 0 ) , E 3 = ( 0 0 1 ) . {\displaystyle E_{1}={\begin{pmatrix}1\\0\\0\end{pmatrix}},E_{2}={\begin{pmatrix}0\\1\\0\end{pmatrix}},E_{3}={\begin{pmatrix}0\\0\\1\end{pmatrix}}.} Therefore R 3 {\displaystyle \mathbb {R} ^{3}} can be viewed as

2412-534: Is the Levi-Civita symbol . It has the property that A × B = − B × A {\displaystyle \mathbf {A} \times \mathbf {B} =-\mathbf {B} \times \mathbf {A} } . Its magnitude is related to the angle θ {\displaystyle \theta } between A {\displaystyle \mathbf {A} } and B {\displaystyle \mathbf {B} } by

2479-418: Is the fact that del is not simply a vector; it is a vector operator . Whereas a vector is an object with both a magnitude and direction, del has neither a magnitude nor a direction until it operates on a function. For that reason, identities involving del must be derived with care, using both vector identities and differentiation identities such as the product rule. Three-dimensional In geometry ,

2546-563: Is the following, where u ⊗ v {\displaystyle \mathbf {u} \otimes \mathbf {v} } is the outer product tensor: When del operates on a scalar or vector, either a scalar or vector is returned. Because of the diversity of vector products (scalar, dot, cross) one application of del already gives rise to three major derivatives: the gradient (scalar product), divergence (dot product), and curl (cross product). Applying these three sorts of derivatives again to each other gives five possible second derivatives, for

2613-418: Is to model physical space as a three-dimensional affine space E ( 3 ) {\displaystyle E(3)} over the real numbers. This is unique up to affine isomorphism. It is sometimes referred to as three-dimensional Euclidean space. Just as the vector space description came from 'forgetting the preferred basis' of R 3 {\displaystyle \mathbb {R} ^{3}} ,

2680-473: Is written as As a vector operator, del naturally acts on scalar fields via scalar multiplication, and naturally acts on vector fields via dot products and cross products. More specifically, for any scalar field f {\displaystyle f} and any vector field F = ( F x , F y , F z ) {\displaystyle \mathbf {F} =(F_{x},F_{y},F_{z})} , if one defines then using

2747-572: The Schrödinger equation . While ∇ 2 {\displaystyle \nabla ^{2}} usually represents the Laplacian , sometimes ∇ 2 {\displaystyle \nabla ^{2}} also represents the Hessian matrix . The former refers to the inner product of ∇ {\displaystyle \nabla } , while the latter refers to

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2814-443: The dyadic product of ∇ {\displaystyle \nabla } : So whether ∇ 2 {\displaystyle \nabla ^{2}} refers to a Laplacian or a Hessian matrix depends on the context. Del can also be applied to a vector field with the result being a tensor . The tensor derivative of a vector field v {\displaystyle \mathbf {v} } (in three dimensions)

2881-513: The 19th century, developments of the geometry of three-dimensional space came with William Rowan Hamilton 's development of the quaternions . In fact, it was Hamilton who coined the terms scalar and vector , and they were first defined within his geometric framework for quaternions . Three dimensional space could then be described by quaternions q = a + u i + v j + w k {\displaystyle q=a+ui+vj+wk} which had vanishing scalar component, that is,

2948-474: The above definition of ∇ {\displaystyle \nabla } , one may write and and Del can also be expressed in other coordinate systems, see for example del in cylindrical and spherical coordinates . Del is used as a shorthand form to simplify many long mathematical expressions. It is most commonly used to simplify expressions for the gradient , divergence , curl , directional derivative , and Laplacian . The vector derivative of

3015-399: The above-mentioned systems. Two distinct points always determine a (straight) line . Three distinct points are either collinear or determine a unique plane . On the other hand, four distinct points can either be collinear, coplanar , or determine the entire space. Two distinct lines can either intersect, be parallel or be skew . Two parallel lines, or two intersecting lines , lie in

3082-495: The abstract vector space, together with the additional structure of a choice of basis. Conversely, V {\displaystyle V} can be obtained by starting with R 3 {\displaystyle \mathbb {R} ^{3}} and 'forgetting' the Cartesian product structure, or equivalently the standard choice of basis. As opposed to a general vector space V {\displaystyle V} ,

3149-775: The axioms of a Lie algebra, instead of associativity the cross product satisfies the Jacobi identity . For any three vectors A , B {\displaystyle \mathbf {A} ,\mathbf {B} } and C {\displaystyle \mathbf {C} } A × ( B × C ) + B × ( C × A ) + C × ( A × B ) = 0 {\displaystyle \mathbf {A} \times (\mathbf {B} \times \mathbf {C} )+\mathbf {B} \times (\mathbf {C} \times \mathbf {A} )+\mathbf {C} \times (\mathbf {A} \times \mathbf {B} )=0} One can in n dimensions take

3216-630: The construction for the isomorphism is found here . However, there is no 'preferred' or 'canonical basis' for V {\displaystyle V} . On the other hand, there is a preferred basis for R 3 {\displaystyle \mathbb {R} ^{3}} , which is due to its description as a Cartesian product of copies of R {\displaystyle \mathbb {R} } , that is, R 3 = R × R × R {\displaystyle \mathbb {R} ^{3}=\mathbb {R} \times \mathbb {R} \times \mathbb {R} } . This allows

3283-491: The construction of the five regular Platonic solids in a sphere. In the 17th century, three-dimensional space was described with Cartesian coordinates , with the advent of analytic geometry developed by René Descartes in his work La Géométrie and Pierre de Fermat in the manuscript Ad locos planos et solidos isagoge (Introduction to Plane and Solid Loci), which was unpublished during Fermat's lifetime. However, only Fermat's work dealt with three-dimensional space. In

3350-880: The cross product are A × B = [ A 2 B 3 − B 2 A 3 , A 3 B 1 − B 3 A 1 , A 1 B 2 − B 1 A 2 ] {\displaystyle \mathbf {A} \times \mathbf {B} =[A_{2}B_{3}-B_{2}A_{3},A_{3}B_{1}-B_{3}A_{1},A_{1}B_{2}-B_{1}A_{2}]} , and can also be written in components, using Einstein summation convention as ( A × B ) i = ε i j k A j B k {\displaystyle (\mathbf {A} \times \mathbf {B} )_{i}=\varepsilon _{ijk}A_{j}B_{k}} where ε i j k {\displaystyle \varepsilon _{ijk}}

3417-516: The definition of canonical projections, π i : R 3 → R {\displaystyle \pi _{i}:\mathbb {R} ^{3}\rightarrow \mathbb {R} } , where 1 ≤ i ≤ 3 {\displaystyle 1\leq i\leq 3} . For example, π 1 ( x 1 , x 2 , x 3 ) = x {\displaystyle \pi _{1}(x_{1},x_{2},x_{3})=x} . This then allows

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3484-493: The definition of the standard basis B Standard = { E 1 , E 2 , E 3 } {\displaystyle {\mathcal {B}}_{\text{Standard}}=\{E_{1},E_{2},E_{3}\}} defined by π i ( E j ) = δ i j {\displaystyle \pi _{i}(E_{j})=\delta _{ij}} where δ i j {\displaystyle \delta _{ij}}

3551-457: The equation: And one of them can even be expressed with the tensor product, if the functions are well-behaved: Most of the above vector properties (except for those that rely explicitly on del's differential properties—for example, the product rule) rely only on symbol rearrangement, and must necessarily hold if the del symbol is replaced by any other vector. This is part of the value to be gained in notationally representing this operator as

3618-503: The expression in parentheses is a row vector. In three-dimensional Cartesian coordinate system R 3 {\displaystyle \mathbb {R} ^{3}} with coordinates ( x , y , z ) {\displaystyle (x,y,z)} and standard basis or unit vectors of axes { e x , e y , e z } {\displaystyle \{\mathbf {e} _{x},\mathbf {e} _{y},\mathbf {e} _{z}\}} , del

3685-421: The hyperboloid of one sheet and the hyperbolic paraboloid are ruled surfaces , meaning that they can be made up from a family of straight lines. In fact, each has two families of generating lines, the members of each family are disjoint and each member one family intersects, with just one exception, every member of the other family. Each family is called a regulus . Another way of viewing three-dimensional space

3752-460: The identity ‖ A × B ‖ = ‖ A ‖ ⋅ ‖ B ‖ ⋅ | sin ⁡ θ | . {\displaystyle \left\|\mathbf {A} \times \mathbf {B} \right\|=\left\|\mathbf {A} \right\|\cdot \left\|\mathbf {B} \right\|\cdot \left|\sin \theta \right|.} The space and product form an algebra over

3819-428: The most compelling and useful way to model the world as it is experienced, it is only one example of a 3-manifold. In this classical example, when the three values refer to measurements in different directions ( coordinates ), any three directions can be chosen, provided that these directions do not lie in the same plane . Furthermore, if these directions are pairwise perpendicular , the three values are often labeled by

3886-555: The position of any point in three-dimensional space is given by an ordered triple of real numbers , each number giving the distance of that point from the origin measured along the given axis, which is equal to the distance of that point from the plane determined by the other two axes. Other popular methods of describing the location of a point in three-dimensional space include cylindrical coordinates and spherical coordinates , though there are an infinite number of possible methods. For more, see Euclidean space . Below are images of

3953-472: The product of n − 1 vectors to produce a vector perpendicular to all of them. But if the product is limited to non-trivial binary products with vector results, it exists only in three and seven dimensions . It can be useful to describe three-dimensional space as a three-dimensional vector space V {\displaystyle V} over the real numbers. This differs from R 3 {\displaystyle \mathbb {R} ^{3}} in

4020-462: The rules for dot products do not turn out to be simple, as illustrated by: The divergence of a vector field v ( x , y , z ) = v x x ^ + v y y ^ + v z z ^ {\displaystyle \mathbf {v} (x,y,z)=v_{x}{\hat {\mathbf {x} }}+v_{y}{\hat {\mathbf {y} }}+v_{z}{\hat {\mathbf {z} }}}

4087-530: The space R 3 {\displaystyle \mathbb {R} ^{3}} is sometimes referred to as a coordinate space. Physically, it is conceptually desirable to use the abstract formalism in order to assume as little structure as possible if it is not given by the parameters of a particular problem. For example, in a problem with rotational symmetry, working with the more concrete description of three-dimensional space R 3 {\displaystyle \mathbb {R} ^{3}} assumes

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4154-418: The surface area of the sphere is A = 4 π r 2 . {\displaystyle A=4\pi r^{2}.} Another type of sphere arises from a 4-ball, whose three-dimensional surface is the 3-sphere : points equidistant to the origin of the euclidean space R . If a point has coordinates, P ( x , y , z , w ) , then x + y + z + w = 1 characterizes those points on

4221-404: The terms width /breadth , height /depth , and length . Books XI to XIII of Euclid's Elements dealt with three-dimensional geometry. Book XI develops notions of orthogonality and parallelism of lines and planes, and defines solids including parallelpipeds, pyramids, prisms, spheres, octahedra, icosahedra and dodecahedra. Book XII develops notions of similarity of solids. Book XIII describes

4288-446: The unit 3-sphere centered at the origin. This 3-sphere is an example of a 3-manifold: a space which is 'looks locally' like 3-D space. In precise topological terms, each point of the 3-sphere has a neighborhood which is homeomorphic to an open subset of 3-D space. In three dimensions, there are nine regular polytopes: the five convex Platonic solids and the four nonconvex Kepler-Poinsot polyhedra . A surface generated by revolving

4355-568: The vector field can then be expressed as the trace of this matrix. For a small displacement δ r {\displaystyle \delta \mathbf {r} } , the change in the vector field is given by: For vector calculus : For matrix calculus (for which u ⋅ v {\displaystyle \mathbf {u} \cdot \mathbf {v} } can be written u T v {\displaystyle \mathbf {u} ^{\text{T}}\mathbf {v} } ): Another relation of interest (see e.g. Euler equations )

4422-491: The work of Hermann Grassmann and Giuseppe Peano , the latter of whom first gave the modern definition of vector spaces as an algebraic structure. In mathematics, analytic geometry (also called Cartesian geometry) describes every point in three-dimensional space by means of three coordinates. Three coordinate axes are given, each perpendicular to the other two at the origin , the point at which they cross. They are usually labeled x , y , and z . Relative to these axes,

4489-473: Was not until Josiah Willard Gibbs that these two products were identified in their own right, and the modern notation for the dot and cross product were introduced in his classroom teaching notes, found also in the 1901 textbook Vector Analysis written by Edwin Bidwell Wilson based on Gibbs' lectures. Also during the 19th century came developments in the abstract formalism of vector spaces, with

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