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dot mathbf a sqrt a_ 1 2 a_ 2 2 a_ 3 2 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 39 a b a b cos θ displaystyle mathbf a cdot mathbf b mathbf a mathbf b cos theta where θ is the angle between a and b for a physical example consider a block on an inclined plane that is being pulled downward by a gravitational force the dot product can be used to compute the work w displaystyle w performed by the constant force vector g displaystyle mathbf g that is applied at an angle θ displaystyle theta to the downslope direction of motion d displaystyle mathbf d that is 40 w g d g d cos θ displaystyle w mathbf g cdot mathbf d mathbf g mathbf d cos theta cross product edit main article cross product the cross product or vector product 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 41 for example it can be used to compute the amount of torque on a bolt being turned by a wrench or the lorentz force on an electron travelling through a magnetic field 42 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 43 the cross product in respect to a right handed coordinate system the components of 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 is the levi civita symbol 44 it has the property that a b b a displaystyle mathbf a times mathbf b mathbf b times mathbf a 41 its magnitude is related to the angle θ displaystyle theta between a displaystyle mathbf a and b displaystyle mathbf b by the identity 41 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 a field which is not commutative nor associative but is a lie algebra with the cross product being the lie bracket 45 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 43 in order to satisfy 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 45 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 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 46 abstract description edit see also vector space 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 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 38 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 the three dimensional euclidean space 47 this allows 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 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 is the kronecker delta written out in full the standard basis is 48 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 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 the space r 3 displaystyle mathbb r 3 is sometimes referred to as a coordinate space 49 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 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 affine description edit see also affine space and euclidean space a more abstract description still 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 50 just as the vector space description came from forgetting the preferred basis of r 3 displaystyle mathbb r 3 the affine space description comes from forgetting the origin of the vector space euclidean spaces are sometimes called euclidean affine spaces for distinguishing them from euclidean vector spaces 51 this is physically appealing as it makes the translation invariance of physical space manifest a preferred origin breaks the translational invariance 50 inner product space edit see also inner product space the above discussion does not involve the dot product the dot product is an example of an inner product physical space can be modelled as a vector space which additionally has the structure of an inner product the inner product defines notions of length and angle and therefore in particular the notion of orthogonality 52 for any inner product there exist bases under which the inner product agrees with the dot product citation needed but again there are many different possible bases none of which are preferred they differ from one another by a rotation an element of the group of rotations so 3 in calculus edit main article vector calculus vector calculus is concerned with infinitesimal and cumulative changes to vector fields primarily in three dimensional euclidean space r 3 displaystyle mathbb r 3 for differentiation the del displaystyle nabla or nabla operator is used gradient divergence and curl edit the gradient indicates the direction of greatest increase of a function and its magnitude an example is a flow of particles with the gradient being the magnitude and direction of the flow at a location 53 in a rectangular coordinate system the gradient of a differentiable function f r 3 r displaystyle f mathbb r 3 rightarrow mathbb r is given by 54 f f x i f y j f z k displaystyle nabla f frac partial f partial x mathbf i frac partial f partial y mathbf j frac partial f partial z mathbf k where i j and k are the unit vectors for the x y and z axes respectively in index notation it is written 55 f i i f displaystyle nabla f _ i partial _ i f the divergence indicates the net flux of a vector field around a point such as an increase or decrease of particle density that is whether the location is a source or sink 56 the divergence of a differentiable vector field f u i v j w k that is a function f r 3 r 3 displaystyle mathbf f mathbb r 3 rightarrow mathbb r 3 is equal to the scalar valued function 54 div f f u x v y w z displaystyle operatorname div mathbf f nabla cdot mathbf f frac partial u partial x frac partial v partial y frac partial w partial z in index notation with einstein summation convention this is 55 f i f i displaystyle nabla cdot mathbf f partial _ i f_ i the curl or rotor is a vector indicating the rotational circulation of a vector field expanded in cartesian coordinates see del in cylindrical and spherical coordinates for spherical and cylindrical coordinate representations the curl f is for f composed of f x f y f z 57 i j k x y z f x f y f z displaystyle begin vmatrix mathbf i mathbf j mathbf k frac partial partial x frac partial partial y frac partial partial z f_ x f_ y f_ z end vmatrix this expands as follows 54 curl f f f z y f y z i f x z f z x j f y x f x y k displaystyle operatorname curl mathbf f nabla times mathbf f left frac partial f_ z partial y frac partial f_ y partial z right mathbf i left frac partial f_ x partial z frac partial f_ z partial x right mathbf j left frac partial f_ y partial x frac partial f_ x partial y right mathbf k in index notation with einstein summation convention this is 55 f i ϵ i j k j f k displaystyle nabla times mathbf f _ i epsilon _ ijk partial _ j f_ k where ϵ i j k displaystyle epsilon _ ijk is the totally antisymmetric symbol the levi civita symbol line surface and volume integrals edit illustration of a line integral along curve c in a vector field f a line integral of a function along a curve can be thought of as a continuous summation of the function value along every infinitesimal increment of that curve for some scalar field f u r n r the line integral along a piecewise smooth curve c u is defined as 58 c f d s a b f r t r t d t displaystyle int limits _ c f ds int _ a b f mathbf r t mathbf r t dt where r a b c is an arbitrary bijective one to one correspondence parametrization of the curve c such that r a and r b give the endpoints of c and a b displaystyle a b for a vector field f u r n r n the line integral along a piecewise smooth curve c u in the direction of r is defined as 58 c f r d r a b f r t r t d t displaystyle int limits _ c mathbf f mathbf r cdot d mathbf r int _ a b mathbf f mathbf r t cdot mathbf r t dt where displaystyle cdot is the dot product and r a b c is a bijective parametrization of the curve c such that r a and r b give the endpoints of c a subtype of line integral found in physics is the plane closed loop which determines the circulation of the function around the loop 59 c f r d r displaystyle oint _ c mathbf f mathbf r cdot d mathbf r a surface integral is a generalization of multiple integrals to integration over surfaces it can be thought of as the double integral analog of the line integral to find an explicit formula for the surface integral we need to parameterize the surface of interest s by considering a system of curvilinear coordinates on s like the latitude and longitude on a sphere let such a parameterization be x s t where s t varies in some region t in the plane then the surface integral is given by citation needed s f d s t f x s t x s x t d s d t displaystyle iint _ s f mathrm d s iint _ t f mathbf x s t left partial mathbf x over partial s times partial mathbf x over partial t right mathrm d s mathrm d t where the expression between bars on the right hand side is the magnitude of the cross product of the partial derivatives of x s t and is known as the surface element given a vector field v on s that is a function that assigns to each x in s a vector v x the surface integral can be defined component wise according to the definition of the surface integral of a scalar field the result is a vector a volume integral is an integral over a three dimensional domain or region when the integrand is trivial unity the volume integral is simply the region s volume 60 1 it can also mean a triple integral within a region d in r 3 of a function f x y z displaystyle f x y z and is usually written as d f x y z d x d y d z displaystyle iiint limits _ d f x y z dx dy dz fundamental theorem of line integrals edit main article fundamental theorem of line integrals the fundamental theorem of line integrals says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve 61 let φ u r n r displaystyle varphi u subseteq mathbb r n to mathbb r then φ q φ p γ p q φ r d r displaystyle varphi left mathbf q right varphi left mathbf p right int _ gamma mathbf p mathbf q nabla varphi mathbf r cdot d mathbf r stokes theorem edit main article stokes theorem stokes theorem relates the surface integral of the curl of a vector field f over a surface σ in euclidean three space to the line integral of the vector field over its boundary σ 62 σ f d σ σ f d r displaystyle iint _ sigma nabla times mathbf f cdot mathrm d mathbf sigma oint _ partial sigma mathbf f cdot mathrm d mathbf r divergence theorem edit main article divergence theorem suppose v is a subset of r n displaystyle mathbb r n in the case of n 3 v represents a volume in 3d space which is compact and has a piecewise smooth boundary s also indicated with v s if f is a continuously differentiable vector field defined on a neighborhood of v then the divergence theorem says 63 v f d v displaystyle iiint _ v left mathbf nabla cdot mathbf f right dv s displaystyle scriptstyle s f n d s displaystyle mathbf f cdot mathbf n ds the left side is a volume integral over the volume v the right side is the surface integral over the boundary of the volume v the closed manifold v is quite generally the boundary of v oriented by outward pointing normals and n is the outward pointing unit normal field of the boundary v d s may be used as a shorthand for n ds in topology edit wikipedia s globe logo in 3 d three dimensional space has a number of topological properties that distinguish it from spaces of other dimension numbers for example at least three dimensions are required to tie a knot in a piece of string 64 page needed in differential geometry the generic three dimensional spaces are 3 manifolds which locally resemble r 3 displaystyle mathbb r 3 globally the same 3 manifold can curve in a variety of manners...
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