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79-454: D f = ∇ f ⋅ d r {\displaystyle df=\nabla f\cdot d\mathbf {r} } where d f {\displaystyle df} is the total infinitesimal change in f {\displaystyle f} for an infinitesimal displacement d r {\displaystyle d\mathbf {r} } , and is seen to be maximal when d r {\displaystyle d\mathbf {r} }

158-415: A ) ) , {\displaystyle \nabla (f\circ g)(c)={\big (}Dg(c){\big )}^{\mathsf {T}}{\big (}\nabla f(a){\big )},} where ( Dg ) denotes the transpose Jacobian matrix . Nabla symbol The nabla is a triangular symbol resembling an inverted Greek delta : ∇ {\displaystyle \nabla } or ∇. The name comes, by reason of the symbol's shape, from

237-408: A 1 a 2 ] and b = [ b 1 b 2 ] which are vectors in the uv -plane. That is, put This is a symmetric function in a and b , meaning that It is also bilinear , meaning that it is linear in each variable a and b separately. That is, for any vectors a , a ′ , b , and b ′ in the uv plane, and any real numbers μ and λ . In particular, the length of

316-437: A metric space . Conversely, the metric tensor itself is the derivative of the distance function (taken in a suitable manner). While the notion of a metric tensor was known in some sense to mathematicians such as Gauss from the early 19th century, it was not until the early 20th century that its properties as a tensor were understood by, in particular, Gregorio Ricci-Curbastro and Tullio Levi-Civita , who first codified

395-467: A nondegenerate symmetric bilinear form on each tangent space that varies smoothly from point to point. Carl Friedrich Gauss in his 1827 Disquisitiones generales circa superficies curvas ( General investigations of curved surfaces ) considered a surface parametrically , with the Cartesian coordinates x , y , and z of points on the surface depending on two auxiliary variables u and v . Thus

474-459: A differentiable function f : R n → R {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} } at a point x {\displaystyle x} in R n {\displaystyle \mathbb {R} ^{n}} is a linear map from R n {\displaystyle \mathbb {R} ^{n}} to R {\displaystyle \mathbb {R} } which

553-557: A function can be expressed in terms of the gradient, rather than the derivative. The gradient of a function f {\displaystyle f} from the Euclidean space R n {\displaystyle \mathbb {R} ^{n}} to R {\displaystyle \mathbb {R} } at any particular point x 0 {\displaystyle x_{0}} in R n {\displaystyle \mathbb {R} ^{n}} characterizes

632-407: A metric tensor g p in the tangent space at p in a way that varies smoothly with p . More precisely, given any open subset U of manifold M and any (smooth) vector fields X and Y on U , the real function g ( X , Y ) ( p ) = g p ( X p , Y p ) {\displaystyle g(X,Y)(p)=g_{p}(X_{p},Y_{p})}

711-422: A metric tensor at a point p of M is a bilinear form defined on the tangent space at p (that is, a bilinear function that maps pairs of tangent vectors to real numbers ), and a metric field on M consists of a metric tensor at each point p of M that varies smoothly with p . A metric tensor g is positive-definite if g ( v , v ) > 0 for every nonzero vector v . A manifold equipped with

790-421: A number of times, no inconvenience to the speaker or listener arises from the repetition. ∇ V is read simply as "del V ". This book is responsible for the form in which the mathematics of the operator in question is now usually expressed—most notably in undergraduate physics, and especially electrodynamics, textbooks. The nabla is used in vector calculus as part of three distinct differential operators:

869-414: A parametric surface is (in today's terms) a vector-valued function depending on an ordered pair of real variables ( u , v ) , and defined in an open set D in the uv -plane. One of the chief aims of Gauss's investigations was to deduce those features of the surface which could be described by a function which would remain unchanged if the surface underwent a transformation in space (such as bending

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948-400: A point of the parametric surface M can be written in the form for suitable real numbers p 1 and p 2 . If two tangent vectors are given: then using the bilinearity of the dot product, This is plainly a function of the four variables a 1 , b 1 , a 2 , and b 2 . It is more profitably viewed, however, as a function that takes a pair of arguments a = [

1027-408: A positive-definite metric tensor is known as a Riemannian manifold . Such a metric tensor can be thought of as specifying infinitesimal distance on the manifold. On a Riemannian manifold M , the length of a smooth curve between two points p and q can be defined by integration, and the distance between p and q can be defined as the infimum of the lengths of all such curves; this makes M

1106-436: A room where the temperature is given by a scalar field , T , so at each point ( x , y , z ) the temperature is T ( x , y , z ) , independent of time. At each point in the room, the gradient of T at that point will show the direction in which the temperature rises most quickly, moving away from ( x , y , z ) . The magnitude of the gradient will determine how fast the temperature rises in that direction. Consider

1185-673: A scalar function f ( x 1 , x 2 , x 3 , …, x n ) is denoted ∇ f or ∇ → f where ∇ ( nabla ) denotes the vector differential operator , del . The notation grad f is also commonly used to represent the gradient. The gradient of f is defined as the unique vector field whose dot product with any vector v at each point x is the directional derivative of f along v . That is, ( ∇ f ( x ) ) ⋅ v = D v f ( x ) {\displaystyle {\big (}\nabla f(x){\big )}\cdot \mathbf {v} =D_{\mathbf {v} }f(x)} where

1264-436: A surface led Gauss to introduce the predecessor of the modern notion of the metric tensor. The metric tensor is [ E F F G ] {\textstyle {\begin{bmatrix}E&F\\F&G\end{bmatrix}}} in the description below; E, F, and G in the matrix can contain any number as long as the matrix is positive definite. If the variables u and v are taken to depend on

1343-404: A surface whose height above sea level at point ( x , y ) is H ( x , y ) . The gradient of H at a point is a plane vector pointing in the direction of the steepest slope or grade at that point. The steepness of the slope at that point is given by the magnitude of the gradient vector. The gradient can also be used to measure how a scalar field changes in other directions, rather than just

1422-429: A tangent vector a is given by and the angle θ between two vectors a and b is calculated by The surface area is another numerical quantity which should depend only on the surface itself, and not on how it is parameterized. If the surface M is parameterized by the function r → ( u , v ) over the domain D in the uv -plane, then the surface area of M is given by the integral where × denotes

1501-463: A third variable, t , taking values in an interval [ a , b ] , then r → ( u ( t ), v ( t )) will trace out a parametric curve in parametric surface M . The arc length of that curve is given by the integral where ‖ ⋅ ‖ {\displaystyle \left\|\cdot \right\|} represents the Euclidean norm . Here the chain rule has been applied, and

1580-519: A vector at each point; while the value of the derivative at a point is a co tangent vector – a linear functional on vectors. They are related in that the dot product of the gradient of f {\displaystyle f} at a point p {\displaystyle p} with another tangent vector v {\displaystyle \mathbf {v} } equals the directional derivative of f {\displaystyle f} at p {\displaystyle p} of

1659-544: Is d f = ∂ f ∂ x i e i {\textstyle \mathrm {d} f={\frac {\partial f}{\partial x^{i}}}\mathbf {e} ^{i}} ), where e i = ∂ x / ∂ x i {\displaystyle \mathbf {e} _{i}=\partial \mathbf {x} /\partial x^{i}} and e i = d x i {\displaystyle \mathbf {e} ^{i}=\mathrm {d} x^{i}} refer to

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1738-467: Is very important. Physical mathematics is very largely the mathematics of ∇. The name Nabla seems, therefore, ludicrously inefficient. Heaviside and Josiah Willard Gibbs (independently) are credited with the development of the version of vector calculus most popular today. The influential 1901 text Vector Analysis , written by Edwin Bidwell Wilson and based on the lectures of Gibbs, advocates

1817-459: Is a vector space T p M , called the tangent space , consisting of all tangent vectors to the manifold at the point p . A metric tensor at p is a function g p ( X p , Y p ) which takes as inputs a pair of tangent vectors X p and Y p at p , and produces as an output a real number ( scalar ), so that the following conditions are satisfied: A metric tensor field g on M assigns to each point p of M

1896-424: Is a smooth function of p . The components of the metric in any basis of vector fields , or frame , f = ( X 1 , ..., X n ) are given by The n functions g ij [ f ] form the entries of an n × n symmetric matrix , G [ f ] . If are two vectors at p ∈ U , then the value of the metric applied to v and w is determined by the coefficients ( 4 ) by bilinearity: Denoting

1975-684: Is also called del . The differential operator given in Cartesian coordinates { x , y , z } {\displaystyle \{x,y,z\}} on three-dimensional Euclidean space by was introduced in 1831 by the Irish mathematician and physicist William Rowan Hamilton , who called it ◁. (The unit vectors { i , j , k } {\displaystyle \{\mathbf {i} ,\mathbf {j} ,\mathbf {k} \}} were originally right versors in Hamilton's quaternions .) The mathematics of ∇ received its full exposition at

2054-466: Is also used in differential geometry to denote a connection . A symbol of the same form, though presumably not genealogically related, appears in other areas, e.g.: My dear Sir, The name I propose for ∇ is, as you will remember, Nabla... In Greek the leading form is ναβλᾰ... As to the thing it is a sort of harp and is said by Hieronymus and other authorities to have had the figure of ∇ (an inverted Δ). We can represent cases of this form, cases where it

2133-400: Is called the total differential or exterior derivative of f {\displaystyle f} and is an example of a differential 1-form . Much as the derivative of a function of a single variable represents the slope of the tangent to the graph of the function, the directional derivative of a function in several variables represents the slope of the tangent hyperplane in

2212-961: Is customary to represent the gradient as a row vector or column vector of its components in a rectangular coordinate system; this article follows the convention of the gradient being a column vector, while the derivative is a row vector. In cylindrical coordinates with a Euclidean metric, the gradient is given by: ∇ f ( ρ , φ , z ) = ∂ f ∂ ρ e ρ + 1 ρ ∂ f ∂ φ e φ + ∂ f ∂ z e z , {\displaystyle \nabla f(\rho ,\varphi ,z)={\frac {\partial f}{\partial \rho }}\mathbf {e} _{\rho }+{\frac {1}{\rho }}{\frac {\partial f}{\partial \varphi }}\mathbf {e} _{\varphi }+{\frac {\partial f}{\partial z}}\mathbf {e} _{z},} where ρ

2291-652: Is defined at the point p = ( x 1 , … , x n ) {\displaystyle p=(x_{1},\ldots ,x_{n})} in n -dimensional space as the vector ∇ f ( p ) = [ ∂ f ∂ x 1 ( p ) ⋮ ∂ f ∂ x n ( p ) ] . {\displaystyle \nabla f(p)={\begin{bmatrix}{\frac {\partial f}{\partial x_{1}}}(p)\\\vdots \\{\frac {\partial f}{\partial x_{n}}}(p)\end{bmatrix}}.} Note that

2370-419: Is given by matrix multiplication . Assuming the standard Euclidean metric on R n {\displaystyle \mathbb {R} ^{n}} , the gradient is then the corresponding column vector, that is, ( ∇ f ) i = d f i T . {\displaystyle (\nabla f)_{i}=df_{i}^{\mathsf {T}}.} The best linear approximation to

2449-466: Is impossible to avoid the repetition of more than two indices. Despite the use of upper and lower indices, e ^ i {\displaystyle \mathbf {\hat {e}} _{i}} , e ^ i {\displaystyle \mathbf {\hat {e}} ^{i}} , and h i {\displaystyle h_{i}} are neither contravariant nor covariant. The latter expression evaluates to

Gradient - Misplaced Pages Continue

2528-401: Is in the direction of the gradient ∇ f {\displaystyle \nabla f} . The nabla symbol ∇ {\displaystyle \nabla } , written as an upside-down triangle and pronounced "del", denotes the vector differential operator . When a coordinate system is used in which the basis vectors are not functions of position, the gradient is given by

2607-427: Is increased by du units, and v is increased by dv units. Using matrix notation, the first fundamental form becomes Suppose now that a different parameterization is selected, by allowing u and v to depend on another pair of variables u ′ and v ′ . Then the analog of ( 2 ) for the new variables is The chain rule relates E ′ , F ′ , and G ′ to E , F , and G via the matrix equation where

2686-429: Is indeterminate whether in fiction f : a = b , as follows: (A) ∇[ a = b ] . Metric tensor In the mathematical field of differential geometry , a metric tensor (or simply metric ) is an additional structure on a manifold M (such as a surface ) that allows defining distances and angles, just as the inner product on a Euclidean space allows defining distances and angles there. More precisely,

2765-401: Is not differentiable at the origin as it does not have a well defined tangent plane despite having well defined partial derivatives in every direction at the origin. In this particular example, under rotation of x-y coordinate system, the above formula for gradient fails to transform like a vector (gradient becomes dependent on choice of basis for coordinate system) and also fails to point towards

2844-508: Is often denoted by d f x {\displaystyle df_{x}} or D f ( x ) {\displaystyle Df(x)} and called the differential or total derivative of f {\displaystyle f} at x {\displaystyle x} . The function d f {\displaystyle df} , which maps x {\displaystyle x} to d f x {\displaystyle df_{x}} ,

2923-535: Is only valid when the basis of the coordinate system is orthonormal. For any other basis, the Metric tensor at that point needs to be taken into account. For example, the function f ( x , y ) = x 2 y x 2 + y 2 {\displaystyle f(x,y)={\frac {x^{2}y}{x^{2}+y^{2}}}} unless at origin where f ( 0 , 0 ) = 0 {\displaystyle f(0,0)=0} ,

3002-417: Is the dot product : taking the dot product of a vector with the gradient is the same as taking the directional derivative along the vector. If R n {\displaystyle \mathbb {R} ^{n}} is viewed as the space of (dimension n {\displaystyle n} ) column vectors (of real numbers), then one can regard d f {\displaystyle df} as

3081-969: Is the axial distance, φ is the azimuthal or azimuth angle, z is the axial coordinate, and e ρ , e φ and e z are unit vectors pointing along the coordinate directions. In spherical coordinates , the gradient is given by: ∇ f ( r , θ , φ ) = ∂ f ∂ r e r + 1 r ∂ f ∂ θ e θ + 1 r sin ⁡ θ ∂ f ∂ φ e φ , {\displaystyle \nabla f(r,\theta ,\varphi )={\frac {\partial f}{\partial r}}\mathbf {e} _{r}+{\frac {1}{r}}{\frac {\partial f}{\partial \theta }}\mathbf {e} _{\theta }+{\frac {1}{r\sin \theta }}{\frac {\partial f}{\partial \varphi }}\mathbf {e} _{\varphi },} where r

3160-429: Is the dot product. As a consequence, the usual properties of the derivative hold for the gradient, though the gradient is not a derivative itself, but rather dual to the derivative: More generally, if instead I ⊂ R , then the following holds: ∇ ( f ∘ g ) ( c ) = ( D g ( c ) ) T ( ∇ f (

3239-514: Is the gradient of f {\displaystyle f} computed at x 0 {\displaystyle x_{0}} , and the dot denotes the dot product on R n {\displaystyle \mathbb {R} ^{n}} . This equation is equivalent to the first two terms in the multivariable Taylor series expansion of f {\displaystyle f} at x 0 {\displaystyle x_{0}} . Let U be an open set in R . If

Gradient - Misplaced Pages Continue

3318-593: Is the number of dimensions of the domain. Here, the upper index refers to the position in the list of the coordinate or component, so x refers to the second component—not the quantity x squared. The index variable i refers to an arbitrary element x . Using Einstein notation , the gradient can then be written as: ∇ f = ∂ f ∂ x i g i j e j {\displaystyle \nabla f={\frac {\partial f}{\partial x^{i}}}g^{ij}\mathbf {e} _{j}} (Note that its dual

3397-465: Is the radial distance, φ is the azimuthal angle and θ is the polar angle, and e r , e θ and e φ are again local unit vectors pointing in the coordinate directions (that is, the normalized covariant basis ). For the gradient in other orthogonal coordinate systems , see Orthogonal coordinates (Differential operators in three dimensions) . We consider general coordinates , which we write as x , …, x , …, x , where n

3476-580: The Hellenistic Greek word νάβλα for a Phoenician harp , and was suggested by the encyclopedist William Robertson Smith in an 1870 letter to Peter Guthrie Tait . The nabla symbol is available in standard HTML as ∇ and in LaTeX as \nabla . In Unicode , it is the character at code point U+2207, or 8711 in decimal notation, in the Mathematical Operators block. It

3555-472: The cross product , and the absolute value denotes the length of a vector in Euclidean space. By Lagrange's identity for the cross product, the integral can be written where det is the determinant . Let M be a smooth manifold of dimension n ; for instance a surface (in the case n = 2 ) or hypersurface in the Cartesian space R n + 1 {\displaystyle \mathbb {R} ^{n+1}} . At each point p ∈ M there

3634-510: The gradient (∇), the divergence (∇⋅), and the curl (∇×). The last of these uses the cross product and thus makes sense only in three dimensions; the first two are fully general. They were all originally studied in the context of the classical theory of electromagnetism, and contemporary university physics curricula typically treat the material using approximately the concepts and notation found in Gibbs and Wilson's Vector Analysis . The symbol

3713-399: The matrix ( g ij [ f ]) by G [ f ] and arranging the components of the vectors v and w into column vectors v [ f ] and w [ f ] , where v [ f ] and w [ f ] denote the transpose of the vectors v [ f ] and w [ f ] , respectively. Under a change of basis of the form for some invertible n × n matrix A = ( a ij ) , the matrix of components of

3792-897: The standard unit vectors in the directions of the x , y and z coordinates, respectively. For example, the gradient of the function f ( x , y , z ) = 2 x + 3 y 2 − sin ⁡ ( z ) {\displaystyle f(x,y,z)=2x+3y^{2}-\sin(z)} is ∇ f ( x , y , z ) = 2 i + 6 y j − cos ⁡ ( z ) k . {\displaystyle \nabla f(x,y,z)=2\mathbf {i} +6y\mathbf {j} -\cos(z)\mathbf {k} .} or ∇ f ( x , y , z ) = [ 2 6 y − cos ⁡ z ] . {\displaystyle \nabla f(x,y,z)={\begin{bmatrix}2\\6y\\-\cos z\end{bmatrix}}.} In some applications it

3871-498: The vector whose components are the partial derivatives of f {\displaystyle f} at p {\displaystyle p} . That is, for f : R n → R {\displaystyle f\colon \mathbb {R} ^{n}\to \mathbb {R} } , its gradient ∇ f : R n → R n {\displaystyle \nabla f\colon \mathbb {R} ^{n}\to \mathbb {R} ^{n}}

3950-407: The 'steepest ascent' in some orientations. For differentiable functions where the formula for gradient holds, it can be shown to always transform as a vector under transformation of the basis so as to always point towards the fastest increase. The gradient is dual to the total derivative d f {\displaystyle df} : the value of the gradient at a point is a tangent vector –

4029-404: The above definition for gradient is defined for the function f {\displaystyle f} only if f {\displaystyle f} is differentiable at p {\displaystyle p} . There can be functions for which partial derivatives exist in every direction but fail to be differentiable. Furthermore, this definition as the vector of partial derivatives

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4108-695: The best linear approximation to f {\displaystyle f} at x 0 {\displaystyle x_{0}} . The approximation is as follows: f ( x ) ≈ f ( x 0 ) + ( ∇ f ) x 0 ⋅ ( x − x 0 ) {\displaystyle f(x)\approx f(x_{0})+(\nabla f)_{x_{0}}\cdot (x-x_{0})} for x {\displaystyle x} close to x 0 {\displaystyle x_{0}} , where ( ∇ f ) x 0 {\displaystyle (\nabla f)_{x_{0}}}

4187-399: The direction of greatest change, by taking a dot product . Suppose that the steepest slope on a hill is 40%. A road going directly uphill has slope 40%, but a road going around the hill at an angle will have a shallower slope. For example, if the road is at a 60° angle from the uphill direction (when both directions are projected onto the horizontal plane), then the slope along the road will be

4266-428: The direction of the vector. The gradient is related to the differential by the formula ( ∇ f ) x ⋅ v = d f x ( v ) {\displaystyle (\nabla f)_{x}\cdot v=df_{x}(v)} for any v ∈ R n {\displaystyle v\in \mathbb {R} ^{n}} , where ⋅ {\displaystyle \cdot }

4345-405: The dot product between the gradient vector and a unit vector along the road, as the dot product measures how much the unit vector along the road aligns with the steepest slope, which is 40% times the cosine of 60°, or 20%. More generally, if the hill height function H is differentiable , then the gradient of H dotted with a unit vector gives the slope of the hill in the direction of

4424-586: The expressions given above for cylindrical and spherical coordinates. The gradient is closely related to the total derivative ( total differential ) d f {\displaystyle df} : they are transpose ( dual ) to each other. Using the convention that vectors in R n {\displaystyle \mathbb {R} ^{n}} are represented by column vectors , and that covectors (linear maps R n → R {\displaystyle \mathbb {R} ^{n}\to \mathbb {R} } ) are represented by row vectors ,

4503-596: The function f  : U → R is differentiable, then the differential of f is the Fréchet derivative of f . Thus ∇ f is a function from U to the space R such that lim h → 0 | f ( x + h ) − f ( x ) − ∇ f ( x ) ⋅ h | ‖ h ‖ = 0 , {\displaystyle \lim _{h\to 0}{\frac {|f(x+h)-f(x)-\nabla f(x)\cdot h|}{\|h\|}}=0,} where ·

4582-531: The function along v {\displaystyle \mathbf {v} } ; that is, ∇ f ( p ) ⋅ v = ∂ f ∂ v ( p ) = d f p ( v ) {\textstyle \nabla f(p)\cdot \mathbf {v} ={\frac {\partial f}{\partial \mathbf {v} }}(p)=df_{p}(\mathbf {v} )} . The gradient admits multiple generalizations to more general functions on manifolds ; see § Generalizations . Consider

4661-1136: The gradient ∇ f {\displaystyle \nabla f} and the derivative d f {\displaystyle df} are expressed as a column and row vector, respectively, with the same components, but transpose of each other: ∇ f ( p ) = [ ∂ f ∂ x 1 ( p ) ⋮ ∂ f ∂ x n ( p ) ] ; {\displaystyle \nabla f(p)={\begin{bmatrix}{\frac {\partial f}{\partial x_{1}}}(p)\\\vdots \\{\frac {\partial f}{\partial x_{n}}}(p)\end{bmatrix}};} d f p = [ ∂ f ∂ x 1 ( p ) ⋯ ∂ f ∂ x n ( p ) ] . {\displaystyle df_{p}={\begin{bmatrix}{\frac {\partial f}{\partial x_{1}}}(p)&\cdots &{\frac {\partial f}{\partial x_{n}}}(p)\end{bmatrix}}.} While these both have

4740-689: The gradient is an element of the tangent space at a point, ∇ f ( p ) ∈ T p R n {\displaystyle \nabla f(p)\in T_{p}\mathbb {R} ^{n}} , while the derivative is a map from the tangent space to the real numbers, d f p : T p R n → R {\displaystyle df_{p}\colon T_{p}\mathbb {R} ^{n}\to \mathbb {R} } . The tangent spaces at each point of R n {\displaystyle \mathbb {R} ^{n}} can be "naturally" identified with

4819-445: The hands of P. G. Tait . After receiving Smith's suggestion, Tait and James Clerk Maxwell referred to the operator as nabla in their extensive private correspondence; most of these references are of a humorous character. C. G. Knott's Life and Scientific Work of Peter Guthrie Tait (p. 145): It was probably this reluctance on the part of Maxwell to use the term Nabla in serious writings which prevented Tait from introducing

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4898-475: The metric changes by A as well. That is, or, in terms of the entries of this matrix, For this reason, the system of quantities g ij [ f ] is said to transform covariantly with respect to changes in the frame f . A system of n real-valued functions ( x , ..., x ) , giving a local coordinate system on an open set U in M , determines a basis of vector fields on U The metric g has components relative to this frame given by Relative to

4977-404: The metric tensor, also considered by Gauss, is that it provides a way in which to compute the length of tangent vectors to the surface, as well as the angle between two tangent vectors. In contemporary terms, the metric tensor allows one to compute the dot product (non-euclidean geometry) of tangent vectors in a manner independent of the parametric description of the surface. Any tangent vector at

5056-420: The name "del": This symbolic operator ∇ was introduced by Sir W. R. Hamilton and is now in universal employment. There seems, however, to be no universally recognized name for it, although owing to the frequent occurrence of the symbol some name is a practical necessity. It has been found by experience that the monosyllable del is so short and easy to pronounce that even in complicated formulae in which ∇ occurs

5135-402: The notion of a tensor. The metric tensor is an example of a tensor field . The components of a metric tensor in a coordinate basis take on the form of a symmetric matrix whose entries transform covariantly under changes to the coordinate system. Thus a metric tensor is a covariant symmetric tensor . From the coordinate-independent point of view, a metric tensor field is defined to be

5214-1703: The original R n {\displaystyle \mathbb {R} ^{n}} , not just as a tangent vector. Computationally, given a tangent vector, the vector can be multiplied by the derivative (as matrices), which is equal to taking the dot product with the gradient: ( d f p ) ( v ) = [ ∂ f ∂ x 1 ( p ) ⋯ ∂ f ∂ x n ( p ) ] [ v 1 ⋮ v n ] = ∑ i = 1 n ∂ f ∂ x i ( p ) v i = [ ∂ f ∂ x 1 ( p ) ⋮ ∂ f ∂ x n ( p ) ] ⋅ [ v 1 ⋮ v n ] = ∇ f ( p ) ⋅ v {\displaystyle (df_{p})(v)={\begin{bmatrix}{\frac {\partial f}{\partial x_{1}}}(p)&\cdots &{\frac {\partial f}{\partial x_{n}}}(p)\end{bmatrix}}{\begin{bmatrix}v_{1}\\\vdots \\v_{n}\end{bmatrix}}=\sum _{i=1}^{n}{\frac {\partial f}{\partial x_{i}}}(p)v_{i}={\begin{bmatrix}{\frac {\partial f}{\partial x_{1}}}(p)\\\vdots \\{\frac {\partial f}{\partial x_{n}}}(p)\end{bmatrix}}\cdot {\begin{bmatrix}v_{1}\\\vdots \\v_{n}\end{bmatrix}}=\nabla f(p)\cdot v} The best linear approximation to

5293-606: The particular coordinate representation . In the three-dimensional Cartesian coordinate system with a Euclidean metric , the gradient, if it exists, is given by ∇ 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 , k are

5372-419: The right-hand side is the directional derivative and there are many ways to represent it. Formally, the derivative is dual to the gradient; see relationship with derivative . When a function also depends on a parameter such as time, the gradient often refers simply to the vector of its spatial derivatives only (see Spatial gradient ). The magnitude and direction of the gradient vector are independent of

5451-426: The row vector with components ( ∂ f ∂ x 1 , … , ∂ f ∂ x n ) , {\displaystyle \left({\frac {\partial f}{\partial x_{1}}},\dots ,{\frac {\partial f}{\partial x_{n}}}\right),} so that d f x ( v ) {\displaystyle df_{x}(v)}

5530-403: The same components, they differ in what kind of mathematical object they represent: at each point, the derivative is a cotangent vector , a linear form (or covector) which expresses how much the (scalar) output changes for a given infinitesimal change in (vector) input, while at each point, the gradient is a tangent vector , which represents an infinitesimal change in (vector) input. In symbols,

5609-1381: The scale factors (also known as Lamé coefficients ) h i = ‖ e i ‖ = g i i = 1 / ‖ e i ‖ {\displaystyle h_{i}=\lVert \mathbf {e} _{i}\rVert ={\sqrt {g_{ii}}}=1\,/\lVert \mathbf {e} ^{i}\rVert }  : ∇ f = ∂ f ∂ x i g i j e ^ j g j j = ∑ i = 1 n ∂ f ∂ x i 1 h i e ^ i {\displaystyle \nabla f={\frac {\partial f}{\partial x^{i}}}g^{ij}{\hat {\mathbf {e} }}_{j}{\sqrt {g_{jj}}}=\sum _{i=1}^{n}\,{\frac {\partial f}{\partial x^{i}}}{\frac {1}{h_{i}}}\mathbf {\hat {e}} _{i}} (and d f = ∑ i = 1 n ∂ f ∂ x i 1 h i e ^ i {\textstyle \mathrm {d} f=\sum _{i=1}^{n}\,{\frac {\partial f}{\partial x^{i}}}{\frac {1}{h_{i}}}\mathbf {\hat {e}} ^{i}} ), where we cannot use Einstein notation, since it

5688-411: The significance of a system of coefficients E , F , and G , that transformed in this way on passing from one system of coordinates to another. The upshot is that the first fundamental form ( 1 ) is invariant under changes in the coordinate system, and that this follows exclusively from the transformation properties of E , F , and G . Indeed, by the chain rule, so that Another interpretation of

5767-403: The subscripts denote partial derivatives : The integrand is the restriction to the curve of the square root of the ( quadratic ) differential where The quantity ds in ( 1 ) is called the line element , while ds is called the first fundamental form of M . Intuitively, it represents the principal part of the square of the displacement undergone by r → ( u , v ) when u

5846-479: The superscript T denotes the matrix transpose . The matrix with the coefficients E , F , and G arranged in this way therefore transforms by the Jacobian matrix of the coordinate change A matrix which transforms in this way is one kind of what is called a tensor . The matrix with the transformation law ( 3 ) is known as the metric tensor of the surface. Ricci-Curbastro & Levi-Civita (1900) first observed

5925-403: The surface without stretching it), or a change in the particular parametric form of the same geometrical surface. One natural such invariant quantity is the length of a curve drawn along the surface. Another is the angle between a pair of curves drawn along the surface and meeting at a common point. A third such quantity is the area of a piece of the surface. The study of these invariants of

6004-687: The unnormalized local covariant and contravariant bases respectively, g i j {\displaystyle g^{ij}} is the inverse metric tensor , and the Einstein summation convention implies summation over i and j . If the coordinates are orthogonal we can easily express the gradient (and the differential ) in terms of the normalized bases, which we refer to as e ^ i {\displaystyle {\hat {\mathbf {e} }}_{i}} and e ^ i {\displaystyle {\hat {\mathbf {e} }}^{i}} , using

6083-408: The vector space R n {\displaystyle \mathbb {R} ^{n}} itself, and similarly the cotangent space at each point can be naturally identified with the dual vector space ( R n ) ∗ {\displaystyle (\mathbb {R} ^{n})^{*}} of covectors; thus the value of the gradient at a point can be thought of a vector in

6162-402: The vector, the directional derivative of H along the unit vector. The gradient of a function f {\displaystyle f} at point a {\displaystyle a} is usually written as ∇ f ( a ) {\displaystyle \nabla f(a)} . It may also be denoted by any of the following: The gradient (or gradient vector field) of

6241-510: The word earlier than he did. The one published use of the word by Maxwell is in the title to his humorous Tyndallic Ode, which is dedicated to the "Chief Musician upon Nabla", that is, Tait. William Thomson (Lord Kelvin) introduced the term to an American audience in an 1884 lecture; the notes were published in Britain and the U.S. in 1904. The name is acknowledged, and criticized, by Oliver Heaviside in 1891: The fictitious vector ∇ given by

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