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Directional derivative

From Wikipedia, the free encyclopedia
Instantaneous rate of change of the function
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Calculus
∫abf′(t)dt=f(b)−f(a){\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}

Inmultivariable calculus, thedirectional derivative measures the rate at which a function changes in a particular direction at a given point.[citation needed]

The directional derivative of a multivariabledifferentiable scalar function along a givenvectorv at a given pointx represents the instantaneous rate of change of the function in the directionv throughx.

Many mathematical texts assume that the directional vector isnormalized (a unit vector), meaning that its magnitude is equivalent to one. This is by convention and not required for proper calculation. In order to adjust a formula for the directional derivative to work for any vector, one must divide the expression by the magnitude of the vector. Normalized vectors are denoted with acircumflex (hat) symbol:^{\displaystyle \mathbf {\widehat {}} }.

The directional derivative of ascalar functionf with respect to a vectorv (denoted asv^{\displaystyle \mathbf {\hat {v}} } whennormalized) at a point (e.g., position) (x,f(x)) may be denoted by any of the following:∇vf(x)=fv′(x)=Dvf(x)=Df(x)(v)=∂vf(x)=∂f(x)∂v=v^⋅∇f(x)=v^⋅∂f(x)∂x.{\displaystyle {\begin{aligned}\nabla _{\mathbf {v} }{f}(\mathbf {x} )&=f'_{\mathbf {v} }(\mathbf {x} )\\&=D_{\mathbf {v} }f(\mathbf {x} )\\&=Df(\mathbf {x} )(\mathbf {v} )\\&=\partial _{\mathbf {v} }f(\mathbf {x} )\\&={\frac {\partial f(\mathbf {x} )}{\partial \mathbf {v} }}\\&=\mathbf {\hat {v}} \cdot {\nabla f(\mathbf {x} )}\\&=\mathbf {\hat {v}} \cdot {\frac {\partial f(\mathbf {x} )}{\partial \mathbf {x} }}.\\\end{aligned}}}

It therefore generalizes the notion of apartial derivative, in which the rate of change is taken along one of thecurvilinearcoordinate curves, all other coordinates being constant.The directional derivative is a special case of theGateaux derivative.

Definition

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Acontour plot off(x,y)=x2+y2{\displaystyle f(x,y)=x^{2}+y^{2}}, showing the gradient vector in black, and the unit vectoru{\displaystyle \mathbf {u} } scaled by the directional derivative in the direction ofu{\displaystyle \mathbf {u} } in orange. The gradient vector is longer because the gradient points in the direction of greatest rate of increase of a function.

Thedirectional derivative of ascalar functionf(x)=f(x1,x2,…,xn){\displaystyle f(\mathbf {x} )=f(x_{1},x_{2},\ldots ,x_{n})}along a vectorv=(v1,…,vn){\displaystyle \mathbf {v} =(v_{1},\ldots ,v_{n})}is thefunction∇vf{\displaystyle \nabla _{\mathbf {v} }{f}} defined by thelimit[1]∇vf(x)=limh→0f(x+hv)−f(x)h||v||=1||v||ddtf(x+tv)|t=0.{\displaystyle \nabla _{\mathbf {v} }{f}(\mathbf {x} )=\lim _{h\to 0}{\frac {f(\mathbf {x} +h\mathbf {v} )-f(\mathbf {x} )}{h||\mathbf {v} ||}}=\left.{\frac {1}{||\mathbf {v} ||}}{\frac {\mathrm {d} }{\mathrm {d} t}}f(\mathbf {x} +t\mathbf {v} )\right|_{t=0}.}

This definition is valid in a broad range of contexts, for example, where thenorm of a vector (and hence a unit vector) is defined.[2]

For differentiable functions

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If the functionf isdifferentiable atx, then the directional derivative exists along any vectorv atx, and one has

∇vf(x)=∇f(x)⋅v||v||{\displaystyle \nabla _{\mathbf {v} }{f}(\mathbf {x} )=\nabla f(\mathbf {x} )\cdot {\frac {\mathbf {v} }{||\mathbf {v} ||}}}

where the∇{\displaystyle \nabla } on the right denotes thegradient and⋅{\displaystyle \cdot } is thedot product.[3]

It can be derived by using the property that all directional derivatives at a point make up a single tangent plane which can be defined using partial derivatives. This can be used to find a formula for the gradient vector and an alternative formula for the directional derivative, the latter of which can be rewritten as shown above for convenience.

It also follows from defining a pathh(t)=x+tv{\displaystyle h(t)=x+tv} and using the definition of the derivative as a limit which can be calculated along this path to get:0=limt→0f(x+tv^)−f(x)−t∇f(x)⋅v^t=limt→0f(x+tv^)−f(x)t−∇f(x)⋅v^=∇vf(x)−∇f(x)⋅v^.∇f(x)⋅v^=∇vf(x){\displaystyle {\begin{aligned}0&=\lim _{t\to 0}{\frac {f(x+t{\hat {v}})-f(x)-t\nabla f(x)\cdot {\hat {v}}}{t}}\\&=\lim _{t\to 0}{\frac {f(x+t{\hat {v}})-f(x)}{t}}-\nabla f(x)\cdot {\hat {v}}\\&=\nabla _{v}f(x)-\nabla f(x)\cdot {\hat {v}}.\\&\nabla f(x)\cdot {\hat {v}}=\nabla _{v}f(x)\end{aligned}}}

Using only direction of vector

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The angleα between the tangentA and the horizontal will be maximum if the cutting plane contains the direction of the gradientA.

In aEuclidean space, some authors[4] define the directional derivative to be with respect to an arbitrary nonzero vectorv afternormalization, thus being independent of its magnitude and depending only on its direction.[5]

This definition gives the rate of increase off per unit of distance moved in the direction given byv. In this case, one has∇vf(x)=limh→0f(x+hv)−f(x)h,{\displaystyle \nabla _{\mathbf {v} }{f}(\mathbf {x} )=\lim _{h\to 0}{\frac {f(\mathbf {x} +h\mathbf {v} )-f(\mathbf {x} )}{h}},}or in casef is differentiable atx,∇vf(x)=∇f(x)⋅v.{\displaystyle \nabla _{\mathbf {v} }{f}(\mathbf {x} )=\nabla f(\mathbf {x} )\cdot \mathbf {v} .}

Restriction to a unit vector

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In the context of a function on aEuclidean space, some texts restrict the vectorv to being aunit vector for convention. Both of the above equations remain true, though redundant, when a vector is normalized.[6]

Properties

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Many of the familiar properties of the ordinaryderivative hold for the directional derivative. These include, for any functionsf andg defined in aneighborhood of, anddifferentiable at,p:

  1. sum rule:∇v(f+g)=∇vf+∇vg.{\displaystyle \nabla _{\mathbf {v} }(f+g)=\nabla _{\mathbf {v} }f+\nabla _{\mathbf {v} }g.}
  2. constant factor rule: For any constantc,∇v(cf)=c∇vf.{\displaystyle \nabla _{\mathbf {v} }(cf)=c\nabla _{\mathbf {v} }f.}
  3. product rule (orLeibniz's rule):∇v(fg)=g∇vf+f∇vg.{\displaystyle \nabla _{\mathbf {v} }(fg)=g\nabla _{\mathbf {v} }f+f\nabla _{\mathbf {v} }g.}
  4. chain rule: Ifg is differentiable atp andh is differentiable atg(p), then∇v(h∘g)(p)=h′(g(p))∇vg(p).{\displaystyle \nabla _{\mathbf {v} }(h\circ g)(\mathbf {p} )=h'(g(\mathbf {p} ))\nabla _{\mathbf {v} }g(\mathbf {p} ).}

In differential geometry

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See also:Tangent space § Tangent vectors as directional derivatives

LetM be adifferentiable manifold andp a point ofM. Suppose thatf is a function defined in a neighborhood ofp, anddifferentiable atp. Ifv is atangent vector toM atp, then thedirectional derivative off alongv, denoted variously asdf(v) (seeExterior derivative),∇vf(p){\displaystyle \nabla _{\mathbf {v} }f(\mathbf {p} )} (seeCovariant derivative),Lvf(p){\displaystyle L_{\mathbf {v} }f(\mathbf {p} )} (seeLie derivative), orvp(f){\displaystyle {\mathbf {v} }_{\mathbf {p} }(f)} (seeTangent space § Definition via derivations), can be defined as follows. Letγ : [−1, 1] →M be a differentiable curve withγ(0) =p andγ′(0) =v. Then the directional derivative is defined by∇vf(p)=ddτf∘γ(τ)|τ=0.{\displaystyle \nabla _{\mathbf {v} }f(\mathbf {p} )=\left.{\frac {d}{d\tau }}f\circ \gamma (\tau )\right|_{\tau =0}.}This definition can be proven independent of the choice ofγ, providedγ is selected in the prescribed manner so thatγ(0) =p andγ′(0) =v.

The Lie derivative

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TheLie derivative of a vector fieldWμ(x){\displaystyle W^{\mu }(x)} along a vector fieldVμ(x){\displaystyle V^{\mu }(x)} is given by the difference of two directional derivatives (with vanishing torsion):LVWμ=(V⋅∇)Wμ−(W⋅∇)Vμ.{\displaystyle {\mathcal {L}}_{V}W^{\mu }=(V\cdot \nabla )W^{\mu }-(W\cdot \nabla )V^{\mu }.}In particular, for a scalar fieldϕ(x){\displaystyle \phi (x)}, the Lie derivative reduces to the standard directional derivative:LVϕ=(V⋅∇)ϕ.{\displaystyle {\mathcal {L}}_{V}\phi =(V\cdot \nabla )\phi .}

The Riemann tensor

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Directional derivatives are often used in introductory derivations of theRiemann curvature tensor. Consider a curved rectangle with an infinitesimal vectorδ{\displaystyle \delta } along one edge andδ′{\displaystyle \delta '} along the other. We translate a covectorS{\displaystyle S} alongδ{\displaystyle \delta } thenδ′{\displaystyle \delta '} and then subtract the translation alongδ′{\displaystyle \delta '} and thenδ{\displaystyle \delta }. Instead of building the directional derivative using partial derivatives, we use thecovariant derivative. The translation operator forδ{\displaystyle \delta } is thus1+∑νδνDν=1+δ⋅D,{\displaystyle 1+\sum _{\nu }\delta ^{\nu }D_{\nu }=1+\delta \cdot D,}and forδ′{\displaystyle \delta '},1+∑μδ′μDμ=1+δ′⋅D.{\displaystyle 1+\sum _{\mu }\delta '^{\mu }D_{\mu }=1+\delta '\cdot D.}The difference between the two paths is then(1+δ′⋅D)(1+δ⋅D)Sρ−(1+δ⋅D)(1+δ′⋅D)Sρ=∑μ,νδ′μδν[Dμ,Dν]Sρ.{\displaystyle (1+\delta '\cdot D)(1+\delta \cdot D)S^{\rho }-(1+\delta \cdot D)(1+\delta '\cdot D)S^{\rho }=\sum _{\mu ,\nu }\delta '^{\mu }\delta ^{\nu }[D_{\mu },D_{\nu }]S_{\rho }.}It can be argued[7] that the noncommutativity of the covariant derivatives measures the curvature of the manifold:[Dμ,Dν]Sρ=±∑σRσρμνSσ,{\displaystyle [D_{\mu },D_{\nu }]S_{\rho }=\pm \sum _{\sigma }R^{\sigma }{}_{\rho \mu \nu }S_{\sigma },}whereR{\displaystyle R} is the Riemann curvature tensor and the sign depends on thesign convention of the author.

In group theory

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Translations

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In thePoincaré algebra, we can define an infinitesimal translation operatorP asP=i∇.{\displaystyle \mathbf {P} =i\nabla .}(thei ensures thatP is aself-adjoint operator) For a finite displacementλ, theunitaryHilbert spacerepresentation for translations is[8]U(λ)=exp⁡(−iλ⋅P).{\displaystyle U({\boldsymbol {\lambda }})=\exp \left(-i{\boldsymbol {\lambda }}\cdot \mathbf {P} \right).}By using the above definition of the infinitesimal translation operator, we see that the finite translation operator is an exponentiated directional derivative:U(λ)=exp⁡(λ⋅∇).{\displaystyle U({\boldsymbol {\lambda }})=\exp \left({\boldsymbol {\lambda }}\cdot \nabla \right).}This is a translation operator in the sense that it acts on multivariable functionsf(x) asU(λ)f(x)=exp⁡(λ⋅∇)f(x)=f(x+λ).{\displaystyle U({\boldsymbol {\lambda }})f(\mathbf {x} )=\exp \left({\boldsymbol {\lambda }}\cdot \nabla \right)f(\mathbf {x} )=f(\mathbf {x} +{\boldsymbol {\lambda }}).}

Proof of the last equation

In standard single-variable calculus, the derivative of a smooth functionf(x) is defined by (for smallε)dfdx=f(x+ε)−f(x)ε.{\displaystyle {\frac {df}{dx}}={\frac {f(x+\varepsilon )-f(x)}{\varepsilon }}.}This can be rearranged to findf(x+ε):f(x+ε)=f(x)+εdfdx=(1+εddx)f(x).{\displaystyle f(x+\varepsilon )=f(x)+\varepsilon \,{\frac {df}{dx}}=\left(1+\varepsilon \,{\frac {d}{dx}}\right)f(x).}It follows that[1+ε(d/dx)]{\displaystyle [1+\varepsilon \,(d/dx)]} is a translation operator. This is instantly generalized[9] to multivariable functionsf(x)f(x+ε)=(1+ε⋅∇)f(x).{\displaystyle f(\mathbf {x} +{\boldsymbol {\varepsilon }})=\left(1+{\boldsymbol {\varepsilon }}\cdot \nabla \right)f(\mathbf {x} ).}Hereε⋅∇{\displaystyle {\boldsymbol {\varepsilon }}\cdot \nabla } is the directional derivative along the infinitesimal displacementε. We have found the infinitesimal version of the translation operator:U(ε)=1+ε⋅∇.{\displaystyle U({\boldsymbol {\varepsilon }})=1+{\boldsymbol {\varepsilon }}\cdot \nabla .}It is evident that the group multiplication law[10]U(g)U(f)=U(gf) takes the formU(a)U(b)=U(a+b).{\displaystyle U(\mathbf {a} )U(\mathbf {b} )=U(\mathbf {a+b} ).}So suppose that we take the finite displacementλ and divide it intoN parts (N→∞ is implied everywhere), so thatλ/N=ε. In other words,λ=Nε.{\displaystyle {\boldsymbol {\lambda }}=N{\boldsymbol {\varepsilon }}.}Then by applyingU(ε)N times, we can constructU(λ):[U(ε)]N=U(Nε)=U(λ).{\displaystyle [U({\boldsymbol {\varepsilon }})]^{N}=U(N{\boldsymbol {\varepsilon }})=U({\boldsymbol {\lambda }}).}We can now plug in our above expression for U(ε):[U(ε)]N=[1+ε⋅∇]N=[1+λ⋅∇N]N.{\displaystyle [U({\boldsymbol {\varepsilon }})]^{N}=\left[1+{\boldsymbol {\varepsilon }}\cdot \nabla \right]^{N}=\left[1+{\frac {{\boldsymbol {\lambda }}\cdot \nabla }{N}}\right]^{N}.}Using the identity[11]exp⁡(x)=[1+xN]N,{\displaystyle \exp(x)=\left[1+{\frac {x}{N}}\right]^{N},}we haveU(λ)=exp⁡(λ⋅∇).{\displaystyle U({\boldsymbol {\lambda }})=\exp \left({\boldsymbol {\lambda }}\cdot \nabla \right).}And sinceU(ε)f(x) =f(x+ε) we have[U(ε)]Nf(x)=f(x+Nε)=f(x+λ)=U(λ)f(x)=exp⁡(λ⋅∇)f(x),{\displaystyle [U({\boldsymbol {\varepsilon }})]^{N}f(\mathbf {x} )=f(\mathbf {x} +N{\boldsymbol {\varepsilon }})=f(\mathbf {x} +{\boldsymbol {\lambda }})=U({\boldsymbol {\lambda }})f(\mathbf {x} )=\exp \left({\boldsymbol {\lambda }}\cdot \nabla \right)f(\mathbf {x} ),}Q.E.D.

As a technical note, this procedure is only possible because the translation group forms anAbeliansubgroup (Cartan subalgebra) in the Poincaré algebra. In particular, the group multiplication lawU(a)U(b) =U(a+b) should not be taken for granted. We also note that Poincaré is a connectedLie group. It is a group of transformationsT(ξ) that are described by a continuous set of real parametersξa{\displaystyle \xi ^{a}}. The group multiplication law takes the formT(ξ¯)T(ξ)=T(f(ξ¯,ξ)).{\displaystyle T({\bar {\xi }})T(\xi )=T(f({\bar {\xi }},\xi )).}Takingξa=0{\displaystyle \xi ^{a}=0} as the coordinates of the identity, we must havefa(ξ,0)=fa(0,ξ)=ξa.{\displaystyle f^{a}(\xi ,0)=f^{a}(0,\xi )=\xi ^{a}.}The actual operators on the Hilbert space are represented by unitary operatorsU(T(ξ)). In the above notation we suppressed theT; we now writeU(λ) asU(P(λ)). For a small neighborhood around the identity, thepower series representationU(T(ξ))=1+i∑aξata+12∑b,cξbξctbc+⋯{\displaystyle U(T(\xi ))=1+i\sum _{a}\xi ^{a}t_{a}+{\frac {1}{2}}\sum _{b,c}\xi ^{b}\xi ^{c}t_{bc}+\cdots }is quite good. Suppose that U(T(ξ)) form a non-projective representation, i.e.,U(T(ξ¯))U(T(ξ))=U(T(f(ξ¯,ξ))).{\displaystyle U(T({\bar {\xi }}))U(T(\xi ))=U(T(f({\bar {\xi }},\xi ))).}The expansion of f to second power isfa(ξ¯,ξ)=ξa+ξ¯a+∑b,cfabcξ¯bξc.{\displaystyle f^{a}({\bar {\xi }},\xi )=\xi ^{a}+{\bar {\xi }}^{a}+\sum _{b,c}f^{abc}{\bar {\xi }}^{b}\xi ^{c}.}After expanding the representation multiplication equation and equating coefficients, we have the nontrivial conditiontbc=−tbtc−i∑afabcta.{\displaystyle t_{bc}=-t_{b}t_{c}-i\sum _{a}f^{abc}t_{a}.}Sincetab{\displaystyle t_{ab}} is by definition symmetric in its indices, we have the standardLie algebra commutator:[tb,tc]=i∑a(−fabc+facb)ta=i∑aCabcta,{\displaystyle [t_{b},t_{c}]=i\sum _{a}(-f^{abc}+f^{acb})t_{a}=i\sum _{a}C^{abc}t_{a},}withC thestructure constant. The generators for translations are partial derivative operators, which commute:[∂∂xb,∂∂xc]=0.{\displaystyle \left[{\frac {\partial }{\partial x^{b}}},{\frac {\partial }{\partial x^{c}}}\right]=0.}This implies that the structure constants vanish and thus the quadratic coefficients in the f expansion vanish as well. This means thatf is simply additive:fabeliana(ξ¯,ξ)=ξa+ξ¯a,{\displaystyle f_{\text{abelian}}^{a}({\bar {\xi }},\xi )=\xi ^{a}+{\bar {\xi }}^{a},}and thus for abelian groups,U(T(ξ¯))U(T(ξ))=U(T(ξ¯+ξ)).{\displaystyle U(T({\bar {\xi }}))U(T(\xi ))=U(T({\bar {\xi }}+\xi )).}Q.E.D.

Rotations

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Therotation operator also contains a directional derivative. The rotation operator for an angleθ, i.e. by an amountθ = |θ| about an axis parallel toθ^=θ/θ{\displaystyle {\hat {\theta }}={\boldsymbol {\theta }}/\theta } isU(R(θ))=exp⁡(−iθ⋅L).{\displaystyle U(R(\mathbf {\theta } ))=\exp(-i\mathbf {\theta } \cdot \mathbf {L} ).}HereL is the vector operator that generatesSO(3):L=(0000010−10)i+(00−1000100)j+(010−100000)k.{\displaystyle \mathbf {L} ={\begin{pmatrix}0&0&0\\0&0&1\\0&-1&0\end{pmatrix}}\mathbf {i} +{\begin{pmatrix}0&0&-1\\0&0&0\\1&0&0\end{pmatrix}}\mathbf {j} +{\begin{pmatrix}0&1&0\\-1&0&0\\0&0&0\end{pmatrix}}\mathbf {k} .}It may be shown geometrically that an infinitesimal right-handed rotation changes the position vectorx byx→x−δθ×x.{\displaystyle \mathbf {x} \rightarrow \mathbf {x} -\delta {\boldsymbol {\theta }}\times \mathbf {x} .}So we would expect under infinitesimal rotation:U(R(δθ))f(x)=f(x−δθ×x)=f(x)−(δθ×x)⋅∇f.{\displaystyle U(R(\delta {\boldsymbol {\theta }}))f(\mathbf {x} )=f(\mathbf {x} -\delta {\boldsymbol {\theta }}\times \mathbf {x} )=f(\mathbf {x} )-(\delta {\boldsymbol {\theta }}\times \mathbf {x} )\cdot \nabla f.}It follows thatU(R(δθ))=1−(δθ×x)⋅∇.{\displaystyle U(R(\delta \mathbf {\theta } ))=1-(\delta \mathbf {\theta } \times \mathbf {x} )\cdot \nabla .}Following the same exponentiation procedure as above, we arrive at the rotation operator in the position basis, which is an exponentiated directional derivative:[12]U(R(θ))=exp⁡(−(θ×x)⋅∇).{\displaystyle U(R(\mathbf {\theta } ))=\exp(-(\mathbf {\theta } \times \mathbf {x} )\cdot \nabla ).}

Normal derivative

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Anormal derivative is a directional derivative taken in the direction normal (that is,orthogonal) to some surface in space, or more generally along anormal vector field orthogonal to somehypersurface. See for exampleNeumann boundary condition. If the normal direction is denoted byn{\displaystyle \mathbf {n} }, then the normal derivative of a functionf is sometimes denoted as∂f∂n{\textstyle {\frac {\partial f}{\partial \mathbf {n} }}}. In other notations,∂f∂n=∇f(x)⋅n=∇nf(x)=∂f∂x⋅n=Df(x)[n].{\displaystyle {\frac {\partial f}{\partial \mathbf {n} }}=\nabla f(\mathbf {x} )\cdot \mathbf {n} =\nabla _{\mathbf {n} }{f}(\mathbf {x} )={\frac {\partial f}{\partial \mathbf {x} }}\cdot \mathbf {n} =Df(\mathbf {x} )[\mathbf {n} ].}

In the continuum mechanics of solids

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Several important results in continuum mechanics require the derivatives of vectors with respect to vectors and oftensors with respect to vectors and tensors.[13] Thedirectional directive provides a systematic way of finding these derivatives.

This section is an excerpt fromTensor derivative (continuum mechanics) § Derivatives with respect to vectors and second-order tensors.[edit]

The definitions of directional derivatives for various situations are given below. It is assumed that the functions are sufficiently smooth that derivatives can be taken.

Derivatives of scalar valued functions of vectors

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Letf(v) be a real valued function of the vector v. Then the derivative off(v) with respect to v (or at v) is the vector defined through itsdot product with any vector u being

∂f∂v⋅u=Df(v)[u]=[ddα f(v+α u)]α=0{\displaystyle {\frac {\partial f}{\partial \mathbf {v} }}\cdot \mathbf {u} =Df(\mathbf {v} )[\mathbf {u} ]=\left[{\frac {d}{d\alpha }}~f(\mathbf {v} +\alpha ~\mathbf {u} )\right]_{\alpha =0}}

for all vectors u. The above dot product yields a scalar, and if u is aunit vector gives the directional derivative off at v, in the u direction.

Properties:

  1. Iff(v)=f1(v)+f2(v){\displaystyle f(\mathbf {v} )=f_{1}(\mathbf {v} )+f_{2}(\mathbf {v} )} then∂f∂v⋅u=(∂f1∂v+∂f2∂v)⋅u{\displaystyle {\frac {\partial f}{\partial \mathbf {v} }}\cdot \mathbf {u} =\left({\frac {\partial f_{1}}{\partial \mathbf {v} }}+{\frac {\partial f_{2}}{\partial \mathbf {v} }}\right)\cdot \mathbf {u} }
  2. Iff(v)=f1(v) f2(v){\displaystyle f(\mathbf {v} )=f_{1}(\mathbf {v} )~f_{2}(\mathbf {v} )} then∂f∂v⋅u=(∂f1∂v⋅u) f2(v)+f1(v) (∂f2∂v⋅u){\displaystyle {\frac {\partial f}{\partial \mathbf {v} }}\cdot \mathbf {u} =\left({\frac {\partial f_{1}}{\partial \mathbf {v} }}\cdot \mathbf {u} \right)~f_{2}(\mathbf {v} )+f_{1}(\mathbf {v} )~\left({\frac {\partial f_{2}}{\partial \mathbf {v} }}\cdot \mathbf {u} \right)}
  3. Iff(v)=f1(f2(v)){\displaystyle f(\mathbf {v} )=f_{1}(f_{2}(\mathbf {v} ))} then∂f∂v⋅u=∂f1∂f2 ∂f2∂v⋅u{\displaystyle {\frac {\partial f}{\partial \mathbf {v} }}\cdot \mathbf {u} ={\frac {\partial f_{1}}{\partial f_{2}}}~{\frac {\partial f_{2}}{\partial \mathbf {v} }}\cdot \mathbf {u} }

Derivatives of vector valued functions of vectors

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Let f(v) be a vector valued function of the vector v. Then the derivative of f(v) with respect to v (or at v) is the second order tensor defined through its dot product with any vector u being

∂f∂v⋅u=Df(v)[u]=[ddα f(v+α u)]α=0{\displaystyle {\frac {\partial \mathbf {f} }{\partial \mathbf {v} }}\cdot \mathbf {u} =D\mathbf {f} (\mathbf {v} )[\mathbf {u} ]=\left[{\frac {d}{d\alpha }}~\mathbf {f} (\mathbf {v} +\alpha ~\mathbf {u} )\right]_{\alpha =0}}

for all vectors u. The above dot product yields a vector, and if u is a unit vector gives the direction derivative of f at v, in the directional u.

Properties:

  1. Iff(v)=f1(v)+f2(v){\displaystyle \mathbf {f} (\mathbf {v} )=\mathbf {f} _{1}(\mathbf {v} )+\mathbf {f} _{2}(\mathbf {v} )} then∂f∂v⋅u=(∂f1∂v+∂f2∂v)⋅u{\displaystyle {\frac {\partial \mathbf {f} }{\partial \mathbf {v} }}\cdot \mathbf {u} =\left({\frac {\partial \mathbf {f} _{1}}{\partial \mathbf {v} }}+{\frac {\partial \mathbf {f} _{2}}{\partial \mathbf {v} }}\right)\cdot \mathbf {u} }
  2. Iff(v)=f1(v)×f2(v){\displaystyle \mathbf {f} (\mathbf {v} )=\mathbf {f} _{1}(\mathbf {v} )\times \mathbf {f} _{2}(\mathbf {v} )} then∂f∂v⋅u=(∂f1∂v⋅u)×f2(v)+f1(v)×(∂f2∂v⋅u){\displaystyle {\frac {\partial \mathbf {f} }{\partial \mathbf {v} }}\cdot \mathbf {u} =\left({\frac {\partial \mathbf {f} _{1}}{\partial \mathbf {v} }}\cdot \mathbf {u} \right)\times \mathbf {f} _{2}(\mathbf {v} )+\mathbf {f} _{1}(\mathbf {v} )\times \left({\frac {\partial \mathbf {f} _{2}}{\partial \mathbf {v} }}\cdot \mathbf {u} \right)}
  3. Iff(v)=f1(f2(v)){\displaystyle \mathbf {f} (\mathbf {v} )=\mathbf {f} _{1}(\mathbf {f} _{2}(\mathbf {v} ))} then∂f∂v⋅u=∂f1∂f2⋅(∂f2∂v⋅u){\displaystyle {\frac {\partial \mathbf {f} }{\partial \mathbf {v} }}\cdot \mathbf {u} ={\frac {\partial \mathbf {f} _{1}}{\partial \mathbf {f} _{2}}}\cdot \left({\frac {\partial \mathbf {f} _{2}}{\partial \mathbf {v} }}\cdot \mathbf {u} \right)}

Derivatives of scalar valued functions of second-order tensors

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Letf(S){\displaystyle f({\boldsymbol {S}})} be a real valued function of the second order tensorS{\displaystyle {\boldsymbol {S}}}. Then the derivative off(S){\displaystyle f({\boldsymbol {S}})} with respect toS{\displaystyle {\boldsymbol {S}}} (or atS{\displaystyle {\boldsymbol {S}}}) in the directionT{\displaystyle {\boldsymbol {T}}} is the second order tensor defined as∂f∂S:T=Df(S)[T]=[ddα f(S+α T)]α=0{\displaystyle {\frac {\partial f}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}=Df({\boldsymbol {S}})[{\boldsymbol {T}}]=\left[{\frac {d}{d\alpha }}~f({\boldsymbol {S}}+\alpha ~{\boldsymbol {T}})\right]_{\alpha =0}}for all second order tensorsT{\displaystyle {\boldsymbol {T}}}.

Properties:

  1. Iff(S)=f1(S)+f2(S){\displaystyle f({\boldsymbol {S}})=f_{1}({\boldsymbol {S}})+f_{2}({\boldsymbol {S}})} then∂f∂S:T=(∂f1∂S+∂f2∂S):T{\displaystyle {\frac {\partial f}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}=\left({\frac {\partial f_{1}}{\partial {\boldsymbol {S}}}}+{\frac {\partial f_{2}}{\partial {\boldsymbol {S}}}}\right):{\boldsymbol {T}}}
  2. Iff(S)=f1(S) f2(S){\displaystyle f({\boldsymbol {S}})=f_{1}({\boldsymbol {S}})~f_{2}({\boldsymbol {S}})} then∂f∂S:T=(∂f1∂S:T) f2(S)+f1(S) (∂f2∂S:T){\displaystyle {\frac {\partial f}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}=\left({\frac {\partial f_{1}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}\right)~f_{2}({\boldsymbol {S}})+f_{1}({\boldsymbol {S}})~\left({\frac {\partial f_{2}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}\right)}
  3. Iff(S)=f1(f2(S)){\displaystyle f({\boldsymbol {S}})=f_{1}(f_{2}({\boldsymbol {S}}))} then∂f∂S:T=∂f1∂f2 (∂f2∂S:T){\displaystyle {\frac {\partial f}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}={\frac {\partial f_{1}}{\partial f_{2}}}~\left({\frac {\partial f_{2}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}\right)}

Derivatives of tensor valued functions of second-order tensors

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LetF(S){\displaystyle {\boldsymbol {F}}({\boldsymbol {S}})} be a second order tensor valued function of the second order tensorS{\displaystyle {\boldsymbol {S}}}. Then the derivative ofF(S){\displaystyle {\boldsymbol {F}}({\boldsymbol {S}})} with respect toS{\displaystyle {\boldsymbol {S}}} (or atS{\displaystyle {\boldsymbol {S}}}) in the directionT{\displaystyle {\boldsymbol {T}}} is the fourth order tensor defined as∂F∂S:T=DF(S)[T]=[ddα F(S+α T)]α=0{\displaystyle {\frac {\partial {\boldsymbol {F}}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}=D{\boldsymbol {F}}({\boldsymbol {S}})[{\boldsymbol {T}}]=\left[{\frac {d}{d\alpha }}~{\boldsymbol {F}}({\boldsymbol {S}}+\alpha ~{\boldsymbol {T}})\right]_{\alpha =0}}for all second order tensorsT{\displaystyle {\boldsymbol {T}}}.

Properties:

  1. IfF(S)=F1(S)+F2(S){\displaystyle {\boldsymbol {F}}({\boldsymbol {S}})={\boldsymbol {F}}_{1}({\boldsymbol {S}})+{\boldsymbol {F}}_{2}({\boldsymbol {S}})} then∂F∂S:T=(∂F1∂S+∂F2∂S):T{\displaystyle {\frac {\partial {\boldsymbol {F}}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}=\left({\frac {\partial {\boldsymbol {F}}_{1}}{\partial {\boldsymbol {S}}}}+{\frac {\partial {\boldsymbol {F}}_{2}}{\partial {\boldsymbol {S}}}}\right):{\boldsymbol {T}}}
  2. IfF(S)=F1(S)⋅F2(S){\displaystyle {\boldsymbol {F}}({\boldsymbol {S}})={\boldsymbol {F}}_{1}({\boldsymbol {S}})\cdot {\boldsymbol {F}}_{2}({\boldsymbol {S}})} then∂F∂S:T=(∂F1∂S:T)⋅F2(S)+F1(S)⋅(∂F2∂S:T){\displaystyle {\frac {\partial {\boldsymbol {F}}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}=\left({\frac {\partial {\boldsymbol {F}}_{1}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}\right)\cdot {\boldsymbol {F}}_{2}({\boldsymbol {S}})+{\boldsymbol {F}}_{1}({\boldsymbol {S}})\cdot \left({\frac {\partial {\boldsymbol {F}}_{2}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}\right)}
  3. IfF(S)=F1(F2(S)){\displaystyle {\boldsymbol {F}}({\boldsymbol {S}})={\boldsymbol {F}}_{1}({\boldsymbol {F}}_{2}({\boldsymbol {S}}))} then∂F∂S:T=∂F1∂F2:(∂F2∂S:T){\displaystyle {\frac {\partial {\boldsymbol {F}}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}={\frac {\partial {\boldsymbol {F}}_{1}}{\partial {\boldsymbol {F}}_{2}}}:\left({\frac {\partial {\boldsymbol {F}}_{2}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}\right)}
  4. Iff(S)=f1(F2(S)){\displaystyle f({\boldsymbol {S}})=f_{1}({\boldsymbol {F}}_{2}({\boldsymbol {S}}))} then∂f∂S:T=∂f1∂F2:(∂F2∂S:T){\displaystyle {\frac {\partial f}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}={\frac {\partial f_{1}}{\partial {\boldsymbol {F}}_{2}}}:\left({\frac {\partial {\boldsymbol {F}}_{2}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}\right)}

See also

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Notes

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  1. ^R. Wrede; M.R. Spiegel (2010).Advanced Calculus (3rd ed.). Schaum's Outline Series.ISBN 978-0-07-162366-7.
  2. ^The applicability extends to functions over spaces without ametric and todifferentiable manifolds, such as ingeneral relativity.
  3. ^If the dot product is undefined, thegradient is also undefined; however, for differentiablef, the directional derivative is still defined, and a similar relation exists with the exterior derivative.
  4. ^Thomas, George B. Jr.; and Finney, Ross L. (1979)Calculus and Analytic Geometry, Addison-Wesley Publ. Co., fifth edition, p. 593.
  5. ^This typically assumes aEuclidean space – for example, a function of several variables typically has no definition of the magnitude of a vector, and hence of a unit vector.
  6. ^Hughes Hallett, Deborah;McCallum, William G.;Gleason, Andrew M. (2012-01-01).Calculus : Single and multivariable. John wiley. p. 780.ISBN 9780470888612.OCLC 828768012.
  7. ^Zee, A. (2013).Einstein Gravity in a Nutshell. Princeton: Princeton University Press. p. 341.ISBN 9780691145587.
  8. ^Weinberg, Steven (1999).The quantum theory of fields (Reprinted (with corr.). ed.). Cambridge [u.a.]: Cambridge Univ. Press.ISBN 9780521550017.
  9. ^Zee, A. (2013).Einstein gravity in a nutshell. Princeton: Princeton University Press.ISBN 9780691145587.
  10. ^Cahill, Kevin Cahill (2013).Physical mathematics (Repr. ed.). Cambridge: Cambridge University Press.ISBN 978-1107005211.
  11. ^Larson, Ron; Edwards, Bruce H. (2010).Calculus of a single variable (9th ed.). Belmont: Brooks/Cole.ISBN 9780547209982.
  12. ^Shankar, R. (1994).Principles of quantum mechanics (2nd ed.). New York: Kluwer Academic / Plenum. p. 318.ISBN 9780306447907.
  13. ^J. E. Marsden and T. J. R. Hughes, 2000,Mathematical Foundations of Elasticity, Dover.

References

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External links

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