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Directional derivative
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<h2 id="definition">Definition</h2> <p>The <i>directional derivative</i> of a <a href="/facts/Scalar_function/8MtPTGob">scalar function</a> f ( x ) = f ( x 1 , x 2 , … , x n ) {\displaystyle f(\mathbf {x} )=f(x_{1},x_{2},\ldots ,x_{n})} along a vector v = ( v 1 , … , v n ) {\displaystyle \mathbf {v} =(v_{1},\ldots ,v_{n})} is the <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> ∇ v f {\displaystyle \nabla _{\mathbf {v} }{f}} defined by the <a href="/facts/Limit_(mathematics)/j8JBhj4e">limit</a><a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> ∇ v f ( x ) = lim h → 0 f ( x + h v ) − 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}}.} </p><p>This definition is valid in a broad range of contexts, for example where the <a href="/facts/Euclidean_norm/R2UbzmzM">norm</a> of a vector (and hence a unit vector) is undefined.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> </p> <h3>For differentiable functions</h3> <p>If the function <i>f</i> is <a href="/facts/Differentiable_function/jQqTgLk1">differentiable</a> at x, then the directional derivative exists along any unit vector v at x, and one has </p><p> ∇ v f ( x ) = ∇ f ( x ) ⋅ v {\displaystyle \nabla _{\mathbf {v} }{f}(\mathbf {x} )=\nabla f(\mathbf {x} )\cdot \mathbf {v} } </p><p>where the ∇ {\displaystyle \nabla } on the right denotes the <i><a href="/facts/Gradient/TsR7uukj">gradient</a></i>, ⋅ {\displaystyle \cdot } is the <a href="/facts/Dot_product/tNz8MLjT">dot product</a> and v is a unit vector.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> This follows from defining a path h ( t ) = x + t v {\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 = lim t → 0 f ( x + t v ) − f ( x ) − t D f ( x ) ( v ) t = lim t → 0 f ( x + t v ) − f ( x ) t − D f ( x ) ( v ) = ∇ v f ( x ) − D f ( x ) ( v ) . {\displaystyle {\begin{aligned}0&=\lim _{t\to 0}{\frac {f(x+tv)-f(x)-tDf(x)(v)}{t}}\\&=\lim _{t\to 0}{\frac {f(x+tv)-f(x)}{t}}-Df(x)(v)\\&=\nabla _{v}f(x)-Df(x)(v).\end{aligned}}} </p><p>Intuitively, the directional derivative of <i>f</i> at a point x represents the <a href="/facts/Derivative/7Ito70Dh">rate of change</a> of <i>f</i>, in the direction of v. </p> <h3>Using only direction of vector</h3> <p>In a <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean space</a>, some authors<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> define the directional derivative to be with respect to an arbitrary nonzero vector v after <a href="/facts/Normalized_vector/GyaXneDn">normalization</a>, thus being independent of its magnitude and depending only on its direction.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> </p><p>This definition gives the rate of increase of <i>f</i> per unit of distance moved in the direction given by v. In this case, one has ∇ v f ( x ) = lim h → 0 f ( x + h v ) − f ( x ) h | v | , {\displaystyle \nabla _{\mathbf {v} }{f}(\mathbf {x} )=\lim _{h\to 0}{\frac {f(\mathbf {x} +h\mathbf {v} )-f(\mathbf {x} )}{h|\mathbf {v} |}},} or in case <i>f</i> is differentiable at x, ∇ v f ( x ) = ∇ f ( x ) ⋅ v | v | . {\displaystyle \nabla _{\mathbf {v} }{f}(\mathbf {x} )=\nabla f(\mathbf {x} )\cdot {\frac {\mathbf {v} }{|\mathbf {v} |}}.} </p> <h3>Restriction to a unit vector</h3> <p>In the context of a function on a <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean space</a>, some texts restrict the vector v to being a <a href="/facts/Unit_vector/GyaXneDn">unit vector</a>. With this restriction, both the above definitions are equivalent.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> </p> <h2 id="properties">Properties</h2> <p>Many of the familiar properties of the ordinary <a href="/facts/Derivative/7Ito70Dh">derivative</a> hold for the directional derivative. These include, for any functions <i>f</i> and <i>g</i> defined in a <a href="/facts/Neighborhood_(mathematics)/XHKgrpkp">neighborhood</a> of, and <a href="/facts/Total_derivative/cPwsm9Wf">differentiable</a> at, p: </p> <ol><li><a href="/facts/Sum_rule_in_differentiation/9q1b7Ev6">sum rule</a>: ∇ v ( f + g ) = ∇ v f + ∇ v g . {\displaystyle \nabla _{\mathbf {v} }(f+g)=\nabla _{\mathbf {v} }f+\nabla _{\mathbf {v} }g.} </li> <li><a href="/facts/Constant_factor_rule_in_differentiation/9q1b7Ev6">constant factor rule</a>: For any constant <i>c</i>, ∇ v ( c f ) = c ∇ v f . {\displaystyle \nabla _{\mathbf {v} }(cf)=c\nabla _{\mathbf {v} }f.} </li> <li><a href="/facts/Product_rule/MyKxsP5t">product rule</a> (or Leibniz's rule): ∇ v ( f g ) = g ∇ v f + f ∇ v g . {\displaystyle \nabla _{\mathbf {v} }(fg)=g\nabla _{\mathbf {v} }f+f\nabla _{\mathbf {v} }g.} </li> <li><a href="/facts/Chain_rule/aMLYxP0x">chain rule</a>: If <i>g</i> is differentiable at p and <i>h</i> is differentiable at <i>g</i>(p), then ∇ v ( h ∘ g ) ( p ) = h ′ ( g ( p ) ) ∇ v g ( p ) . {\displaystyle \nabla _{\mathbf {v} }(h\circ g)(\mathbf {p} )=h'(g(\mathbf {p} ))\nabla _{\mathbf {v} }g(\mathbf {p} ).} </li></ol> <h2 id="in-differential-geometry">In differential geometry</h2> <p class="note">See also: <a href="/facts/Tangent_space/crvpbSeG">Tangent space § Tangent vectors as directional derivatives</a></p> <p>Let <i>M</i> be a <a href="/facts/Differentiable_manifold/aSnbXKcV">differentiable manifold</a> and p a point of <i>M</i>. Suppose that <i>f</i> is a function defined in a neighborhood of p, and <a href="/facts/Total_derivative/cPwsm9Wf">differentiable</a> at p. If v is a <a href="/facts/Tangent_vector/UEq5yffm">tangent vector</a> to <i>M</i> at p, then the directional derivative of <i>f</i> along v, denoted variously as <i>df</i>(v) (see <a href="/facts/Exterior_derivative/p7HA0wze">Exterior derivative</a>), ∇ v f ( p ) {\displaystyle \nabla _{\mathbf {v} }f(\mathbf {p} )} (see <a href="/facts/Covariant_derivative/sRGN0Xh6">Covariant derivative</a>), L v f ( p ) {\displaystyle L_{\mathbf {v} }f(\mathbf {p} )} (see <a href="/facts/Lie_derivative/1KSJvrzk">Lie derivative</a>), or v p ( f ) {\displaystyle {\mathbf {v} }_{\mathbf {p} }(f)} (see <a href="/facts/Tangent_space/crvpbSeG">Tangent space § Definition via derivations</a>), can be defined as follows. Let <i>γ</i> : [−1, 1] → <i>M</i> be a differentiable curve with <i>γ</i>(0) = p and <i>γ</i>′(0) = v. Then the directional derivative is defined by ∇ v f ( p ) = d d τ f ∘ 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 of <i>γ</i>, provided <i>γ</i> is selected in the prescribed manner so that <i>γ</i>(0) = p and <i>γ</i>′(0) = v. </p> <h3>The Lie derivative</h3> <p>The <a href="/facts/Lie_derivative/1KSJvrzk">Lie derivative</a> of a vector field W μ ( x ) {\displaystyle W^{\mu }(x)} along a vector field V μ ( x ) {\displaystyle V^{\mu }(x)} is given by the difference of two directional derivatives (with vanishing torsion): L V W μ = ( V ⋅ ∇ ) W μ − ( W ⋅ ∇ ) V μ . {\displaystyle {\mathcal {L}}_{V}W^{\mu }=(V\cdot \nabla )W^{\mu }-(W\cdot \nabla )V^{\mu }.} In particular, for a scalar field ϕ ( x ) {\displaystyle \phi (x)} , the Lie derivative reduces to the standard directional derivative: L V ϕ = ( V ⋅ ∇ ) ϕ . {\displaystyle {\mathcal {L}}_{V}\phi =(V\cdot \nabla )\phi .} </p> <h3>The Riemann tensor</h3> <p>Directional derivatives are often used in introductory derivations of the <a href="/facts/Riemann_curvature_tensor/KyeHCWEh">Riemann curvature tensor</a>. Consider a curved rectangle with an infinitesimal vector δ {\displaystyle \delta } along one edge and δ ′ {\displaystyle \delta '} along the other. We translate a covector S {\displaystyle S} along δ {\displaystyle \delta } then δ ′ {\displaystyle \delta '} and then subtract the translation along δ ′ {\displaystyle \delta '} and then δ {\displaystyle \delta } . Instead of building the directional derivative using partial derivatives, we use the <a href="/facts/Covariant_derivative/sRGN0Xh6">covariant derivative</a>. The translation operator for δ {\displaystyle \delta } is thus 1 + ∑ ν δ ν D ν = 1 + δ ⋅ D , {\displaystyle 1+\sum _{\nu }\delta ^{\nu }D_{\nu }=1+\delta \cdot D,} and for δ ′ {\displaystyle \delta '} , 1 + ∑ μ δ ′ μ D μ = 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 ) S ρ − ( 1 + δ ⋅ D ) ( 1 + δ ′ ⋅ D ) S ρ = ∑ μ , ν δ ′ μ δ ν [ D μ , D ν ] S ρ . {\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<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> that the noncommutativity of the covariant derivatives measures the curvature of the manifold: [ D μ , D ν ] S ρ = ± ∑ σ R σ ρ μ ν S σ , {\displaystyle [D_{\mu },D_{\nu }]S_{\rho }=\pm \sum _{\sigma }R^{\sigma }{}_{\rho \mu \nu }S_{\sigma },} where R {\displaystyle R} is the Riemann curvature tensor and the sign depends on the <a href="/facts/Sign_convention/0yGl2bzD">sign convention</a> of the author. </p> <h2 id="in-group-theory">In group theory</h2> <h3>Translations</h3> <p>In the <a href="/facts/Poincar%C3%A9_algebra/3IerRYG1">Poincaré algebra</a>, we can define an infinitesimal translation operator P as P = i ∇ . {\displaystyle \mathbf {P} =i\nabla .} (the <i>i</i> ensures that P is a <a href="/facts/Self-adjoint_operator/VrH1nEAK">self-adjoint operator</a>) For a finite displacement λ, the <a href="/facts/Unitary_operator/rkypfQQ9">unitary</a> <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert space</a> <a href="/facts/Group_representation/FsuuDagQ">representation</a> for translations is<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> U ( λ ) = exp ( − i λ ⋅ 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 functions <i>f</i>(x) as U ( 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 }}).} </p> Proof of the last equation <p>In standard single-variable calculus, the derivative of a smooth function <i>f</i>(<i>x</i>) is defined by (for small <i>ε</i>) d f d x = f ( x + ε ) − f ( x ) ε . {\displaystyle {\frac {df}{dx}}={\frac {f(x+\varepsilon )-f(x)}{\varepsilon }}.} This can be rearranged to find <i>f</i>(<i>x</i>+<i>ε</i>): f ( x + ε ) = f ( x ) + ε d f d x = ( 1 + ε d d x ) 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 / d x ) ] {\displaystyle [1+\varepsilon \,(d/dx)]} is a translation operator. This is instantly generalized<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> to multivariable functions <i>f</i>(x) f ( x + ε ) = ( 1 + ε ⋅ ∇ ) f ( x ) . {\displaystyle f(\mathbf {x} +{\boldsymbol {\varepsilon }})=\left(1+{\boldsymbol {\varepsilon }}\cdot \nabla \right)f(\mathbf {x} ).} Here ε ⋅ ∇ {\displaystyle {\boldsymbol {\varepsilon }}\cdot \nabla } is the directional derivative along the infinitesimal displacement <i>ε</i>. 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<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> <i>U</i>(<i>g</i>)<i>U</i>(<i>f</i>)=<i>U</i>(<i>gf</i>) takes the form U ( a ) U ( b ) = U ( a + b ) . {\displaystyle U(\mathbf {a} )U(\mathbf {b} )=U(\mathbf {a+b} ).} So suppose that we take the finite displacement <i>λ</i> and divide it into <i>N</i> parts (<i>N</i>→∞ is implied everywhere), so that <i>λ</i>/<i>N</i>=<i>ε</i>. In other words, λ = N ε . {\displaystyle {\boldsymbol {\lambda }}=N{\boldsymbol {\varepsilon }}.} Then by applying <i>U</i>(<i>ε</i>) <i>N</i> times, we can construct <i>U</i>(<i>λ</i>): [ U ( ε ) ] N = U ( N ε ) = 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<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> exp ( x ) = [ 1 + x N ] N , {\displaystyle \exp(x)=\left[1+{\frac {x}{N}}\right]^{N},} we have U ( λ ) = exp ( λ ⋅ ∇ ) . {\displaystyle U({\boldsymbol {\lambda }})=\exp \left({\boldsymbol {\lambda }}\cdot \nabla \right).} And since <i>U</i>(<i>ε</i>)<i>f</i>(x) = <i>f</i>(x+<i>ε</i>) we have [ U ( ε ) ] N f ( x ) = f ( x + N ε ) = 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. </p><p>As a technical note, this procedure is only possible because the translation group forms an <a href="/facts/Abelian_group/FfWwmx4R">Abelian</a> <a href="/facts/Subgroup/r91mBgpa">subgroup</a> (<a href="/facts/Cartan_subalgebra/8gymnPU0">Cartan subalgebra</a>) in the Poincaré algebra. In particular, the group multiplication law <i>U</i>(a)<i>U</i>(b) = <i>U</i>(a+b) should not be taken for granted. We also note that Poincaré is a connected <a href="/facts/Lie_group/a9jCQyjF">Lie group</a>. It is a group of transformations <i>T</i>(<i>ξ</i>) that are described by a continuous set of real parameters ξ a {\displaystyle \xi ^{a}} . The group multiplication law takes the form T ( ξ ¯ ) T ( ξ ) = T ( f ( ξ ¯ , ξ ) ) . {\displaystyle T({\bar {\xi }})T(\xi )=T(f({\bar {\xi }},\xi )).} Taking ξ a = 0 {\displaystyle \xi ^{a}=0} as the coordinates of the identity, we must have f a ( ξ , 0 ) = f a ( 0 , ξ ) = ξ a . {\displaystyle f^{a}(\xi ,0)=f^{a}(0,\xi )=\xi ^{a}.} The actual operators on the Hilbert space are represented by unitary operators <i>U</i>(<i>T</i>(<i>ξ</i>)). In the above notation we suppressed the <i>T</i>; we now write <i>U</i>(λ) as <i>U</i>(P(λ)). For a small neighborhood around the identity, the <a href="/facts/Power_series/vYh6sTps">power series</a> representation U ( T ( ξ ) ) = 1 + i ∑ a ξ a t a + 1 2 ∑ b , c ξ b ξ c t b c + ⋯ {\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 is f a ( ξ ¯ , ξ ) = ξ a + ξ ¯ a + ∑ b , c f a b c ξ ¯ b ξ c . {\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 condition t b c = − t b t c − i ∑ a f a b c t a . {\displaystyle t_{bc}=-t_{b}t_{c}-i\sum _{a}f^{abc}t_{a}.} Since t a b {\displaystyle t_{ab}} is by definition symmetric in its indices, we have the standard <a href="/facts/Lie_algebra/n2gAUkNq">Lie algebra</a> commutator: [ t b , t c ] = i ∑ a ( − f a b c + f a c b ) t a = i ∑ a C a b c t a , {\displaystyle [t_{b},t_{c}]=i\sum _{a}(-f^{abc}+f^{acb})t_{a}=i\sum _{a}C^{abc}t_{a},} with <i>C</i> the <a href="/facts/Structure_constant/9TVS57qS">structure constant</a>. The generators for translations are partial derivative operators, which commute: [ ∂ ∂ x b , ∂ ∂ x c ] = 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 that <i>f</i> is simply additive: f abelian a ( ξ ¯ , ξ ) = ξ a + 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. </p> <h3>Rotations</h3> <p>The <a href="/facts/Rotation_operator_(quantum_mechanics)/vdUA900a">rotation operator</a> also contains a directional derivative. The rotation operator for an angle <i>θ</i>, i.e. by an amount <i>θ</i> = |<i>θ</i>| about an axis parallel to θ ^ = θ / θ {\displaystyle {\hat {\theta }}={\boldsymbol {\theta }}/\theta } is U ( R ( θ ) ) = exp ( − i θ ⋅ L ) . {\displaystyle U(R(\mathbf {\theta } ))=\exp(-i\mathbf {\theta } \cdot \mathbf {L} ).} Here L is the vector operator that generates <a href="/facts/SO(3)/yXbKPzhT">SO(3)</a>: L = ( 0 0 0 0 0 1 0 − 1 0 ) i + ( 0 0 − 1 0 0 0 1 0 0 ) j + ( 0 1 0 − 1 0 0 0 0 0 ) 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 vector x by x → x − δ θ × x . {\displaystyle \mathbf {x} \rightarrow \mathbf {x} -\delta {\boldsymbol {\theta }}\times \mathbf {x} .} So we would expect under infinitesimal rotation: U ( R ( δ θ ) ) f ( x ) = f ( x − δ θ × x ) = f ( x ) − ( δ θ × 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 that U ( R ( δ θ ) ) = 1 − ( δ θ × 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:<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a> U ( R ( θ ) ) = exp ( − ( θ × x ) ⋅ ∇ ) . {\displaystyle U(R(\mathbf {\theta } ))=\exp(-(\mathbf {\theta } \times \mathbf {x} )\cdot \nabla ).} </p> <h2 id="normal-derivative">Normal derivative</h2> <p>A normal derivative is a directional derivative taken in the direction normal (that is, <a href="/facts/Orthogonal/JVIjYPog">orthogonal</a>) to some surface in space, or more generally along a <a href="/facts/Normal_vector/jhtLIhcu">normal vector</a> field orthogonal to some <a href="/facts/Hypersurface/cTvtLVsV">hypersurface</a>. See for example <a href="/facts/Neumann_boundary_condition/0e0YBW4G">Neumann boundary condition</a>. If the normal direction is denoted by n {\displaystyle \mathbf {n} } , then the normal derivative of a function <i>f</i> is sometimes denoted as ∂ f ∂ n {\textstyle {\frac {\partial f}{\partial \mathbf {n} }}} . In other notations, ∂ f ∂ n = ∇ f ( x ) ⋅ n = ∇ n f ( x ) = ∂ f ∂ x ⋅ n = D f ( 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} ].} </p> <h2 id="in-the-continuum-mechanics-of-solids">In the continuum mechanics of solids</h2> <p>Several important results in continuum mechanics require the derivatives of vectors with respect to vectors and of <a href="/facts/Tensors/pNjFo5tV">tensors</a> with respect to vectors and tensors.<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a> The directional directive provides a systematic way of finding these derivatives. </p> <p class="note">This section is an excerpt from <a href="/facts/Tensor_derivative_(continuum_mechanics)/2Hd0KUa5">Tensor derivative (continuum mechanics) § Derivatives with respect to vectors and second-order tensors</a>.[<a href="https://en.wikipedia.org/w/index.php?title=Tensor_derivative_(continuum_mechanics)&action=edit">edit</a>]</p> <p>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. </p> <h3>Derivatives of scalar valued functions of vectors</h3> <p>Let <i>f</i>(v) be a real valued function of the vector v. Then the derivative of <i>f</i>(v) with respect to v (or at v) is the vector defined through its <a href="/facts/Dot_product/tNz8MLjT">dot product</a> with any vector u being </p><p> ∂ f ∂ v ⋅ u = D f ( v ) [ u ] = [ d d α 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}} </p><p>for all vectors u. The above dot product yields a scalar, and if u is a unit vector gives the directional derivative of <i>f</i> at v, in the u direction. </p><p>Properties: </p> <ol><li>If f ( v ) = f 1 ( v ) + f 2 ( v ) {\displaystyle f(\mathbf {v} )=f_{1}(\mathbf {v} )+f_{2}(\mathbf {v} )} then ∂ f ∂ v ⋅ u = ( ∂ f 1 ∂ v + ∂ f 2 ∂ 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} } </li> <li>If f ( v ) = f 1 ( v ) f 2 ( v ) {\displaystyle f(\mathbf {v} )=f_{1}(\mathbf {v} )~f_{2}(\mathbf {v} )} then ∂ f ∂ v ⋅ u = ( ∂ f 1 ∂ v ⋅ u ) f 2 ( v ) + f 1 ( v ) ( ∂ f 2 ∂ 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)} </li> <li>If f ( v ) = f 1 ( f 2 ( v ) ) {\displaystyle f(\mathbf {v} )=f_{1}(f_{2}(\mathbf {v} ))} then ∂ f ∂ v ⋅ u = ∂ f 1 ∂ f 2 ∂ f 2 ∂ 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} } </li></ol> <h3>Derivatives of vector valued functions of vectors</h3> <p>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 </p><p> ∂ f ∂ v ⋅ u = D f ( v ) [ u ] = [ d d α 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}} </p><p>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. </p><p>Properties: </p> <ol><li>If f ( v ) = f 1 ( v ) + f 2 ( v ) {\displaystyle \mathbf {f} (\mathbf {v} )=\mathbf {f} _{1}(\mathbf {v} )+\mathbf {f} _{2}(\mathbf {v} )} then ∂ f ∂ v ⋅ u = ( ∂ f 1 ∂ v + ∂ f 2 ∂ 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} } </li> <li>If f ( v ) = f 1 ( v ) × f 2 ( v ) {\displaystyle \mathbf {f} (\mathbf {v} )=\mathbf {f} _{1}(\mathbf {v} )\times \mathbf {f} _{2}(\mathbf {v} )} then ∂ f ∂ v ⋅ u = ( ∂ f 1 ∂ v ⋅ u ) × f 2 ( v ) + f 1 ( v ) × ( ∂ f 2 ∂ 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)} </li> <li>If f ( v ) = f 1 ( f 2 ( v ) ) {\displaystyle \mathbf {f} (\mathbf {v} )=\mathbf {f} _{1}(\mathbf {f} _{2}(\mathbf {v} ))} then ∂ f ∂ v ⋅ u = ∂ f 1 ∂ f 2 ⋅ ( ∂ f 2 ∂ 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)} </li></ol> <h3>Derivatives of scalar valued functions of second-order tensors</h3> <p>Let f ( S ) {\displaystyle f({\boldsymbol {S}})} be a real valued function of the second order tensor S {\displaystyle {\boldsymbol {S}}} . Then the derivative of f ( S ) {\displaystyle f({\boldsymbol {S}})} with respect to S {\displaystyle {\boldsymbol {S}}} (or at S {\displaystyle {\boldsymbol {S}}} ) in the direction T {\displaystyle {\boldsymbol {T}}} is the second order tensor defined as ∂ f ∂ S : T = D f ( S ) [ T ] = [ d d α 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 tensors T {\displaystyle {\boldsymbol {T}}} . </p><p>Properties: </p> <ol><li>If f ( S ) = f 1 ( S ) + f 2 ( S ) {\displaystyle f({\boldsymbol {S}})=f_{1}({\boldsymbol {S}})+f_{2}({\boldsymbol {S}})} then ∂ f ∂ S : T = ( ∂ f 1 ∂ S + ∂ f 2 ∂ 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}}} </li> <li>If f ( S ) = f 1 ( S ) f 2 ( S ) {\displaystyle f({\boldsymbol {S}})=f_{1}({\boldsymbol {S}})~f_{2}({\boldsymbol {S}})} then ∂ f ∂ S : T = ( ∂ f 1 ∂ S : T ) f 2 ( S ) + f 1 ( S ) ( ∂ f 2 ∂ 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)} </li> <li>If f ( S ) = f 1 ( f 2 ( S ) ) {\displaystyle f({\boldsymbol {S}})=f_{1}(f_{2}({\boldsymbol {S}}))} then ∂ f ∂ S : T = ∂ f 1 ∂ f 2 ( ∂ f 2 ∂ 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)} </li></ol> <h3>Derivatives of tensor valued functions of second-order tensors</h3> <p>Let F ( S ) {\displaystyle {\boldsymbol {F}}({\boldsymbol {S}})} be a second order tensor valued function of the second order tensor S {\displaystyle {\boldsymbol {S}}} . Then the derivative of F ( S ) {\displaystyle {\boldsymbol {F}}({\boldsymbol {S}})} with respect to S {\displaystyle {\boldsymbol {S}}} (or at S {\displaystyle {\boldsymbol {S}}} ) in the direction T {\displaystyle {\boldsymbol {T}}} is the fourth order tensor defined as ∂ F ∂ S : T = D F ( S ) [ T ] = [ d d α 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 tensors T {\displaystyle {\boldsymbol {T}}} . </p><p>Properties: </p> <ol><li>If F ( S ) = F 1 ( S ) + F 2 ( S ) {\displaystyle {\boldsymbol {F}}({\boldsymbol {S}})={\boldsymbol {F}}_{1}({\boldsymbol {S}})+{\boldsymbol {F}}_{2}({\boldsymbol {S}})} then ∂ F ∂ S : T = ( ∂ F 1 ∂ S + ∂ F 2 ∂ 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}}} </li> <li>If F ( S ) = F 1 ( S ) ⋅ F 2 ( S ) {\displaystyle {\boldsymbol {F}}({\boldsymbol {S}})={\boldsymbol {F}}_{1}({\boldsymbol {S}})\cdot {\boldsymbol {F}}_{2}({\boldsymbol {S}})} then ∂ F ∂ S : T = ( ∂ F 1 ∂ S : T ) ⋅ F 2 ( S ) + F 1 ( S ) ⋅ ( ∂ F 2 ∂ 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)} </li> <li>If F ( S ) = F 1 ( F 2 ( S ) ) {\displaystyle {\boldsymbol {F}}({\boldsymbol {S}})={\boldsymbol {F}}_{1}({\boldsymbol {F}}_{2}({\boldsymbol {S}}))} then ∂ F ∂ S : T = ∂ F 1 ∂ F 2 : ( ∂ F 2 ∂ 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)} </li> <li>If f ( S ) = f 1 ( F 2 ( S ) ) {\displaystyle f({\boldsymbol {S}})=f_{1}({\boldsymbol {F}}_{2}({\boldsymbol {S}}))} then ∂ f ∂ S : T = ∂ f 1 ∂ F 2 : ( ∂ F 2 ∂ 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)} </li></ol> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Del_in_cylindrical_and_spherical_coordinates/rCFHtIbm">Del in cylindrical and spherical coordinates</a> – Mathematical gradient operator in certain coordinate systems</li> <li><a href="/facts/Differential_form/T9gyHtMv">Differential form</a> – Expression that may be integrated over a region</li> <li><a href="/facts/Ehresmann_connection/qIpSLmFz">Ehresmann connection</a> – Differential geometry construct on fiber bundles</li> <li><a href="/facts/Fr%C3%A9chet_derivative/kyAlz3up">Fréchet derivative</a> – Derivative defined on normed spaces</li> <li><a href="/facts/Gateaux_derivative/dqdtTtpf">Gateaux derivative</a> – Generalization of the concept of directional derivative</li> <li><a href="/facts/Generalizations_of_the_derivative/mgWY23Sd">Generalizations of the derivative</a> – Fundamental construction of differential calculus</li> <li><a href="/facts/Semi-differentiability/UJUGB8hR">Semi-differentiability</a></li> <li><a href="/facts/Hadamard_derivative/eD8aUxVs">Hadamard derivative</a></li> <li><a href="/facts/Lie_derivative/1KSJvrzk">Lie derivative</a> – A derivative in Differential Geometry</li> <li><a href="/facts/Material_derivative/WsI7rKSF">Material derivative</a> – Time rate of change of some physical quantity of a material element in a velocity field</li> <li><a href="/facts/Structure_tensor/FyyELa9n">Structure tensor</a> – Tensor related to gradients</li> <li><a href="/facts/Tensor_derivative_(continuum_mechanics)/2Hd0KUa5">Tensor derivative (continuum mechanics)</a></li> <li><a href="/facts/Total_derivative/cPwsm9Wf">Total derivative</a> – Type of derivative in mathematics</li></ul> <h2 id="notes">Notes</h2> <ul><li>Hildebrand, F. B. (1976). <i>Advanced Calculus for Applications</i>. Prentice Hall. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-13-011189-9.</li> <li>K.F. Riley; M.P. Hobson; S.J. Bence (2010). <a href="https://archive.org/details/mathematicalmeth00rile"><i>Mathematical methods for physics and engineering</i></a>. Cambridge University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-86153-3.</li> <li>Shapiro, A. (1990). "On concepts of directional differentiability". <i>Journal of Optimization Theory and Applications</i>. 66 (3): 477–487. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF00940933">10.1007/BF00940933</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:120253580">120253580</a>.</li></ul> <h2 id="external-links">External links</h2> <p> Media related to Directional derivative at Wikimedia Commons </p> <ul><li><a href="http://mathworld.wolfram.com/DirectionalDerivative.html">Directional derivatives</a> at <a href="/facts/MathWorld/yc1jodbq">MathWorld</a>.</li> <li><a href="https://planetmath.org/directionalderivative">Directional derivative</a> at <a href="/facts/PlanetMath/wphPCtLf">PlanetMath</a>.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>R. Wrede; M.R. Spiegel (2010). Advanced Calculus (3rd ed.). Schaum's Outline Series. ISBN 978-0-07-162366-7. <a href="978-0-07-162366-7" target="_blank">978-0-07-162366-7</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>The applicability extends to functions over spaces without a metric and to differentiable manifolds, such as in general relativity. <a href="/wiki/Metric_(mathematics)" target="_blank">/wiki/Metric_(mathematics)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>If the dot product is undefined, the gradient is also undefined; however, for differentiable f, the directional derivative is still defined, and a similar relation exists with the exterior derivative. <a href="/wiki/Gradient" target="_blank">/wiki/Gradient</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Thomas, George B. Jr.; and Finney, Ross L. (1979) Calculus and Analytic Geometry, Addison-Wesley Publ. Co., fifth edition, p. 593. <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>This typically assumes a Euclidean space – for example, a function of several variables typically has no definition of the magnitude of a vector, and hence of a unit vector. <a href="/wiki/Euclidean_space" target="_blank">/wiki/Euclidean_space</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Hughes Hallett, Deborah; McCallum, William G.; Gleason, Andrew M. (2012-01-01). Calculus : Single and multivariable. John wiley. p. 780. ISBN 9780470888612. OCLC 828768012. <a href="9780470888612" target="_blank">9780470888612</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Zee, A. (2013). Einstein gravity in a nutshell. Princeton: Princeton University Press. p. 341. ISBN 9780691145587. <a href="9780691145587" target="_blank">9780691145587</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Weinberg, Steven (1999). The quantum theory of fields (Reprinted (with corr.). ed.). Cambridge [u.a.]: Cambridge Univ. Press. ISBN 9780521550017. <a href="9780521550017" target="_blank">9780521550017</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Zee, A. (2013). Einstein gravity in a nutshell. Princeton: Princeton University Press. ISBN 9780691145587. <a href="9780691145587" target="_blank">9780691145587</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Cahill, Kevin Cahill (2013). Physical mathematics (Repr. ed.). Cambridge: Cambridge University Press. ISBN 978-1107005211. <a href="978-1107005211" target="_blank">978-1107005211</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Larson, Ron; Edwards, Bruce H. (2010). Calculus of a single variable (9th ed.). Belmont: Brooks/Cole. ISBN 9780547209982. <a href="9780547209982" target="_blank">9780547209982</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Shankar, R. (1994). Principles of quantum mechanics (2nd ed.). New York: Kluwer Academic / Plenum. p. 318. ISBN 9780306447907. <a href="9780306447907" target="_blank">9780306447907</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>J. E. Marsden and T. J. R. Hughes, 2000, Mathematical Foundations of Elasticity, Dover. <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> </ol>