A directional derivative is a concept in multivariable calculus that measures the rate at which a function changes in a particular direction at a given point.[citation needed]
The directional derivative of a multivariable differentiable (scalar) function along a given vectorv at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity specified by v.
The directional derivative of a scalar functionf with respect to a vector v at a point (e.g., position) x may be denoted by any of the following:
This definition is valid in a broad range of contexts, for example where the norm of a vector (and hence a unit vector) is undefined.[2]
For differentiable functionsedit
If the function f is differentiable at x, then the directional derivative exists along any unit vector v at x, and one has
where the on the right denotes the gradient, is the dot product and v is a unit vector.[3] This follows from defining a path and using the definition of the derivative as a limit which can be calculated along this path to get:
Intuitively, the directional derivative of f at a point x represents the rate of change of f, in the direction of v with respect to time, when moving past x.
Using only direction of vectoredit
In a Euclidean space, some authors[4] define the directional derivative to be with respect to an arbitrary nonzero vector v after normalization, thus being independent of its magnitude and depending only on its direction.[5]
This definition gives the rate of increase of f per unit of distance moved in the direction given by v. In this case, one has
or in case f is differentiable at x,
Restriction to a unit vectoredit
In the context of a function on a Euclidean space, some texts restrict the vector v to being a unit vector. With this restriction, both the above definitions are equivalent.[6]
Propertiesedit
Many of the familiar properties of the ordinary derivative hold for the directional derivative. These include, for any functions f and g defined in a neighborhood of, and differentiable at, p:
This definition can be proven independent of the choice of γ, provided γ is selected in the prescribed manner so that γ′(0) = v.
The Lie derivativeedit
The Lie derivative of a vector field along a vector field is given by the difference of two directional derivatives (with vanishing torsion):
In particular, for a scalar field , the Lie derivative reduces to the standard directional derivative:
The Riemann tensoredit
Directional derivatives are often used in introductory derivations of the Riemann curvature tensor. Consider a curved rectangle with an infinitesimal vector along one edge and along the other. We translate a covector along then and then subtract the translation along and then . Instead of building the directional derivative using partial derivatives, we use the covariant derivative. The translation operator for is thus
and for ,
The difference between the two paths is then
It can be argued[7] that the noncommutativity of the covariant derivatives measures the curvature of the manifold:
where is the Riemann curvature tensor and the sign depends on the sign convention of the author.
In group theoryedit
Translationsedit
In the Poincaré algebra, we can define an infinitesimal translation operator P as
By using the above definition of the infinitesimal translation operator, we see that the finite translation operator is an exponentiated directional derivative:
This is a translation operator in the sense that it acts on multivariable functions f(x) as
Proof of the last equation
In standard single-variable calculus, the derivative of a smooth function f(x) is defined by (for small ε)
This can be rearranged to find f(x+ε):
It follows that is a translation operator. This is instantly generalized[9] to multivariable functions f(x)
Here is the directional derivative along the infinitesimal displacement ε. We have found the infinitesimal version of the translation operator:
It is evident that the group multiplication law[10]U(g)U(f)=U(gf) takes the form
So suppose that we take the finite displacement λ and divide it into N parts (N→∞ is implied everywhere), so that λ/N=ε. In other words,
Then by applying U(ε) N times, we can construct U(λ):
As a technical note, this procedure is only possible because the translation group forms an Abeliansubgroup (Cartan subalgebra) in the Poincaré algebra. In particular, the group multiplication law U(a)U(b) = U(a+b) should not be taken for granted. We also note that Poincaré is a connected Lie group. It is a group of transformations T(ξ) that are described by a continuous set of real parameters . The group multiplication law takes the form
Taking as the coordinates of the identity, we must have
The actual operators on the Hilbert space are represented by unitary operators U(T(ξ)). In the above notation we suppressed the T; we now write U(λ) as U(P(λ)). For a small neighborhood around the identity, the power series representation
is quite good. Suppose that U(T(ξ)) form a non-projective representation, i.e.,
The expansion of f to second power is
After expanding the representation multiplication equation and equating coefficients, we have the nontrivial condition
Since is by definition symmetric in its indices, we have the standard Lie algebra commutator:
with C the structure constant. The generators for translations are partial derivative operators, which commute:
This implies that the structure constants vanish and thus the quadratic coefficients in the f expansion vanish as well. This means that f is simply additive:
and thus for abelian groups,
Q.E.D.
Rotationsedit
The rotation operator also contains a directional derivative. The rotation operator for an angle θ, i.e. by an amount θ = |θ| about an axis parallel to is
Here L is the vector operator that generates SO(3):
It may be shown geometrically that an infinitesimal right-handed rotation changes the position vector x by
So we would expect under infinitesimal rotation:
It follows that
Following the same exponentiation procedure as above, we arrive at the rotation operator in the position basis, which is an exponentiated directional derivative:[12]
Normal derivativeedit
A normal derivative is a directional derivative taken in the direction normal (that is, orthogonal) to some surface in space, or more generally along a normal vector field orthogonal to some hypersurface. See for example Neumann boundary condition. If the normal direction is denoted by , then the normal derivative of a function f is sometimes denoted as . In other notations,
In the continuum mechanics of solidsedit
Several important results in continuum mechanics require the derivatives of vectors with respect to vectors and of tensors with respect to vectors and tensors.[13] The directional directive provides a systematic way of finding these derivatives.
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 vectorsedit
Let f(v) be a real valued function of the vector v. Then the derivative of f(v) with respect to v (or at v) is the vector defined through its dot product with any vector u being
for all vectors u. The above dot product yields a scalar, and if u is a unit vector gives the directional derivative of f at v, in the u direction.
Properties:
If then
If then
If then
Derivatives of vector valued functions of vectorsedit
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
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:
If then
If then
If then
Derivatives of scalar valued functions of second-order tensorsedit
Let be a real valued function of the second order tensor . Then the derivative of with respect to (or at ) in the direction is the second order tensor defined as
for all second order tensors .
Properties:
If then
If then
If then
Derivatives of tensor valued functions of second-order tensorsedit
Let be a second order tensor valued function of the second order tensor . Then the derivative of with respect to (or at ) in the direction is the fourth order tensor defined as
^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.
^Thomas, George B. Jr.; and Finney, Ross L. (1979) Calculus and Analytic Geometry, Addison-Wesley Publ. Co., fifth edition, p. 593.
^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.
^Zee, A. (2013). Einstein gravity in a nutshell. Princeton: Princeton University Press. p. 341. ISBN9780691145587.
^Weinberg, Steven (1999). The quantum theory of fields (Reprinted (with corr.). ed.). Cambridge [u.a.]: Cambridge Univ. Press. ISBN9780521550017.
^Zee, A. (2013). Einstein gravity in a nutshell. Princeton: Princeton University Press. ISBN9780691145587.
^Cahill, Kevin Cahill (2013). Physical mathematics (Repr. ed.). Cambridge: Cambridge University Press. ISBN978-1107005211.
^Larson, Ron; Edwards, Bruce H. (2010). Calculus of a single variable (9th ed.). Belmont: Brooks/Cole. ISBN9780547209982.
^Shankar, R. (1994). Principles of quantum mechanics (2nd ed.). New York: Kluwer Academic / Plenum. p. 318. ISBN9780306447907.
^J. E. Marsden and T. J. R. Hughes, 2000, Mathematical Foundations of Elasticity, Dover.
Referencesedit
Hildebrand, F. B. (1976). Advanced Calculus for Applications. Prentice Hall. ISBN0-13-011189-9.
K.F. Riley; M.P. Hobson; S.J. Bence (2010). Mathematical methods for physics and engineering. Cambridge University Press. ISBN978-0-521-86153-3.
Shapiro, A. (1990). "On concepts of directional differentiability". Journal of Optimization Theory and Applications. 66 (3): 477–487. doi:10.1007/BF00940933. S2CID 120253580.
External linksedit
Media related to Directional derivative at Wikimedia Commons
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This section needs additional citations for verification Relevant discussion may be found on the talk page Please help improve this article by adding citations to reliable sources in this section Unsourced material may be challenged and removed October 2012 template removal help A directional derivative is a concept in multivariable calculus that measures the rate at which a function changes in a particular direction at a given point citation needed The directional derivative of a multivariable differentiable scalar function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function moving through x with a velocity specified by v The directional derivative of a scalar function f with respect to a vector v at a point e g position x may be denoted by any of the following v f x f v x D v f x D f x v v f x v f x v f x x displaystyle 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 mathbf v cdot nabla f mathbf x mathbf v cdot frac partial f mathbf x partial mathbf x It therefore generalizes the notion of a partial derivative in which the rate of change is taken along one of the curvilinear coordinate curves all other coordinates being constant The directional derivative is a special case of the Gateaux derivative Contents 1 Definition 1 1 For differentiable functions 1 2 Using only direction of vector 1 3 Restriction to a unit vector 2 Properties 3 In differential geometry 3 1 The Lie derivative 3 2 The Riemann tensor 4 In group theory 4 1 Translations 4 2 Rotations 5 Normal derivative 6 In the continuum mechanics of solids 6 1 Derivatives of scalar valued functions of vectors 6 2 Derivatives of vector valued functions of vectors 6 3 Derivatives of scalar valued functions of second order tensors 6 4 Derivatives of tensor valued functions of second order tensors 7 See also 8 Notes 9 References 10 External linksDefinition edit nbsp A contour plot of f x y x 2 y 2 displaystyle f x y x 2 y 2 nbsp showing the gradient vector in black and the unit vector u displaystyle mathbf u nbsp scaled by the directional derivative in the direction of u displaystyle mathbf u nbsp in orange The gradient vector is longer because the gradient points in the direction of greatest rate of increase of a function The directional derivative of a scalar functionf x f x 1 x 2 x n displaystyle f mathbf x f x 1 x 2 ldots x n nbsp along a vector v v 1 v n displaystyle mathbf v v 1 ldots v n nbsp is the function v f displaystyle nabla mathbf v f nbsp defined by the limit 1 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 nbsp This definition is valid in a broad range of contexts for example where the norm of a vector and hence a unit vector is undefined 2 For differentiable functions edit If the function f is differentiable at x then the directional derivative exists along any unit vector v at x and one has v f x f x v displaystyle nabla mathbf v f mathbf x nabla f mathbf x cdot mathbf v nbsp where the displaystyle nabla nbsp on the right denotes the gradient displaystyle cdot nbsp is the dot product and v is a unit vector 3 This follows from defining a path h t x t v displaystyle h t x tv nbsp 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 amp lim t to 0 frac f x tv f x tDf x v t amp lim t to 0 frac f x tv f x t Df x v amp nabla v f x Df x v end aligned nbsp Intuitively the directional derivative of f at a point x represents the rate of change of f in the direction of v with respect to time when moving past x Using only direction of vector edit nbsp The angle a between the tangent A and the horizontal will be maximum if the cutting plane contains the direction of the gradient A In a Euclidean space some authors 4 define the directional derivative to be with respect to an arbitrary nonzero vector v after normalization thus being independent of its magnitude and depending only on its direction 5 This definition gives the rate of increase of f 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 nbsp or in case f 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 nbsp Restriction to a unit vector edit In the context of a function on a Euclidean space some texts restrict the vector v to being a unit vector With this restriction both the above definitions are equivalent 6 Properties editMany of the familiar properties of the ordinary derivative hold for the directional derivative These include for any functions f and g defined in a neighborhood of and differentiable at p sum rule v f g v f v g displaystyle nabla mathbf v f g nabla mathbf v f nabla mathbf v g nbsp constant factor rule For any constant c v c f c v f displaystyle nabla mathbf v cf c nabla mathbf v f nbsp product rule 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 nbsp chain rule If g is differentiable at p and h is differentiable at g 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 nbsp In differential geometry editSee also Tangent space Tangent vectors as directional derivatives Let M be a differentiable manifold and p a point of M Suppose that f is a function defined in a neighborhood of p and differentiable at p If v is a tangent vector to M at p then the directional derivative of f along v denoted variously as df v see Exterior derivative v f p displaystyle nabla mathbf v f mathbf p nbsp see Covariant derivative L v f p displaystyle L mathbf v f mathbf p nbsp see Lie derivative or v p f displaystyle mathbf v mathbf p f nbsp see Tangent space Definition via derivations can be defined as follows Let g 1 1 M be a differentiable curve with g 0 p and g 0 v Then the directional derivative is defined by v f p d d t f g t t 0 displaystyle nabla mathbf v f mathbf p left frac d d tau f circ gamma tau right tau 0 nbsp This definition can be proven independent of the choice of g provided g is selected in the prescribed manner so that g 0 v The Lie derivative edit The Lie derivative of a vector field W m x displaystyle W mu x nbsp along a vector field V m x displaystyle V mu x nbsp is given by the difference of two directional derivatives with vanishing torsion L V W m V W m W V m displaystyle mathcal L V W mu V cdot nabla W mu W cdot nabla V mu nbsp In particular for a scalar field ϕ x displaystyle phi x nbsp the Lie derivative reduces to the standard directional derivative L V ϕ V ϕ displaystyle mathcal L V phi V cdot nabla phi nbsp The Riemann tensor edit Directional derivatives are often used in introductory derivations of the Riemann curvature tensor Consider a curved rectangle with an infinitesimal vector d displaystyle delta nbsp along one edge and d displaystyle delta nbsp along the other We translate a covector S displaystyle S nbsp along d displaystyle delta nbsp then d displaystyle delta nbsp and then subtract the translation along d displaystyle delta nbsp and then d displaystyle delta nbsp Instead of building the directional derivative using partial derivatives we use the covariant derivative The translation operator for d displaystyle delta nbsp is thus1 n d n D n 1 d D displaystyle 1 sum nu delta nu D nu 1 delta cdot D nbsp and for d displaystyle delta nbsp 1 m d m D m 1 d D displaystyle 1 sum mu delta mu D mu 1 delta cdot D nbsp The difference between the two paths is then 1 d D 1 d D S r 1 d D 1 d D S r m n d m d n D m D n S r 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 nbsp It can be argued 7 that the noncommutativity of the covariant derivatives measures the curvature of the manifold D m D n S r s R s r m n S s displaystyle D mu D nu S rho pm sum sigma R sigma rho mu nu S sigma nbsp where R displaystyle R nbsp is the Riemann curvature tensor and the sign depends on the sign convention of the author In group theory editTranslations edit In the Poincare algebra we can define an infinitesimal translation operator P asP i displaystyle mathbf P i nabla nbsp the i ensures that P is a self adjoint operator For a finite displacement l the unitary Hilbert space representation for translations is 8 U l exp i l P displaystyle U boldsymbol lambda exp left i boldsymbol lambda cdot mathbf P right nbsp By using the above definition of the infinitesimal translation operator we see that the finite translation operator is an exponentiated directional derivative U l exp l displaystyle U boldsymbol lambda exp left boldsymbol lambda cdot nabla right nbsp This is a translation operator in the sense that it acts on multivariable functions f x as U l f x exp l f x f x l displaystyle U boldsymbol lambda f mathbf x exp left boldsymbol lambda cdot nabla right f mathbf x f mathbf x boldsymbol lambda nbsp Proof of the last equation In standard single variable calculus the derivative of a smooth function f x is defined by for small e d f d x f x e f x e displaystyle frac df dx frac f x varepsilon f x varepsilon nbsp This can be rearranged to find f x e f x e f x e d f d x 1 e d d x f x displaystyle f x varepsilon f x varepsilon frac df dx left 1 varepsilon frac d dx right f x nbsp It follows that 1 e d d x displaystyle 1 varepsilon d dx nbsp is a translation operator This is instantly generalized 9 to multivariable functions f x f x e 1 e f x displaystyle f mathbf x boldsymbol varepsilon left 1 boldsymbol varepsilon cdot nabla right f mathbf x nbsp Here e displaystyle boldsymbol varepsilon cdot nabla nbsp is the directional derivative along the infinitesimal displacement e We have found the infinitesimal version of the translation operator U e 1 e displaystyle U boldsymbol varepsilon 1 boldsymbol varepsilon cdot nabla nbsp It is evident that the group multiplication law 10 U g U f U gf takes the form U a U b U a b displaystyle U mathbf a U mathbf b U mathbf a b nbsp So suppose that we take the finite displacement l and divide it into N parts N is implied everywhere so that l N e In other words l N e displaystyle boldsymbol lambda N boldsymbol varepsilon nbsp Then by applying U e N times we can construct U l U e N U N e U l displaystyle U boldsymbol varepsilon N U N boldsymbol varepsilon U boldsymbol lambda nbsp We can now plug in our above expression for U e U e N 1 e N 1 l 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 nbsp Using the identity 11 exp x 1 x N N displaystyle exp x left 1 frac x N right N nbsp we have U l exp l displaystyle U boldsymbol lambda exp left boldsymbol lambda cdot nabla right nbsp And since U e f x f x e we have U e N f x f x N e f x l U l f x exp l 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 nbsp Q E D As a technical note this procedure is only possible because the translation group forms an Abelian subgroup Cartan subalgebra in the Poincare algebra In particular the group multiplication law U a U b U a b should not be taken for granted We also note that Poincare is a connected Lie group It is a group of transformations T 3 that are described by a continuous set of real parameters 3 a displaystyle xi a nbsp The group multiplication law takes the formT 3 T 3 T f 3 3 displaystyle T bar xi T xi T f bar xi xi nbsp Taking 3 a 0 displaystyle xi a 0 nbsp as the coordinates of the identity we must have f a 3 0 f a 0 3 3 a displaystyle f a xi 0 f a 0 xi xi a nbsp The actual operators on the Hilbert space are represented by unitary operators U T 3 In the above notation we suppressed the T we now write U l as U P l For a small neighborhood around the identity the power series representation U T 3 1 i a 3 a t a 1 2 b c 3 b 3 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 nbsp is quite good Suppose that U T 3 form a non projective representation i e U T 3 U T 3 U T f 3 3 displaystyle U T bar xi U T xi U T f bar xi xi nbsp The expansion of f to second power is f a 3 3 3 a 3 a b c f a b c 3 b 3 c displaystyle f a bar xi xi xi a bar xi a sum b c f abc bar xi b xi c nbsp 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 nbsp Since t a b displaystyle t ab nbsp is by definition symmetric in its indices we have the standard Lie algebra 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 nbsp with C the structure constant 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 nbsp This implies that the structure constants vanish and thus the quadratic coefficients in the f expansion vanish as well This means that f is simply additive f abelian a 3 3 3 a 3 a displaystyle f text abelian a bar xi xi xi a bar xi a nbsp and thus for abelian groups U T 3 U T 3 U T 3 3 displaystyle U T bar xi U T xi U T bar xi xi nbsp Q E D Rotations edit The rotation operator also contains a directional derivative The rotation operator for an angle 8 i e by an amount 8 8 about an axis parallel to 8 8 8 displaystyle hat theta boldsymbol theta theta nbsp isU R 8 exp i 8 L displaystyle U R mathbf theta exp i mathbf theta cdot mathbf L nbsp Here L is the vector operator that generates SO 3 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 amp 0 amp 0 0 amp 0 amp 1 0 amp 1 amp 0 end pmatrix mathbf i begin pmatrix 0 amp 0 amp 1 0 amp 0 amp 0 1 amp 0 amp 0 end pmatrix mathbf j begin pmatrix 0 amp 1 amp 0 1 amp 0 amp 0 0 amp 0 amp 0 end pmatrix mathbf k nbsp It may be shown geometrically that an infinitesimal right handed rotation changes the position vector x by x x d 8 x displaystyle mathbf x rightarrow mathbf x delta boldsymbol theta times mathbf x nbsp So we would expect under infinitesimal rotation U R d 8 f x f x d 8 x f x d 8 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 nbsp It follows that U R d 8 1 d 8 x displaystyle U R delta mathbf theta 1 delta mathbf theta times mathbf x cdot nabla nbsp 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 8 exp 8 x displaystyle U R mathbf theta exp mathbf theta times mathbf x cdot nabla nbsp Normal derivative editA normal derivative is a directional derivative taken in the direction normal that is orthogonal to some surface in space or more generally along a normal vector field orthogonal to some hypersurface See for example Neumann boundary condition If the normal direction is denoted by n displaystyle mathbf n nbsp then the normal derivative of a function f is sometimes denoted as f n textstyle frac partial f partial mathbf n nbsp 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 nbsp In the continuum mechanics of solids editSeveral important results in continuum mechanics require the derivatives of vectors with respect to vectors and of tensors with respect to vectors and tensors 13 The directional directive provides a systematic way of finding these derivatives This section is an excerpt from Tensor 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 edit Let f v be a real valued function of the vector v Then the derivative of f v with respect to v or at v is the vector defined through its dot product with any vector u being f v u D f v u d d a f v a u a 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 nbsp for all vectors u The above dot product yields a scalar and if u is a unit vector gives the directional derivative of f at v in the u direction Properties If f v f 1 v f 2 v displaystyle f mathbf v f 1 mathbf v f 2 mathbf v nbsp 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 nbsp If f v f 1 v f 2 v displaystyle f mathbf v f 1 mathbf v f 2 mathbf v nbsp 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 nbsp If f v f 1 f 2 v displaystyle f mathbf v f 1 f 2 mathbf v nbsp 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 nbsp Derivatives of vector valued functions of vectors edit 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 D f v u d d a f v a u a 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 nbsp 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 If f v f 1 v f 2 v displaystyle mathbf f mathbf v mathbf f 1 mathbf v mathbf f 2 mathbf v nbsp 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 nbsp 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 nbsp 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 nbsp If f v f 1 f 2 v displaystyle mathbf f mathbf v mathbf f 1 mathbf f 2 mathbf v nbsp 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 nbsp Derivatives of scalar valued functions of second order tensors edit Let f S displaystyle f boldsymbol S nbsp be a real valued function of the second order tensor S displaystyle boldsymbol S nbsp Then the derivative of f S displaystyle f boldsymbol S nbsp with respect to S displaystyle boldsymbol S nbsp or at S displaystyle boldsymbol S nbsp in the direction T displaystyle boldsymbol T nbsp is the second order tensor defined as f S T D f S T d d a f S a T a 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 nbsp for all second order tensors T displaystyle boldsymbol T nbsp Properties If f S f 1 S f 2 S displaystyle f boldsymbol S f 1 boldsymbol S f 2 boldsymbol S nbsp 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 nbsp If f S f 1 S f 2 S displaystyle f boldsymbol S f 1 boldsymbol S f 2 boldsymbol S nbsp 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 nbsp If f S f 1 f 2 S displaystyle f boldsymbol S f 1 f 2 boldsymbol S nbsp 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 nbsp Derivatives of tensor valued functions of second order tensors edit Let F S displaystyle boldsymbol F boldsymbol S nbsp be a second order tensor valued function of the second order tensor S displaystyle boldsymbol S nbsp Then the derivative of F S displaystyle boldsymbol F boldsymbol S nbsp with respect to S displaystyle boldsymbol S nbsp or at S displaystyle boldsymbol S nbsp in the direction T displaystyle boldsymbol T nbsp is the fourth order tensor defined as F S T D F S T d d a F S a T a 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 nbsp for all second order tensors T displaystyle boldsymbol T nbsp Properties If F S F 1 S F 2 S displaystyle boldsymbol F boldsymbol S boldsymbol F 1 boldsymbol S boldsymbol F 2 boldsymbol S nbsp 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 nbsp 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 nbsp 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 nbsp If F S F 1 F 2 S displaystyle boldsymbol F boldsymbol S boldsymbol F 1 boldsymbol F 2 boldsymbol S nbsp 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 nbsp If f S f 1 F 2 S displaystyle f boldsymbol S f 1 boldsymbol F 2 boldsymbol S nbsp 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 nbsp See also editDel in cylindrical and spherical coordinates Mathematical gradient operator in certain coordinate systems Differential form Expression that may appear after an integral sign Ehresmann connection Differential geometry construct on fiber bundles Frechet derivative Derivative defined on normed spaces Gateaux derivative Generalization of the concept of directional derivative Generalizations of the derivative Fundamental construction of differential calculus Semi differentiability Hadamard derivative Lie derivative A derivative in Differential Geometry Material derivative Time rate of change of some physical quantity of a material element in a velocity field Structure tensor Tensor related to gradients Tensor derivative continuum mechanics Total derivative Type of derivative in mathematicsNotes edit R Wrede M R Spiegel 2010 Advanced Calculus 3rd ed Schaum s Outline Series ISBN 978 0 07 162366 7 The applicability extends to functions over spaces without a metric and to differentiable manifolds such as in general relativity 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 Thomas George B Jr and Finney Ross L 1979 Calculus and Analytic Geometry Addison Wesley Publ Co fifth edition p 593 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 Hughes Hallett Deborah McCallum William G Gleason Andrew M 2012 01 01 Calculus Single and multivariable John wiley p 780 ISBN 9780470888612 OCLC 828768012 Zee A 2013 Einstein gravity in a nutshell Princeton Princeton University Press p 341 ISBN 9780691145587 Weinberg Steven 1999 The quantum theory of fields Reprinted with corr ed Cambridge u a Cambridge Univ Press ISBN 9780521550017 Zee A 2013 Einstein gravity in a nutshell Princeton Princeton University Press ISBN 9780691145587 Cahill Kevin Cahill 2013 Physical mathematics Repr ed Cambridge Cambridge University Press ISBN 978 1107005211 Larson Ron Edwards Bruce H 2010 Calculus of a single variable 9th ed Belmont Brooks Cole ISBN 9780547209982 Shankar R 1994 Principles of quantum mechanics 2nd ed New York Kluwer Academic Plenum p 318 ISBN 9780306447907 J E Marsden and T J R Hughes 2000 Mathematical Foundations of Elasticity Dover References editHildebrand F B 1976 Advanced Calculus for Applications Prentice Hall ISBN 0 13 011189 9 K F Riley M P Hobson S J Bence 2010 Mathematical methods for physics and engineering Cambridge University Press ISBN 978 0 521 86153 3 Shapiro A 1990 On concepts of directional differentiability Journal of Optimization Theory and Applications 66 3 477 487 doi 10 1007 BF00940933 S2CID 120253580 External links edit nbsp Media related to Directional derivative at Wikimedia Commons Directional derivatives at MathWorld Directional derivative at PlanetMath Retrieved from https en wikipedia org w index php title Directional derivative amp oldid 1184280724, wikipedia, wiki, book, books, library,