What is Higher order: Definition and 97 Discussions

In mathematics and computer science, a higher-order function is a function that does at least one of the following:

takes one or more functions as arguments (i.e. procedural parameters),
returns a function as its result.All other functions are first-order functions. In mathematics higher-order functions are also termed operators or functionals. The differential operator in calculus is a common example, since it maps a function to its derivative, also a function. Higher-order functions should not be confused with other uses of the word "functor" throughout mathematics, see Functor (disambiguation).
In the untyped lambda calculus, all functions are higher-order; in a typed lambda calculus, from which most functional programming languages are derived, higher-order functions that take one function as argument are values with types of the form









{\displaystyle (\tau _{1}\to \tau _{2})\to \tau _{3}}

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  1. M

    A Rectangular higher order edge element (finite element method)

    I have solved many finite element problems using nodal based (rectangular element) for higher order. now i am trying to solve electromagnetic problem using vector element (Nedelec or Whitney). I know only triangular edge based element with first order only and not higher order. i am searching...
  2. hagopbul

    I Drawing Direction Fields for Higher Order ODEs

    Hello : Trying to find references on drawing direction fields of higher order differential equation by hand as 1st step then by computer , do you know any reference I can read ( PDF , books ,...) , and hope it is not only some short notes Best regards HB
  3. B

    A Higher order PT terms in molecular parameters

    Hello! Is there a place where I can find the expressions for higher order terms of molecular parameters (in particular the spin rotational parameter, ##\gamma##)? All the papers I find just show them up to second order. Thank you!
  4. C

    A Divergence, Gradient of higher order tensor

    1.) I have the following equation $$\nabla \cdot \left( \mathbf{A} : \nabla_{s}\mathbf{b} \right) - \frac{\partial^2\mathbf{c}}{\partial t^2} = - \nabla \cdot \left( \mathbf{D}^{Transpose} \cdot \nabla \phi \right )$$ Is my index notation correct? $$(A_{ijkl} b_{k,l}),_{j} - c_{i,tt} = -...
  5. T

    Higher order derivatives using the chain rule

    Mentor note: Fixed the LaTeX in the following I have the following statement: \begin{cases} u=x \cos \theta - y\sin \theta \\ v=x\sin \theta + y\cos \theta \end{cases} I wan't to calculate: $$\dfrac{\partial^2}{\partial x^2}$$ My solution for ##\dfrac{\partial^2}{\partial...
  6. bagasme

    I Derivation of a Higher Order Derivative Test

    Hello, In second-order derivative test, the test is inconclusive when ##f''(c)=0##, so we had to generalize to higher-order derivative test. I was wondering how such tests can be generalized and derived? For example, how can I prove that ##f(x)=x^4## have minimum at 0? Bagas
  7. pallab

    A Pauli's spin matrices in higher order

    What are the rules to write Pauli's spin matrices in higher-order matrices (especially in 4x4 matrices)
  8. S

    Is there place for higher order derivatives in mechanics?

    The building of theoretical mechanics can be constructed using only the first and the second derivatives (those of coordinates in case of kinematics: velocity and acceleration and those of energy in case of dynamics: force and gradient thereof). It is obviously unavoidable if one wants to deal...
  9. L

    A Higher Order Derivative Test and Germs

    If ##f'(0) = 0## and ##n## is the smallest natural number such that ##f^{(n)}(0)\neq 0##, then the higher-order derivative test states the following: 1. If ##n## is even and ##f^{(n)}(0)>0##, then ##f## has a local minimum at ##0##. 2. If ##n## is even and ##f^{(n)}(0)<0##, then ##f## has a...
  10. nomadreid

    I Why the emphasis on first-order logic?

    It seems to me that, despite several systems developed for higher-order logics, almost all the attention in Logic is devoted to first-order. I understand that higher-order logics have some drawbacks, such as the compactness theorem and the Löwenheim-Skolem theorems and other such not holding in...
  11. mertcan

    A Higher Order Operator Splitting Method

    Hi, in the link https://www.researchgate.net/profile/Andrew_Sornborger/publication/220662120_Higher-order_operator_splitting_methods_for_deterministic_parabolic_equations/links/568ffaab08aec14fa557b85e/Higher-order-operator-splitting-methods-for-deterministic-parabolic-equations.pdf and equation...
  12. Adgorn

    I Differentials of order 2 or bigger that are equal to 0

    So I've seen in several lectures and explanations the idea that when you have an equation containing a relation between certain expressions ##x## and ##y##, if the expression ##x## approaches 0 (and ##y## is scaled down accordingly) then any power of that expression bigger than 2 (##x^n## where...
  13. L

    I Higher order derivatives rules

    mod: moved from homework Does anyone know why and when this equation holds? I have searched online but cannot find the reason or the rules for the higher order derivatives.
  14. 4

    MHB Higher order derivatives calculation

    Does anyone know why this is true?
  15. F

    I Higher order terms in perturbation theory (QFT)

    I'm fairly new to QFT and I'm currently trying to understand perturbation theory on this context. As I understand it, when one does a perturbative expansion of the S-matrix and subsequently calculates the transition amplitude between two asymptotic states, each order in the perturbative...
  16. M

    Runge Kutta to solve higher order ODE

    Homework Statement Edit* should say F'(0) = F(0) = 0 Homework Equations I know that I typically need 3 equations for a 3rd order ODE, does this apply if the is no F'? In the picture above are the equations I came up with, am I on the right trail? Lastly I am familiar with RK4, however I have...
  17. T

    A Backward finite differences on higher order derivative

    I am trying to solve a system of equations and have a question regarding the validity of my approach when implementing a fifth-order Cash-Karp Runge-Kutta (CKRK) embedded method with the method of lines. To give the questions some context, let me state the problem I am attempting to solve: $$...
  18. G

    I Solving complex formulas and higher order polynomials

    How would you go about solving (4(x^3)+38(x^2)+44x-20)/(20+12x+x^2), without the use of a computer, further, what about functions which have more x components, with higher powers. Also what process do computers use to solve these.
  19. T

    What is the (higher order) time derivative of centripetal acceleration?

    Just using basic dimensional analysis, it appears the time derivative of centripetal acceleration is ## \vec{r} \omega^3 ##, but this intuitive guess would also extend to higher order time derivatives, no? Implying: ## \frac {d^n \vec{r}}{dt^n} = \vec{r} \omega^n ## It seems to follow from the...
  20. M

    Higher Order Linear Homogenous Differential Equation

    Homework Statement Hi, basically I have a boundary value problem and just want to check that my general solution is correct. x'''' + 16x = 0 Homework EquationsThe Attempt at a Solution I'm pretty sure you make a characteristic equation which would be m4 + 16 = 0. Solving this I get m to be...
  21. R

    Higher order diff.eq undetermined coefficient

    What to do here if D=d/dx (D2+2D+4)y= x2e2x ? How to find particular integral by method of undetermined coefficient? if R.H.S would have been x2+ e2x then we could have taken yp= Ax2+ Bx+C + De2x but here in product, what to do?
  22. R

    Higher order differential equation

    Solve y'' - 2y' + 2y = ## e^x tanx ## What concept should we use if we know only solving first order differential equation?
  23. Destroxia

    Higher Order Differential Equation

    Homework Statement ##y^{(4)} + y = 0, y(0)=0, y'(0)=0,y''(0)=-1,y'''(0)=0## My issue with this equation is not with the steps, I don't believe but the solving of the IVP, the derivatives of my solution end up being close to 32 terms long, and I was wondering if there is any shorter method I...
  24. S

    Can one define higher order infinitesimals?

    Say ##A##, ##B##, ##C##,... are finite numbers; real, complex, quarternians, tensors, or what have you. "First Order" infinitesimals are finite variables prepended with the letter ##d##. Infinitesimal of any order, are prepended with ##d^n## where ##n## is the infinitesimal "order". Finite...
  25. Bleakfacade

    Are hyperbolic sines and cosines orthogonal in solving higher order PDE's?

    Hello there! So here's my problem, while you solve the Euler Bernoulli beam Equation by separation of variables, how do I have to prove the separated function of space are orthogonal? If so, are hyperbolic sines and cosines orthogonal when you have a product or a linear combination of them...
  26. M

    Solving Higher Order ODEs: y''''''+y'''=t

    Homework Statement y''''''+y'''=t Homework EquationsThe Attempt at a Solution I got all the roots and solved the homo eq. Then I tried to guess the partial eq and got At+B However, I don't know how to proceed because the 6th derivative or the 3rd would be 0.
  27. O

    How to reduce higher order partial differential equations

    Hello guys, I have the system of PDE below and I want to solve it using finite difference method but I think I have to reduce it first to a system of first order PDE. The problem is that I don't know how to reduce this PDE to a first order system. I will appreciate any hints in this regard...
  28. D

    High order differential equations: undetermined coefficients

    Homework Statement If the method of undetermined coefficients is used to find a particular solution yp (t) to the differential equation y'''-y'=te^(-t)+2cos(t) should have the form: ?Homework EquationsThe Attempt at a Solution LHS r^3-r=0 roots= 0, 1 y_c(t)=c_1e^tRHS te^(-t)+2cos(t)...
  29. alyafey22

    MHB Higher order reflected logarithms

    Let us define the following $$I(n,m) = \int^1_0 \log^n(x)\log^m(1-x)\,dx$$ Our purpose is finding a closed form for the general case. Note: for a given n and m the above formula can be deduced by succesive differentiation of the beta representation $$B(p,q) = \int^1_0 x^{p-1}...
  30. J

    On derivatives of higher order

    let's assume that f(t) is a real function on [a,b], n is a positive integer (n-1)th derivative of f is continuous on [a,b], (n)th derivative exists for all t in (a,b) 1. Then can we say that (n-2)th derivative of f is continuous on [a,b]? 2. (n-2)th derivative of f is defined on a and b?
  31. gfd43tg

    Solving higher order ODE as system of first order

    For this problem, I am stuck on the actual system. I don't see what substitution I can make, and the fact that ##u(v)## is a piece-wise function is tripping me up. How the heck do I approach this?? This doesn't look like a standard problem at all.
  32. E

    Shear deformation higher order plate theroies - variational consistent

    What is the meaning of... 1. Governing differential equations obtained are consistent with the assumed displacement field (If for a particular theory, governing differential equations are derived using variational approach). 2. Obtained boundary conditions are consistent with the governing...
  33. J

    Higher Order Diffraction: Why Do Colors Appear Different and Overlap?

    Hi everybody ,Why arrangement of colors in higher order of diffraction is different from order 1 , 2 ? Thanks
  34. J

    Higher order D.E. to linear system of 1st order D.E.'s

    Homework Statement \textbf{(a)} This is an exercise from a course on numerical analysis. Write the system of differential equations u''' = x^2uu'' - uv' v'' = xvv' + 4u' as a first order system of differential equations, \textbf{y'} = \textbf{y}(x,\textbf{y}). \textbf{(b)} Determine...
  35. N

    Higher order derivatives with help of Taylor expansion?

    Homework Statement Function f(x) = x^2/(x-1) should be expanded by Taylor method around point x=2 and 17th order derivative at that point should be calculated. Homework Equations Taylor formula: f(x)=f(x0)+f'(x0)*(x-x0)+f''(x0)*(x-x0)^2+... The Attempt at a Solution I...
  36. DreamWeaver

    MHB Evaluations of the higher order Polygamma functions

    In this brief tutorial we evaluate the Trigamma, Tetragamma, and other higher order Polygamma functions at small rational arguments:(01) \quad \psi_{m \ge 1}(z) = (-1)^{m+1}m!\, \sum_{k=0}^{\infty} \frac{1}{(k+z)^{m+1}}We will have frequent need of the reflection formula, which is obtained by...
  37. Y

    Multivariate Higher Order Derivatives

    Homework Statement Let h(u,v) = f(u+v, u-v). Show that f_{xx} - f_{yy} = h_{uv} and f_{xx} + f_{yy} = \frac12(h_{uu}+h_{vv}) . Homework Equations The Attempt at a Solution I'm always confused on how to tackle these types of questions because there isn't an actual function to...
  38. A

    Higher Order Differential Equation: Substitution

    Homework Statement Solve x^{2}\times y'' - 4 \times x \times y' + 6 \times y = 0 for y(x) by first using the substitution v = ln(x) to obtain an equation involving y, dy/dv, d^2y/dv^2 and no x. Solve for y(v), then return to y(x). Homework Equations NA The Attempt at a Solution I know how...
  39. K

    Derivatives of a higher order - Satisfying the equation

    Homework Statement Show that y= xex satisfi es A(d2y)/dx2 + B(dy/dx) + Cy = 0 for suitably chosen values of the constants A, B, and C. Homework Equations Y=xex The Attempt at a Solution Please see the attachment. I get to a point where I need to find the value of A, B...
  40. S

    Multivariable Chain rule for higher order derivatives

    Hello, Given is the function f = f(a,b,t), where a=a(b) and b = b(t). Need to express first and second order derivatives. \frac{\partial f}{\partial a} and \frac{\partial f}{\partial b} should be just that, nothing more to it here, correct? But \frac{df}{dt} = \frac{\partial...
  41. C

    Method of Undetermined Coefficients for Higher Order Linear Equations

    Hi, I'd just like to have a quick clarification with regards to the method of undetermined coefficients. I know that if a characteristic equation has the form (r-4)3 = 0 then the characteristic solution will be yc = e4t + te4t + t2e4t + t3e4t and the particular solution ought to be...
  42. V

    Higher order time derivatives of position

    Newton's laws says ## F=ma ##. Which, as far as I can see, states that all physical interactions concern the second time derivative of position. And because there is no other way for two bodies to interact in the physical world, the "worst" I can do to a system is change its acceleration, right...
  43. R

    Solution for higher order wave ODE

    Hi guys, Here is an equation that I have tried for few days to solve and still haven't succeeded, I'm interested to solve this 4th order wave equation to find u(x). ∫∫(A u(x) + B u(x)2 + C u(x)3 +D u''(x)) dx dx=0 the 4th term is second derivative of displacement u(x). I assume...
  44. R

    Is there any meaning to higher order derivatives?

    We know that the first derivative represents the slope of the tangent line to a curve at any particular point. We know that the second derivative represents the concavity of the curve. Or, the first derivative represents the rate of change of a function, and the second derivative represents the...
  45. C

    Higher Order Partial Derivatives and Clairaut's Theorem

    Homework Statement general course question Homework Equations N/A The Attempt at a Solution fx is a first order partial derivative fxy is a second order partial derivative fxyz is a third order partial derivative I understand that Clairaut's Theorem applies to second order...
  46. B

    Linear approximation higher order terms

    My questions are from lecture 9, MIT OCW SV Calculus, Jerison, 2009; At 27:50 he is deriving the linear approximation for the function e^(-3x)(1+x)^(-1/2)≈(1-3x)(1-1/2x)≈1-3x-1/2x+3/2x^2≈1-7/2x, for x near 0. In the last step he drops the x squared term since it is negligible(no questions so...
  47. S

    Compute Higher Order Mixed Derivative.

    Homework Statement Let f(u,v) be an infinitely differentiable function of two variables, and let g(x,y) = (x^2 + y^4, xy). If f_{v} (5,2) = 1, f_{uu} (5,2) = 2, f_{vv} (5,2) = -2 and f_{uv} (5,2) = 1, computer d^2(f o g)/dxdy at (2,1)Homework Equations The Chain Rule The Attempt at a...
  48. T

    Factoring for Higher order ODE

    Solve the differential equation: y(5)+12y(4)+104y(3)+408y''+564y'=0 where the (n) is the nth derivative. So it's a 5th order DE. Now I'm trying to find the roots: One of the roots is r = 0, which I obtain by factoring the equation into this form: r(r4+12r3+104r2+408r+1156) = 0...
  49. R

    Determining wheter or not a non trivial solutions exists for higher order PDE's

    Is there a way to determine if a non trivial solution exists for higher order PDE's? For example suppose I have the following. X''''(x) + \alpha^2X(x)=0 With given conditions U(0,t) = u(1,t) = uxx(0,t) = uxx(1,t) = 0 if t≥0 The general solution will have 4 constants of which I will have...
  50. A

    Higher order DE to State space

    Hey, I've been trying to run a few simulations in Matlab using ODE45. This algorithm requires a function which gives the first order differential as an output i.e a state space format (Correct me if I'm wrong here). If its a normal N order differential such as d2x/dt2 + dx/dt -1 =0 . dx/dt...