What is Time derivative: Definition and 63 Discussions

A time derivative is a derivative of a function with respect to time, usually interpreted as the rate of change of the value of the function. The variable denoting time is usually written as


{\displaystyle t}

View More On Wikipedia.org
  1. S

    Understanding how time derivative = acceleration

    I'm having a hard time understanding some concepts and would really appreciate some help(not super smart so I need some things basically dumbed down). In my physics lab we're going over Newton's Second Law. There's a statement in the lab papers I don't understand. It states "As you should know...
  2. George Keeling

    I Time derivative of the moment of inertia tensor

    I am completely stuck on problem 2.45 of Blennow's book Mathematical Models for Physics and Engineering. @Orodruin It says "We just stated that the moment of inertia tensor ##I_{ij}## satisfies the relation$${\dot{I}}_{ij}\omega_j=\varepsilon_{ijk}\omega_jI_{kl}\omega_l$$Show that this relation...
  3. L

    I The time derivative of kinetic energy

    Lets consider T(\vec{p})=\frac{\vec{p}^2}{2m}=\frac{\vec{p}\cdot \vec{p}}{2m}. Then \frac{dT}{dt}=\vec{v}\cdot \vec{F}. And if we consider T=\frac{p^2}{2m} than \frac{dT}{dt}=\frac{1}{2m}2p\frac{dp}{dt} Could I see from that somehow that this is \vec{v}\cdot \vec{F}?
  4. M

    I Time derivative using the quotient rule

    Hi Guys Sorry for the rudimentary post. I am busy with a numerical solution to a mechanics problem, and the results are just not as expected. I am re-checking the mathematics to ensure that all is in order in doing so I am second guessing a few things Referring to the attached scan, is the...
  5. A

    I Time derivative of the angular momentum as a cross product

    I am trying to find the equations of motion of the angular momentum ##\boldsymbol L## for a system consisting of a particle of mass ##m## and magnetic moment ##\boldsymbol{\mu} \equiv \gamma \boldsymbol{L}## in a magnetic field ##\boldsymbol B##. The Hamiltonian of the system is therefore...
  6. A

    Von Neumann Entropy time derivative(evolution)

    I'm not sure about my proof. So please check my step. I used log as a natural log(ln). Specially, I'm not sure about "d/dt=dρ/dt d/dρ=i/ħ [ρ, H] d/dρ" in the second line. and matrix can differentiate the other matrix? (d/dρ (dρ lnρ))
  7. Arman777

    I Taking the partial time derivative of a functional

    Let us suppose we have a functional of f such that ##f=f((\vec{r}(t),t)## where ##\vec{r}(t) = a(t)\vec{x}(t)##. I am trying to derive an equation such that $$\left.\frac{\partial}{\partial t}\right|_r = \left.\frac{\partial }{\partial t}\right|_x + \left.\frac{\partial \vec{x}}{\partial...
  8. M

    Dyson's series and the time derivative

    I'm having a hard time understanding how exactly to evaluate the expression} $$\partial_t \mathcal{T}\exp\left(-i S(t)\right)\quad \text{where}\quad S(t)\equiv\int_{t_0}^tdu \,H(u) .$$ The confusing part for me is that if we can consider the following: $$\partial_t \mathcal{T}\exp\left(-i...
  9. V

    Time derivative jump of the electric/magnetic field

    So I just wanted to see if anyone could offer some suggestions. So in my mind this seems impossible, in the case of electric field a jump in time derivative of that field would indicated in my mind that electric charge was either introduced or removed from the system instantaneously which would...
  10. maistral

    I Verification regarding Neumann conditions at time derivative

    Hi, just a question regarding neumann conditions, I seem to have forgotten these things already. I think this question is answerable by a yes or a no. So given the 2D heat equation, If I assign a neumann condition at say, x = 0; Does it still follow that at the derivative of t, the...
  11. A

    Trying to calculate the time derivative of a position differential

    here I am trying to find ##\frac{d}{dt}dx## where ##x(t)## is the position vector Now ##\frac{d}{dt}(v_x(x,y,z,t)dt)=\frac{dv_x}{dt}dt=\frac{\partial v_x}{\partial t}dt+\frac{\partial v_x}{\partial x}dx+\frac{\partial v_x}{\partial y}dy+\frac{\partial v_x}{\partial z}dz## Now dividing by ##dx##...
  12. E

    I The partial time derivative of Hamiltonian vs Lagrangian

    I have been reading a book on classical theoretical physics and it claims: -------------- If a Lagrange function depends on a continuous parameter ##\lambda##, then also the generalized momentum ##p_i = \frac{\partial L}{\partial\dot{q}_i}## depends on ##\lambda##, also the velocity...
  13. Matt Chu

    Time Derivative of Expectation Value of Position

    Homework Statement I want to prove that ##\frac{\partial \langle x \rangle}{\partial t} = \frac{\langle p_x \rangle}{m}##. Homework Equations $$i\hbar \frac{\partial \Psi}{\partial t} = -\frac{\hbar^2}{2m} \frac{\partial^2 \Psi}{\partial x^2} + V \Psi$$ The Attempt at a Solution [/B] So...
  14. O

    Time derivative of gravity due to acceleration

    Homework Statement We have the equation for gravity due to the acceleration a = -GM/r2, calculate velocity and position dependent on time and show that v/x = √2GM/r03⋅(r/r0-1) Homework Equations x(t = 0) = x0 and v(t = 0) = 0 The Attempt at a Solution v = -GM∫1/r2 dt v = dr/dt v2 = -GM∫1/r2...
  15. Marcus95

    Time Derivative of Rank 2 Tensor Determinant

    Homework Statement Show that for a second order cartesian tensor A, assumed invertible and dependent on t, the following holds: ## \frac{d}{dt} det(A) = det(a) Tr(A^{-1}\frac{dA}{dt}) ## Homework Equations ## det(a) = \frac{1}{6} \epsilon_{ijk} \epsilon_{lmn} A_{il}A_{jm}A_{kn} ## The...
  16. S

    Calculating the time derivative of <p>

    Homework Statement Problem 1.7 in Griffiths "Quantum Mechanics" asks to prove $$\frac{d\left \langle p \right \rangle}{dt}=\left \langle -\frac{\partial V}{\partial x} \right \rangle$$ Homework Equations Schrödinger equation The Attempt at a Solution I was able to arrive at the correct...
  17. B

    I Force as a Time Derivative of ihk

    If energy is ihw and p is ihk, can force be written as derivatives of these? Might the fundamental forces just be some patterned change in the change of the wave functions of Dirac's equation? Edit: the title should be "Time derivative of ihk" but I can't edit the title.
  18. P

    How to take the time derivative of a potential gradient ?

    I am not that great at vector calculus , etc. Can someone show me how to take the time rate of change of a potential gradient? (Not homework) Thx.
  19. N

    Discontinuities in the time derivative of the magnetic field

    An inductor and resistor are arranged in parallel to a constant voltage source. There is a switch connected to a terminal on the inductor that can create a closed loop that includes either the voltage source, or the resistor. The switch is left connecting the source and inductor for a long...
  20. C

    Time derivative of a time-dependent vector and scalar

    Homework Statement ## \frac{d}{dt}\gamma(t)\vec{u(t)} ## Homework Equations See above The Attempt at a Solution This comes from trying to verify a claim in Chapter 12 of Griffiths Electrodynamics, 4th. edition (specifically Eq. 12.62 -> Eq. 12.63, if anyone has it on hand). I would have...
  21. PhysicsKid0123

    Time Derivative of Unit Vectors

    Quick question (a little rusty on this): Why don't unit vectors in Cartesian Coordinates not change with time? For example, suppose \mathbf{r} (t) = x(t) \mathbf{x} + y(t) \mathbf{y} + z(t) \mathbf{z} How exactly do we know that the unit vectors don't change with time? Or in other words...
  22. 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...
  23. G

    Time derivative of tensor expression

    I was trying to compute the time derivative of the following expression: \mathbf{p_k} = \sum_i e_{ki}\sum_{n=0}^{\infty} \frac{(-1)^n}{(n+1)!} \mathbf{r_{ki}}(\mathbf{r_{ki}\cdot \nabla})^n \delta(\mathbf{R_k}-\mathbf{R}) I am following deGroot in his Foundations of Electrodynamics. He says...
  24. E

    Time derivative of 3D Spherical Coordinate

    When we obtain the velocity vector for position vector (r, θ, φ) Why do we take the time derivative of the radial part in the 3D Spherical Coordinate system only? Don't we need to consider the polar angle and azimuthal angle part like (dr/dt, dθ/dt, dφ/dt)?
  25. L

    Can I pull a time derivative outside of a curl?

    Homework Statement For the equation ∇ x E = -∂B/∂t I took the curl of both sides to get ∇ x (∇ x E) = ∇ x -∂B/∂t I feel like it'd be very wrong to pull out the time derivative. Am I correct?
  26. B

    Differentiate time derivative w/ respect to generalized var.

    Homework Statement Solve ∂v/∂θ and ∂v/∂r. (refer to attached image for equations) Homework Equations Refer to attached image. note that the velocity is expressed in cylindrical coordinates and attention must be paid to the directional unit vectors eθ and eρ.[/B] The Attempt at a Solution...
  27. N

    Taking the time derivative of a curl

    Is the time derivative of a curl commutative? I think I may have answered this question... Only the partial time derivative of a curl is commutative? The total time derivative is not, since for example in cartesian coordinates, x,y,and z can themselves be functions of time. In spherical and...
  28. C

    Kinetic Energy Time Derivative

    Homework Statement So the first part asks to prove the time derivative of kinetic energy is dT/dt=F dot product v which I did not problem. but then the second part of the problem asks to prove that if the mass is changing with time then the time derivative of d(mT)/dt=F dot product m and I'm...
  29. P

    Can the Time Derivative of a Vector Change if its Magnitude or Angle Decreases?

    In chapter 1 of the book "Introduction to Mechanics" by Kleppner and Kolenkow, the derivative of a generic vector ##\vec{A}## is discussed in terms of decomposing an increment in ##\vec{A}##, ##Δ\vec{A}##, into two perpendicular vector vectors; one parallel to ##\vec{A}## and the other...
  30. B

    How Do You Calculate the Time Derivative of a Non-Constant Vector?

    Homework Statement I have somewhat general question about time derivative of a vector. If we have r=at2+b3 it's easy to find instantaneous acceleration and velocity(derivative with respect to dt) v=2at+3bt2 a=2a+6bt But consider this position vector r=b(at-t2) where b is constant vector and a...
  31. A

    What if Newton's laws were shifted by one time derivative?

    What would be some important properties of a universe where Force = Mass * Jerk and objects stay in constant acceleration until acted upon by a net force? (if we ignore the fact that objects would reach the speed of light, and just deal with classical mechanics)
  32. L

    Time derivative of Hubble parameter

    Is rather a question of calculus skills, but how do I get the time derivative of the Hubble parameter here in [1]? Is it the Leibnitz rule, the chain rule, some clever re-arrangement? thank you
  33. M

    Is the energy operator (time derivative) a linear one?

    Typically in mathematics time derivative is linear in the sense that constants are pulled out the operator which then operates on a time dependent function. But in quantum mechanics we say linear to mean that the operator passes over the coefficients of the kets (which themselves might be time...
  34. D

    Time derivative of vector potential - indices help

    Hey guys, So I'm reading something about vector potentials and I've come across this one line which is really annyoing me. Here's how it goes \frac{d}{dt}\mathbf{A}=\frac{\partial \mathbf{A}}{\partial t}+\frac{\partial \mathbf{r}}{\partial t}\cdot \frac{\partial }{\partial...
  35. J

    Sign of the time derivative of the Majorana Lagrangian

    Let \gamma^{\rho} \in M_{4}(\mathbb{R}) be the Majorana representation of the Dirac algebra (in spacetime signature \eta_{00} = -1), and consider the Majorana Lagrangian \mathcal{L} = \mathrm{i} \theta^{\mathrm{T}} \gamma^{0} (\gamma^{\rho} \partial_{\rho} - m) \theta, where \theta is a...
  36. P

    Time derivative of creation/annhilation operators

    Basically is it possible to take a time derivative of a creation/annhilation operator?
  37. F

    Understanding Dirac Delta Function: Time Derivative & Hankel Transformation

    Hi All, I have a problem in understanding the concept of dirac delta function. Let say I have a function, q(r,z,t) and its defined as q(r,z,t)= δ(t)Q(r,z), where δ(t) is dirac delta function and Q(r,z) is just the spatial distribution. My question are: 1. How can I find the time derivative...
  38. O

    Landau Lifshitz - Total time derivative of the Lagrangian

    On page 13 in Landau-Lifgarbagez Mechanics, the total time time derivative of the Lagrangian of a closed system is given to be, \frac{d L}{d t} = \sum_i \frac{\partial L}{\partial q_i} \dot{q_i} + \sum_i \frac{\partial L}{\partial \dot{q_i}} \ddot{q_i} Why does this stop here? I mean, why...
  39. S

    Time derivative of an observable

    According to my book: \frac{d}{dt} \langle Q \rangle = \frac{i}{\hbar} \langle [\hat{H}, \hat{Q}] \rangle + \langle \frac{\partial \hat{Q}}{\partial t} \rangle . No derivation for this is given. How can derive you this?
  40. A

    Finding the time derivative of a trigonometric function

    Homework Statement Finding the time derivative of sin^2( \alpha ), knowing that \dot \alpha ≠ 0 Homework Equations i know that \frac {d}{dt} sin ( \alpha ) = \dot \alpha cos ( \alpha) The Attempt at a Solution That should give \dot \alpha ^2 cos^2( \alpha ) But it's...
  41. K

    Commutation of Curl and the partial time derivative?

    I am curious if there are any issue with commuting the curl of a vector with the partial time derivative? For example if we take Faraday's law: Curl(E)-dB/dt=0 And I take the curl of both sides: Curl(Curl(E))-Curl(dB/dt)=0 Is Curl(dB/dt)=d/dt(Curl(B)) I assume this is only...
  42. W

    Time derivative of rotating vector

    Trying to teach myself physics and I've run into a problem I don't quite understand. "The magnitude of dA/dt can be found by the following geometrical argument. The change of A in the time interval t to Δt is" ΔA = A(t + Δt) - A(t) And then somehow it gets to |ΔA| =...
  43. G

    Time derivative of schrodinger equation

    Why is the TDSE first derivative in time. Now I know that it is required so that the wave functions are complex... but is there any physical interpretation for this requirment??
  44. pellman

    Time derivative of a Schrodinger-picture operator?

    Given an operator Q, how do we derive the relationship \frac{dQ}{dt}=i\left[H,Q \right]+\frac{\partial Q}{\partial t} ? I had thought that this was only true in the Heisenberg picture. But Greiner has it here (eq 8.19) for an operator in the Schrodinger picture. No need to show...
  45. pellman

    Time derivative in the Heisenberg picture?

    On the Wikipedia page for http://en.wikipedia.org/wiki/Heisenberg_picture#Mathematical_details" we find this relation \frac{d}{dt}A(t)=\frac{i}{\hbar}[H,A(t)]+\left(\frac{\partial A}{\partial t}\right) I don't understand what the distinction between \frac{d}{dt}A(t) and...
  46. S

    Calculating time derivative of Magnetic force

    Hi all, I ran into a bit of an issue trying to figure out how to properly differentiate the magnetic force due to particle interactions. To be specific, I'm actually looking for the time derivative of acceleration (jerk) due to the magnetic force, but it's essentially the same problem. For...
  47. G

    Time derivative of electric field? Electromagnetic radiation energy emitted

    Homework Statement Electromagnetic radiation is emitted by accelerating charges. The rate at which energy is emitted from an accelerating charge that has charge q and acceleration a is given by: \frac{dE}{dt} = \frac{q^{2}a^{2}}{6\pi\epsilon_{0}c^{3}} where c is the speed of light...
  48. T

    Time derivative of momentum expectation?

    Hello, I am trying to learn about some basic quantum mechanics. http://farside.ph.utexas.edu/teaching/qmech/lectures/node35.html this website shows that the time derivative of the momentum expectation d<p>/dt = -<dV/dx> The part that i am not getting is how the writer goes from the...
  49. D

    Time Derivative: How Does 2x Differ from x² Differ?

    I'm wondering, how does 2 multiplied by the first and second time derivatives of x equal the time derivative of the time derivative of x squared. Thanks.
  50. I

    Coulomb Gauge Fixing: Adding Gradient & Subtracting Partial Time Derivative

    Coulomb gauge fixes gauge by setting div(A)=0. What has it to do with adding a gradient to A and subtract a partial time derivative from V?