Questions tagged [differentiation]

Differentiation is the set of techniques and results from Differential Calculus, concerning the calculation of derivatives of functions or distributions.

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Partial derivatives and Galilean transforms

I have trouble understanding the following: \begin{equation} \frac{\partial}{\partial x'} = \frac {\partial x}{\partial x'} \frac{\partial}{\partial x} + \frac{\partial t}{\partial x'} \frac{\partial}...
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Partial derivatives in a $PVT$ system

So, I have been studying thermodynamics for a while and some mathematical steps involving partial derivatives have now started to hurt my head. First of all, I understand that whenever we take a $PVT$ ...
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An identity between the d'Alembertian and the covariant derivative

Suppose $f$ is function which depends on $\phi$, $f = f(\phi)$; and $\phi$ is a scalar field. We define $$\square \equiv g^{\mu\nu} \nabla_\mu \nabla_\nu$$ and $$\nabla_\mu f(\phi) \equiv f_{;\nu}$$ ...
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Covariant derivatives in a rank 2 tensor

I was trying to prove that for any second order tensor: $$A^{\mu\nu}_{;\mu\nu}=A^{\mu\nu}_{;\nu\mu}$$ considering the torsion free property and locally flat coordinates. Considering the point where ...
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Is $ d \mathbf v · d \mathbf v = d \mathit v^2 $?

My teacher has proved the following: $$ \mathit v^2 = \mathbf v·\mathbf v = \frac{d\mathbf r}{dt}·\frac{d\mathbf r}{dt} = \left(\frac {ds}{dt}\right)^2 \Rightarrow \mathit v = \frac{ds}{dt} $$ Because ...
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What is $D$ or $D$-with-a-slash-through-it in the Standard Model equation(s)?

In the mathematical formulation of the Standard Model, which I do not understand yet, there is a capital letter $D$ or $D$-with-a-slash-through-it that I can't find an explanation for. Flip Tanedo (a ...
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Differential form of Gauss's law from Coulomb's law in spherical coordinates [duplicate]

Coulomb's law for the static electric field of a point charge is given by $$\overrightarrow{E}=\frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{r}$$ Now if we take the divergence of both sides of the above ...
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What does “Just before” and “Just after” really mean in physics problems?

So I'm stuck in a dynamics problem that asks what is the acceleration of a body just after A, where A is the point that separates the motion of the body from a curvilinear path to projectile motion. ...
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What does $d$ stand for in this formula?

Context: I am building a tennis ball machine and am having trouble interpreting the following formula for the flight path of the ball. I know all of the other values in the formula but the source I am ...
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Confusion about successive derivatives of position in circular motion

Suppose we define a unit vector $\vec r$ along radial direction for a particle in uniform circular motion at an angular frequency $\omega$. Then we can write: $$\vec r = \cos(\omega t)\hat i + \sin(\...
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What is meant by a partial derivative of a ket?

In my QM book I often see partial derivatives mixed with kets, like $$ \frac{\partial}{\partial i} |\psi \rangle $$ where $i \in {x, y, z}$. Here I'm assuming that $| \psi \rangle \in \mathbb{C}^n$ ...
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How to show functional derivative as a limit of ordinary derivative?

I found this footnote in the appendix (on path integral page 333) of J. Polchinski’s string theory book. can you explain this?
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Why isn't $g^{\mu \nu}\partial_{\nu}$ equal to $\partial^{\mu}$ in General Relativity?

The question is pretty short. I have been told that unlike for flat spacetime, in curved spacetime we cannot write $g^{\mu \nu}\partial_{\nu}f=\partial^{\mu}f$ With $f$ an scalar function, but I don't ...
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Goldstein's generalized forces

In Goldstein's $$ Q_{j}=\sum_{i} \mathbf{F}_{i} \cdot \frac{\partial \mathbf{r}_{i}}{\partial q_{j}}=-\sum_{i} \nabla_{i} V \cdot \frac{\partial \mathbf{r}_{i}}{\partial q_{j}} $$ which is exactly ...
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Calculating equation of motion in gauge theories: using ordinary derivatives or covariant derivatives?

For general gauge theories, the total Lagrangian density is given as $$L=-\frac{1}{4}F^2+L_M(\psi, D\psi)$$ where $L_M(\psi, D\psi)$ is the matter field with the ordinary derivative replaced by the ...
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Is double derivative of a tensor component with respect to time itself a tensor?

Can you please clear my doubt that if we take double derivative of a tensor component with respect to time, will the resulting quantity be a tensor or not?
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Geometric and mathematical interpretation about a derivative of $H$ and related questions

I would like to check whether my analogy is right and ask some related questions. So, I am studying Thermodynamics right now and there are many calculus techniques are used for the thermodynamic ...
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Navier Stokes: $(u⋅∇)u$ vs $u⋅∇u$

I can find this term stated both ways in different literature. Are they equivalent? It's weird because the dot is a dot product in (u⋅∇), but ∇u being a gradient of a vector field, would (presumably) ...
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1answer
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Derivatives in the Lorentz Transformation

I am trying to better understand the Lorentz Transformation on a fundamental level and gain some intuition of it. In the Lorentz Transformation, the derivative of x' with respect to x must be a ...
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Explicit expression of gradient, laplacian, divergence and curl using covariant derivatives

For my course in General Relativity I am given the problem to find the expressions for the gradient, laplacian, divergence and curl in spherical coordinates using covariant derivatives. I have tried ...
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How do total time derivatives of partial derivatives of functions work?

Say im trying to prove $\frac{\partial \dot{T}}{\partial \dot{q}^i} - 2\frac{\partial {T}}{\partial {q^i}} = - \frac{\partial {V}}{\partial {q^i}}$ from the Lagrangian equation: $L = T - V$, and the ...
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Multivariable chain rule in classical mechanics; example of physical system [closed]

I'm a teaching assistant in calculus and my students who are studying mechanical engineering asked me to explain the multivariable chain rule. So I thought it could be fun if I could give an example ...
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Computing derivatives “at constant” quantities in thermodynamics

What does it mean in thermodynamics when a derivative is computed "at constant $X$"? If I see $\left.\frac{\partial S(E, N)}{\partial E}\middle| \right._N$ how is the derivation performed ...
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Interpretation of Variation Notes

I would like an explanation to how this Lagragian partial derivative was taken (eq. 3). This probably is more suited for the math Stack Exchange, however this is for a physics course which is why I am ...
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What is the theoretical mathematical justification for differential arithmetic? [migrated]

Throughout undergraduate physics textbooks, you will see informal math with differentials where elements like $dx$ and $dy$ are multiplied around like scalar constants, and differentiation in terms of ...
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Definition of parallel transport

The definition of parallel transport is $t^iD_i u^j=0$, where $\vec{t}$ is the tangent vector to the curve and $\vec{u}$ is the vector being parallel transported along the curve. In flat space, using ...
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Relation between time-derivatives observed from fixed and rotating frames

I will begin by stating the question and then I will explain my doubt. The relation between time-derivatives of a vector $\vec{u}$ observed from fixed and rotating frames (with a common origin) is $$ \...
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How a 'variation' $\delta x$ of an independent parameter differs from $dx$? [closed]

I have been reading the Classical Field theory part from The Quantum field theory book of Lewis H Ryder. After defining classical field $\phi(x^\mu)$ he says something about adding variations on both ...
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Alternative formula for the affine connection in a new coordinate basis

In Hobsons's General Relativity: An Introduction for Physicists, pg. 64, he gave two different expressions for the affine connection $\Gamma'^a_{bc}$ in a transformed coordinate basis $x'^a$ (the ...
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Isn't the following addition wrong on manifold as done in Frankel book?

In ch-$4$ when calculating expression of Lie derivative using Hadamard's Lemma before $(4.4)$ Frankel's do following manipulation: $$\lim_{t\rightarrow0}\frac{\textbf{Y}_{\phi_tx}(f)-\textbf{Y}_x(f)}{...
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What does $\overset\leftrightarrow{\partial_{\mu}}$ means?

I have a scalar complex field: $\phi(x) = \phi_{1} + i \phi_{2}\;$ so $\;\phi^{*}(x) = \phi_{1} - i \phi_{2}$ where $\phi_{1}, \; \phi_{2}$ are real scalar fields. Then I have something like $\;\phi^{...
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What motivates defining vectors as first order differential operators?

I have read some introductions to geometrical ideas and tensors and physics and what some of them do (see, for example, Frankel's Geometry of Physics) is define a vector as a first order differential ...
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Covariant derivative in $f(Q,T)$ field equations

I am trying to evaluate the field equations in $f(Q,T)$-gravity and I'm not sure how to evaluate this term: $$-\dfrac{-2}{\sqrt{-g}}\nabla_{\alpha}(f_{Q} \sqrt{-g}\enspace P^{\alpha}_{\enspace \mu \nu}...
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How is d'Alembertian operator is defined in differential geometry?

Which general formula for the box operator is correct, $\Box=g^{ij}\partial_i\partial_j$ or $\Box=\frac{1}{\sqrt{g}}\partial_i(\sqrt{g}g^{ij}\partial_j)$? I have seen both the definition being used ...
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How do I show that $\dfrac{dE}{dt} = \dfrac{\partial U}{\partial t}$ where $U(\mathbf{r}_1,…,\mathbf{r}_N,t)$ is the potential energy?

I'm working through Chapter 1 of Analytical Mechanics for Relativity and Quantum Mechanics, and in Section 1.8, they derive the equality in the question. To show this, they claim that $$\dfrac{dT}{dt} ...
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Feynman lectures on physics vol 1: Speed as a derivative (8-3)

I have some trouble understanding what Feynman says here: "The procedure we have just carried out is performed so often in mathematics that for convenience special notations have been assigned ...
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Significance of $\frac{dv}{dx}=0$

Suppose an object is moving with varying acceleration in time. What does it mean when it hits a point where $\frac{dv}{dx}=0$? Does it mean the object has hit maximum velocity? Assume the object ...
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Partial derivatives when the function's inputs are dependent on the same variable

at 53:13 of this lecture by mit ocw, the prof. Moungi Bowendi writes, $$ (\frac{\partial U}{\partial T})_{p} = (\frac{\partial U}{\partial T})_{v} + (\frac{\partial U}{\partial T})_{T} (\frac{\...
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Invariance of differential operators

How do we prove that del operator is invariant under any kind of change of coordinates, specifically under galilean transformations? I am getting an extra term containing the relative velocity of two ...
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Solving the Euler-Lagrange equations for a complex scalar field in which the time derivatives and gradient are separate

This is found at the bottom of page 9 of David Tong's QFT lectures. The Euler-Lagrange equations for the complex scalar field: $$\mathcal L=\frac{i}{2}(\psi^*\dot\psi-\dot{\psi^*}\psi)-\nabla\psi^*\...
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How to (“geometrically”) differentiate unit vectors of spherical coordinates?

I have been trying to derive the expressions of partial derivatives of unit vectors with respect to each other in the spherical coordinate system. I was able to get all of them except $\frac{\partial \...
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Force and Accleration

It's just a basic question I had when I was studying physics years back, So acceleration have two equations $$a=\frac{F}{m}$$ and $$a=\frac{\text{d}v}{\text{d}t}$$ So by the first equation, if I'm ...
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Having trouble taking derivative of a cross product when finding Lagrangian to find force equation for rotating non-inertial frame

I've been working on a problem for my classical mechanics 2 course and I am stuck on a little math problem. Basically, I am trying to prove this equation of motion with a Lagrangian: $$m\ddot{r} = F + ...
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Accelerometer readings simulation

I wrote software which generates a trajectory in an inertial frame. I'm trying to simulate an accelerometer reading, as if this accelerometer were moving along this trajectory. If this trajectory does ...
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1answer
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How do we calculate the directional derivative of a static vector field? (If there is such a thing.)

So, for a static scalar field $T(x,y,z)$, the derivative along $d\vec l$ is given by $$\frac {dT}{|d\vec l|} = |\vec \nabla T| cos\theta$$where $\theta$ is the angle between $\vec \nabla T$ and $d\...
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Question about the math notation when studying Euler equation

I am trying to work out the Euler equation. However, I find some difficulties of the following step. Am I doing the right things and how could I get into the last step? Thanks a lot. $$ \nabla (\vec{v}...
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Tensor Question (Klein–Gordon equation) [closed]

I have a question following the derivation of the Klein-Gordon equation from a lagrangian. From Eq. (13d), where does $\delta^\mu_\nu$ come from? I guess it's a conversion factor of some sort.
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How Total derivative got converted into partial derivative?

While studying the book Heat and Thermodynamics by Zemansky and RH Dittman, in the topic 'equation for a hydrostatic system' (page no. 88) it was given in equation 4.12, when we take Pressure P ...
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Rewriting the Laplacian on a curved manifold

I guess there is a sense in which the following is true: "The Laplacian written on a Riemannian manifold $(M,g)$ can be seen as adding a coordinate dependent mass field to the Laplacian on ...

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