# 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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### Finding the maximum electric field strength above a ring with a hole in the middle

I'm doing a problem (not homework, by the way) which asks for the electric field strength on the axis of symmetry a distance $x$ above the centre of a circular disc, which has uniform surface charge ...
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### Intuitive Definition of Curl and Stokes' Theorem

I am asking this question on SE Physics because it's about an intuitive derivation, not a rigorously made proof of a theorem. Reading Mathematical Methods for Physics and Engineering by Riley, et al. (...
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### Is there any difference in superscript and subscript notation in finite difference method

Is there any difference in superscript and subscript notation in the finite difference method? I see the same paper use (superscript for $x$ and superscript for $y$ notation) and (subscript for x and ...
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### Do partial derivation respect to velocity and total derivation respect to time commute? [duplicate]

Imagine we have a function of position $x^i$ and velocity $v^i$ $f(x,v)$. Position and velocity are both functions of time $t$. If the function doesn't depend explicitely on time, then we have the ...
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### Generalization of straight line motion under constant acceleration

My question is that, we all know the three equations of straight line motion under constant acceleration, \begin{align} x & =x_{\rm o}+v_{\rm o}\,t+\tfrac12 \mathrm a\,t^2 \tag{1d-a}\label{1d-a}\\ ...
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### Defining the exterior derivative with torsion [duplicate]

As introduced in most standard GR textbooks, the exterior derivative of a p-form $X$ is defined in a coordinate basis as $(dX)_{\mu_1....\mu_{{p+1}}}:= (p+1) \partial_{[\mu_1}X_{\mu_2....\mu_{{p+1}]}}$...
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### Expressing acceleration in terms of velocity and derivative of velocity with respect to position

we know that $$a = \dfrac{dv}{dt}$$ dividing numerator and denominator by $dx$, we get $$a=v\dfrac{dv}{dx}$$ provided that $dx$ is not equal to zero or instantaneous velocity not equal to zero when I ...
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### Second derivative of a function in a manifold [closed]

Suppose we have a function curve $\gamma(t)$ on a manifold $M$ . Define the function $$f(t)=\frac{1}{2}g_{ab}\dot\gamma(t)^a\dot\gamma(t)^b$$ Introducing coordinates $x^i$ the first derivative of the ...
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My assumption here is that in the definition of elementary work : $dW = F ds$ symbol $d$ represents a differential. But a differential implies a function : $dy =_{df} d[f(x)] = f'(x) \Delta x = f'(... 3answers 148 views ### Why is$ \text{div}(\vec{r}/r^3) = 0 $? Let's begin with some context. I was reading Griffiths's introduction to electrodynamics. This makes sense given all the content one can find online in order to understand visually the divergence. ... 1answer 64 views ### Applying gradient in spherical coordinates to vector in cartesian coordinates [closed] I am trying to calculate the gradient of a vector field$\boldsymbol{u}$. In cartesian coordinates, I would normally do $$\left(\nabla\boldsymbol{u}\right)_{ij}=\partial_{i}u_{j}=\left(\begin{array}{... 1answer 52 views ### Spatial derivative of the Hamiltonian Operator I have a Hamiltonian of a semiclassical wavepacket in some external potential (say an EM field). I would like to linearize this Hamiltonian around a point x_c. Intuitively, I would simply try to do ... 1answer 42 views ### The contravariant derivative of a substitution for the de Sitter metric Consider the de Sitter metric:$$ds^2 = (1-\frac{r^2}{a^2})dt^2 -(1-\frac{r^2}{a^2})^{-1}dr^2-r^2d\Omega^2$$I know that we rewrite the metric as (u,r,\theta \phi) using the substitution$$u = t-... 2answers 50 views ### Where does one more '$\rm m$' come from in the units? $$\nabla \times A = B$$$A$is vector magnetic potential,$\mathrm{Wb/m}B$is magnetic field intensity,$\mathrm{Wb/m^2}$Where does one more m come from for$B$? Is that from the gradient operator ... 3answers 87 views ### Derivative with respect to vector of a function depending on vectors I've been trying to understand this concept for hours without any success. I found similar questions on this forum (Derivative with respect to a vector is a gradient?) but I still don't understand. ... 1answer 53 views ### Given that$m \dot v \cdot v = 0$, how is it equal to$m \frac{d}{dt} (v \cdot v)/2$? [closed] While studying about scalar triple product in vector algebra, I stumbled upon the following question with the solution. I want know how is$m \dot v \cdot v $=$m \frac{d}{dt} (v \cdot v)/2$? 3answers 168 views ### Covariant gradient - What am I missing? I know that the components of the gradient should transform covariantly, and have used this many times in special relativity, etc. However, I also know that covariant components transform like the ... 1answer 55 views ### Killing equation in coordinates In proving that it is possible to write the killing equation in coordinates as $$L_X g=0\iff X_{\alpha;\beta}+X_{\beta;\alpha}=0$$ I have read that the key observation, to write the equation in ... 1answer 50 views ### Analogous notation to$\nabla$but for gradient with respect to$\vec{k}$not$\vec{x}\nabla = \frac{\partial}{\partial x_i}$so$\nabla F = (\frac{\partial F}{\partial x}, \frac{\partial F}{\partial y}, \frac{\partial F}{\partial z})$. However, is there a similar equalivalent notion ... 1answer 53 views ### Question about operation with derviative of product in Noether proof I have been studying Noether's theorem proof, I have a problem understanding one of the last steps, because I don't understand the calculation. Noether theorem assures us that the action under a ... 3answers 2k views ### Partial derivative notation in thermodynamics Most thermodynamics textbooks introduce a notation for partial derivatives that seems redundant to students who have already studied multivariable calculus. Moreover, the authors do not dwell on the ... 3answers 467 views ### Why is the covariant derivative of the metric tensor with UPPER indices equal to zero? [closed] I've shown that$\nabla_{\lambda} g_{\mu\nu} = 0 $rigorously by the following method:$ \nabla_{\lambda} g_{\mu\nu} = \partial_{\lambda}g_{\mu\nu} - \Gamma^{\rho}_{\lambda\mu} g_{\rho\nu} - \Gamma^{\...
In Sean Carroll's GR book, pg 84, the exterior derivative $d$ is defined as $$(dA)_{\mu_1\mu_2...\mu_{p+1}} = (p+1) \partial_{[\mu_1}A_{\mu_2...\mu_{p+1}]}\tag{2.76}$$ where $A$ is a $p$-form and the ...