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# 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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### Why does the derivative become $x=r\dot\theta\cos(\theta)$ when calculating angular velocity?

This is not a homework question because I do not want help with solving my homework. I would rather want an explanation of why the derivative of this seems to break math as I know it. Background ...
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### Why the continuous arrangement of point masses (particles) at infinitesimal separations leads to a extended system?

I am basically talking in terms of Newtonian mechanics. The Newton's laws started with a good and easy assumption of particles as point masses. This assumption clearly reformed physics and a great ...
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### Why do we differentiate displacement to get velocity if displacement is given as a function of time [on hold]

Like if the displacement is given as S=(2t³)and if we are asked to find the velocity in 2 seconds then if we put t=2 in the expression we get 16 which isn't correct. The correct must be dS/dt=6t² and ...
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### Physics exercises with solutions that use derivatives and integrals [on hold]

I'm looking for physics exercises, mainly from classical mechanics and power, work, energy, that focus on creating a mathematical model using differentiation and integration to obtain the solution. ...
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### Relationship Between Differentiation in Two Frames

In section 10.3 of Principles of Ideal-Fluid Aerodynamics by Karamcheti, he writes the following: $\qquad$ Denote by $K_1$ a reference frame fixed with respect to the moving body. We shall denote ...
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### Product rule of variations

I am deriving the Einstein equation using the Einstein-Hilbert action: It is obvious that the variation in the Riemann Tensor is calculated from a variational product rule. What is not obvious to ...
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### Covariant derivative contracted with a metric

I would like to calculate $\nabla_\mu(g^{\mu\alpha}g^{\nu\beta}\nabla_\alpha \kappa_\beta)$. How would this expand? Where $\nabla$ is the covariant derivative, g the metric and $\kappa_\beta$ a 1-...
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### exponential autocorrelation function by approximation of derivative

I have been pondering about the following question: Given a time-dependent function $f(t)$, is it possible to show that its autocorrelation function will generally follow a decaying exponential ...
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### How to further expand $\text{grad} \left( \vec{a} \cdot\vec{b} \right ) = \vec{\nabla} \left (\vec{a} \cdot\vec{b} \right )$? [migrated]

With $\vec{a}, \vec{b}: \mathbb{R}^3 \to \mathbb{R}^3$ vector fields: I want to expand $\text{grad} \left( \vec{a} \cdot \vec{b} \right ) = \vec{\nabla} \left (\vec{a} \cdot \vec{b} \right )$. So I ...
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### What does it mean “differentiation with respect to the coordinates of particle 1 or 2”?

I was reading Introduction to Quantum Mechanics by Griffiths. In Chapter 5, Identical Particles, I came across the notation $\nabla_1$ and $\nabla_2$. Griffiths writes that it means "differentiation ...
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### Deriving forces transformation for special relativity using the four-vector energy-momentum

I don't understand how we derive the forces transformation using the four-vector "energy-momentum". Supposing that we have two inertial frame of reference $R := \{x,y,z\}$ and $R' := \{x',y',z'\}$ ...
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### Differentiation of the determinant $g$

Let $g$ be the determinant of the metric tensor. I want to derive the following equation $g_{,\nu}=gg^{\lambda \mu}g_{\lambda \mu,\nu}$. It is said that $gg^{\lambda \mu}$ is a cofactor, but I can't ...
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### Converting velocity vector formula from Cartesian coordinate system to polar coordinate system

I have a little question about converting Velocity formula that is derived as, $$\vec{V}=\frac{d\vec{r}}{dt}=\frac{dx}{dt}\hat{x}+\frac{dy}{dt}\hat{y}+\frac{dz}{dt}\hat{z}$$ in Cartesian Coordinate ...
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### Differentiation of metric tensor in new coordinate

I want to understand the explicit meaning of $g_{\mu'\nu',\lambda}=0$ where unprimed coordinates are coordinates of the the original coordinate systems and primed ones are for new coordinate system. ...
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### Physical significance of divergence

In my textbook They considered a parallelopiped $ABCDEFGH$ with sides $dx,dy,dz$ parallel to $x,y,z$ axis respectively $\vec V$ represents the vector velocity of the fluid at the centre $P$ of f ...
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