# Questions tagged [calculus]

Calculus is the branch of mathematics which deals with the study of rate of change of quantities. This is usually divided into differential calculus and integral calculus which are concerned with derivatives and integrals respectively. DO NOT USE THIS TAG just because your question makes use of calculus.

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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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### How is it possible to differentiate or integrate with respect to discrete time or space?

As far as I have understood, the case is that there is nothing that argues that time or space is continuous, but at the same time we must assume this in order to be able to calculate derivatives or ...
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### How to calculate flux in a helical wire

We have a wire going around in a helix (just like an inductor) and a constant magnetic field exists along its axis throughout the space. How do we calculate the flux through it? I can't understand ...
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### Equation for the velocity of a roller coaster at every point

I have been issued a task to create a roller coaster comprised of a piecewise function. In my research I have come across the an equation to calculate the final velocity of the cart found at https://...
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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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### I need the derivative for this equation [closed]

I want to differentiate this Area equation.
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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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### If current is charge flow through a point then isn't the surface current always zero?

Current is defined as the rate of flow of charge through a point. Now say we've got a surface charge density $\sigma$ which moves around on a surface. According to the above definition the current ...
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### How to find the Taylor expansion of $\vec{r}/r^3$?

I want to show that the Taylor expansion of $\frac{R\vec{e_1}-\vec{y}}{|| R\vec{e_1}-\vec{y} ||^3}$ at $\vec{y}=0$ is equal to $\frac {\vec{e_1}}{R^2}+\frac{3y_1 \vec{e_1}-\vec{y}}{R^3} + O(y^2)$. I ...
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### How did the Lagrangian and Hamiltonian theories of motion inspire the idea that forces should be treated as one-forms instead of vectors?

On page-5 of this paper1 by E. Minguzzi titled "A geometrical introduction to screw theory", he writes: Who adopts this point of view argues that it should also be adopted for forces in ...
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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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### Problem finding Centre of Mass [closed]

My Question: For finding the Center of Mass ($COM$) of a hollow cone, why do we use its area to define its elemental mass ($dm$) and not its volume, which we use to find the $COM$ of a solid cone. The ...
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### Quantum mechanics Dirac delta representation with integral

So I’m doing QM and found bunch of problems for beginners and I’m struggling with this one: $$\lim_{a\rightarrow 0}\int^{\infty}_{-\infty}e^{\frac{ip x}{\hbar}-a x^2}dx=2\pi\hbar\delta(p).$$ If I swap ...
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### Can I use the Gauss divergence theorem in a region whose divergence blows to infinity on its surface boundary?

Say we have a vector function $\vec{D}$ defined in some region on whose boundary its divergence goes to infinity and inside we have $\nabla \cdot \vec{D}=\rho$. Then is it valid to use the Gauss ...
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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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### 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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### Can anyone suggest a math review book for someone interested in beginning physics study as a hobby?

Good day. This is my first post and I was not sure whether to post here or on Math StackExchange. Since the end product of my goal results in ultimately understanding some basic math in physics, I ...
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### Berry Connection Calculation for a 2-Level System [closed]

Suppose we start with a state on the Bloch sphere given by: $$|\psi\rangle = \begin{pmatrix}\cos\left(\frac{\theta}{2}\right)\\e^{i\varphi} \sin\left(\frac{\theta}{2}\right)\end{pmatrix}$$ The Berry ...
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### How to create a position-vs-time graph from this?

I'm given the following problem: At what times is the particle found at $x= 20m$? I know this is a very fundamental problem, but still I cannot see how the answer ends up being $4$ and $12$. Note ...
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### Integration used in Derivations

I've seen many derivations in which Integration is used. But I don't understand the fact that why after going to a distance like $y$ or $x$, we take an element $dy$ or $dx$? Instead can't we take any ...
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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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### How are the SUVAT equations derived? [duplicate]

I hope everyone is doing well and staying safe. So instead of simply memorizing the SUVAT equations, I wanted to find out how the equations are derived to broaden my knowledge. I'm currently a high ...
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### Is expectation value of $p^2$ equivalent to this integral?

Let $\psi(x)=Ne^{iax -\frac{m^2x^2}{2} -ibt}$ and I want to compute the possibility of momentum $p$. By definition : $\langle p^2\rangle=\int_{-\infty}^{\infty}\psi^*p^2\psi dx$. Is that equivalent to ...
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### Having difficulties deriving the formula for the force acting upon a dam with height $H$ and width $L$

I was recently fiddling around with the derivation of the formula for the force acting upon a dam with height $H$ and width $L$, which in my textbook is derived by integrating the term $dF=p(z)Ldz$ ...
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### Acceleration function of position and time

I have an acceleration function in python with position and time parameters and returns the acceleration value. I need the end velocity at a position ,start velocity is zero. how to calculate this ...
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### Does it make sense to take an infinitesimal volume of shape other than a cube?

The question clearer: Is the infinitesimal cube the absolute smallest infinitesimal volume? (Sorry if people thought that it meant: "Is it possible and is it done in daily life to use anything ...
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### Kinematic formulas for constant $n$th derivative of position

I was wondering how to solve for $x(t)$ in the general case of constant $n$th derivative of $x$. This means to solve the equation $$\frac{\mathrm{d}^n x}{\mathrm{d} t^n}=q,$$ where $q$ is a constant. ...
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### Validity of moment of inertia integral in the case of a rod

The wiki says that rotational inertia is defined for point-masses, and by extension continuous bodies. It says: This simple formula generalizes to define moment of inertia for an arbitrarily shaped ...
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### Equation for stationary string

I have some doubts on the following derivation of the EOM of a stationary string. Let $F_x, F_y$ be horizontal and vertical tension of the string $\mu$ be the mass per unit length of the string [kg/m] ...
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### Acceleration due to gravity during its journey up and down

When we throw an object up into the air, ignoring air resistance, etc, we define acceleration to be -9.8 m/s^2. When it goes down after its journey up, like a parabola, do we define the acceleration ...