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Questions tagged [calculus]

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Deformation matrix, Math hack for stability on large simulation steps?

So there is a numeric technique for updating a deformation gradient in MPM that goes as: $$F_{n+1} = (I + \nabla \vec v \Delta t)F_n$$ This works for small time steps but for large time steps ...
Makogan's user avatar
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What happens at the interface between a solid and a fluid?

I am doing an MPM simulation of water colliding against a solid, I am currently encoding the model for collision forces and I am realising something isn't clicking. Let us assume the fluid experiences ...
Makogan's user avatar
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1 answer
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Finding maximums in mesh of graph?

I have a triangle mesh which is an approximation of a smooth graph. i.e. a scalar function of $xy$. I am interested in finding extrema. One naive way I did it was to look at some number of points ...
Makogan's user avatar
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1 answer
268 views

Finding total derivative of a multivariate function in Maple

In Maple, I have a function $f(x(t),y(t),t)$ that I want to differentiate with respect to $t$. I know the command for partial derivative $\frac{\partial f}{\partial x}$,$\frac{\partial f}{\partial y}$,...
ilawid's user avatar
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2 votes
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161 views

Best way to compute given functional with accuracy:

I need to plot the following functional with accuracy: $$ I(x,s) =\int_0^\infty\mathrm dy \frac{F(x + \mathrm iy,s) − F(x −\mathrm iy,s)}{\mathrm e^{2πy}-1}, $$ Where $ F(z,s) = \dfrac{1}{z^s\Gamma(\...
bambi's user avatar
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2 answers
149 views

How to minimize $(x-a)^2+(y-b)^2$ subject to $ \sqrt{a}+\sqrt{b}=\sqrt{2}$?

I am not sure if this is on-topic here, but I am trying. Let $x,y$ be positive real numbers. I am trying to find $$ \min_{\sqrt{a}+\sqrt{b}=\sqrt{2}}(x-a)^2+(y-b)^2$$ I tried using Mathematica for ...
Asaf Shachar's user avatar
2 votes
1 answer
550 views

Using Implicit Euler with second order differential equations

We can numerically integrate first order differential equations using Euler method like this: $$y_{n+1} = y_n + hf(t_n, y_n)$$ And with Implicit Euler like this: $$y_{n+1} = y_n + hf(t_{n+1},y _{n+...
Lenny White's user avatar
2 votes
1 answer
69 views

Numerical integration of the dataset of a function

The energy equation for a spherically symmetric system is given by $$\mathscr{E}=\frac{v^2(r)}{2}+\frac{c_s^2(r)}{\gamma-1}+\phi(r)$$ where $\mathscr{E}$ is the total energy, $v$ is the velocity of ...
Richard's user avatar
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1 vote
1 answer
122 views

Evaluating an indefinite integral that has no closed form

I need to evaluate the following indefinite integral: $$I=\int\frac{x^5+2ax^3+a^2x-4a}{x^7+ax^5+2ax^4}dx=\int\frac{x^5+2ax^3+a^2x-4a}{x^4(x^3+ax+2a)}dx$$ The solution that I obtained while ...
Richard's user avatar
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1 vote
2 answers
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Numerical integral with symbolic integral in exponent

Many times in fourier approximation we come across integrals such as $$\int_0^1 e^{-\gamma\int_0^xu_0(\eta)d\eta}dx$$ where $\gamma$ is a constant and the data for $u_0$ is provided as a discretely ...
Turbo's user avatar
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1 answer
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Change of random variables and check by plot

Question As a test, I transform a uniform distribution over the unit square. But when I check the transformed distribution with Monte Carlo, it is wrong. What went wrong? Thanks. Problem Random ...
R zu's user avatar
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1 vote
0 answers
214 views

How to get the derivatives of the determinant and inverse of 2nd-order tensor wrt itself in SymPy?

I have a second-order tensor for which I need to compute the derivatives of its determinant and inverse w.r.t. itself. The equations are as follows: $$\frac{\partial \, det(\mathbf{F})}{\partial F_{...
Chenna K's user avatar
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1 answer
216 views

SciPy 3d plotting Integral of $\int x^y dx$ for $y$ in $[-4,4]$

Ideally, I would like to get the symbolic/algebraic integral of the function and plot the resulting surface in 3d. I am not sufficiently versed in SciPy to know if this is even really possible.
user20460's user avatar
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1 answer
109 views

What is the expected value of area of intersection of a circle and a rectangle

$r$: cicle C1's radius $w$,$h$: rectangle R1's edges: $x=w$, $y=h$, $x=0$, $y=0$ $(w>2r, h>2r)$ $S(x,y)$: area of ...
Donald Wu's user avatar
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1 answer
473 views

Derivative chain rule

Define a sequence $(\mathbf{y})_{i=0}^N$ in $\mathbb{R}^n$ such that: $$\mathbf{y}_{k+1} = \mathbf{y}_{k} + \lambda \nabla_\mathbf{y} E(\mathbf{y}_k,\mathbf{w}), \quad k=0,1,\ldots,N-1,$$ where $\...
f10w's user avatar
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1 vote
2 answers
2k views

Derivative of the inverse of the Right Cauchy-Green Deformation Tensor wrt itself

In continuum mechanics, we define the Right-Cauchy-Green Deformation Tensor as $\boldsymbol{C}=\boldsymbol{F}^T\boldsymbol{F}$ I want to compute $\frac{\partial \boldsymbol{C}^{-1}}{\partial \...
user1751434's user avatar
4 votes
1 answer
181 views

Does the limit of $\frac{\partial f}{\partial u}$ at $u=0$ exist?

For an optimization routine I needed to compute the derivative of the right-hand side $\: f_u(x_k, u_k)$ of a discrete-time system $x_{k+1} = f(x_k, u_k)$. Since $\: f_u(x_k, u_k)$ includes terms that ...
Max Herrmann's user avatar
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1 answer
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Simpsons Rule 10<-6 Error

I am making a Java function to calculate $\int_{0}^{\pi/2}x/sin(x) dx$ using Simpsons rule. When the function equals $0$ or $0/\sin(0)$ should I just add $1$ (limit) or $10^{-10}$ (limit). My ...
Julius A.'s user avatar
-1 votes
1 answer
69 views

I have a problem in finding the exact area and estimating the error [closed]

The program should first read from the keyboard the values of a and b then estimate the area under f(x) in the interval [a, b] using Simpson’s 1/3rd rule and display the estimated area and the error ...
fatma's user avatar
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4 votes
1 answer
3k views

Is there a relationship between the covariance matrix and the partial derivative?

Suppose that there are $N$ pieces of data, each of which contains $M$ parameter values such that $N >> M$. If we put this information into matrix form ($N$ rows, $M$ columns) and then compute ...
drjrm3's user avatar
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