Questions tagged [calculus-and-analysis]
Questions related to the calculus and analysis branches of Mathematica, including, but not limited to, limits, derivatives, integrals, series, and residues.
2,682
questions
0
votes
0answers
9 views
Explicitly computing antiderivatives
I would like to use Mathematica to compute the antiderivative of $\frac{1}{ \left( x_1 \cos(\phi) + x_2 \sin(\phi) \right)^2}$, which is $\frac{\sin(\phi)}{x_1^2 \cos(\phi) + x_1 x_2 \sin(\phi)} + C$. ...
0
votes
0answers
25 views
1
vote
3answers
38 views
Unexpected limit result (was: How to fix the limit problem?)
MATHEMATICA fails in proving that
Limit[
Integrate[
x^(x - 1)
, {x, t, 2*t}
]
, t -> 0
, Direction -> "FromAbove"
] == Log[2]
How to fix ...
1
vote
1answer
36 views
Plotting cylinders and planes on the same graph
How would I plot the cylinder $y^2 + z^2 = 9$ and the planes $x = 0$, $y = 3x$, and $z = 0$ in the same graph and in the first octant.
...
0
votes
0answers
41 views
How do I plot the region of stability? [on hold]
How do I plot the region of stability in Mathematica? Can any one help out please?
$\qquad \phi=\left|\frac{1+z}{1-z}\right|$
0
votes
1answer
33 views
How to graph a solid
Given a solid described by $S = \{(x,y,z)|0\le x\le4-y^2, -2\le y\le2, 0\le z\le y+2\}$,
how do I graph the solid? I have already tried a RegionPlot3D but I might ...
4
votes
0answers
61 views
Computing differentials on a manifold
Consider $\phi:SO(3) \to \mathbb{R}^3$, $R \mapsto (R^\top e_3)\times e_3$ where $R$ is a real $3\times 3$ orthogonal matrix and $e_3 = [0\ 0\ 1]^\top$.
Can Mathematica compute the differential of $\...
0
votes
0answers
33 views
0
votes
0answers
94 views
How to integrate this expression one time? [duplicate]
How can I integrate this expression? I want to have the second derivative of V i.e. V''[r]. Note that, you can multiply it be ...
2
votes
1answer
100 views
Evaluating an integral symbolically
I am trying to integrate following integral symbolically via integrate command:
$$(0.09)\Bigg[1 + \Bigg\{ \int_{3t-4k-7}^{2k+7/3} (0.09) \text{exp}\Bigg(\int_{3s-...
0
votes
0answers
24 views
curl of tensor in spherical coordinates [closed]
When I use curl in Spherical coordinates in Mathematica it gives me a scalar value but it should give tensor value. What should I do to reach the tensor value(Solution)?
0
votes
0answers
54 views
Finding area between curves formed by csv data
I have a bunch of Differential Pulse Voltammetry data I am trying to import into Mathematica, and find the area between each curve and the blank. So far I have used the following commands:
...
0
votes
2answers
86 views
How to correctly inter-convert between two related infinitesimals?
The following input text
Clear[ρ, ν, λ, T, h, c, kB]
λ[ν_] := c/ν
ρ[ν_, T_] := (8 πh/c^3) (ν^3/(Exp[h ν/(kB T)] - 1)) \[DifferentialD]ν
ρ[ν, T]
ρ[λ, T]
is ...
0
votes
0answers
16 views
Unknown Vector Valued Function
I am trying to evaluate this:
D[f[x[t, k, l], t, k, l], k, l]
where x[t,k,l] is a 2x1 vector and the output of f is 2x1 as well. Currently the function just outputs it as if it were a scalar ...
10
votes
0answers
155 views
Workaround for DSolve in V 12 when it gives undefined as solution to 1D heat PDE?
Reported to WRI CASE: 4278450
Comparing the following, all done from clean kernel
The strange thing is that V 12 can solve this same PDE without the assumptions
...
1
vote
0answers
57 views
Integral of the derivative does not work
In 12.0.0 for Microsoft Windows (64-bit) (April 6, 2019) writing:
Integrate[D[ArcTan[Sqrt[1 + x^5]], x], x]
I get:
ArcTan[...
1
vote
0answers
42 views
Integration evaluation running forever
I am trying to integrate a function but the evaluation goes on and on forever.
Here is the code:
...
0
votes
1answer
37 views
3
votes
0answers
105 views
How to test whether an improper integral converges or diverges?
To verify the improper integral $\int_{1}^{\infty}x\cdot\left | \sin (x^4)\cdot\sin x \right |dx$ converges or diverges,I code
...
3
votes
2answers
63 views
Plotting the derivative of NDSolve divided by the solution
I'm trying to solve the Friedmann equation as is given by:
$$
H^{2}=\Big(\frac{\dot{a}}{a}\Big)^{2}=H_{0}^{2}\big(\Omega_{m,0}a^{-3}+\Omega_{r,0}a^{-4}+\Omega_{\Lambda,0}a^{2.3}+\Omega_{k}a^{-2}\big),...
2
votes
0answers
76 views
Hessian of a loss function for a neural netwrok
Is a way to calculate the Hessian of the loss function in Mathematica. It's easy to define a loss function for neural nets but I couldn't figure out how to calculate the Hessian(more precisely I ...
2
votes
1answer
29 views
How to manipulate a plot of partial derivatives of self-defined bivariate functions?
I have a piecewise linear function (spline) describing a progressive income tax schedule, with higher marginal rates kicking in for higher and higher incomes.
I want to understand the implications of ...
1
vote
1answer
34 views
Evaluate a nested Gaussian integral
I would like to evaluate the following integral. I suspect it just doesn't have an analytic form but thought I'd check to see if there's anything I'm missing,
$\int_{-\infty}^{\infty} \mathcal{N}(z;\...
0
votes
1answer
47 views
How to solve this equation by Solve?
I have an equation to be solved. But Mathematica does not work for it. I hope the solution x can be expressed as a function of a and b
...
1
vote
1answer
36 views
how can I plot two variables (inside a definite integral) against each other
I want to plot x against y in this relation:
2
votes
1answer
27 views
How to mark non-pre-defined functions as functions of a certain variable during calculation of derivative?
If I have some calculation $X=v\cdot\delta$ and I want to calculate $\frac{\partial}{\partial v}X$, the result according to Mathematica is $\delta$. Now, I want to define $\delta$ as a function of $v$ ...
0
votes
0answers
31 views
Why do I get the error message as integral does not converge?
Can anyone please help me to find the integral value of the following expression,
Integrate[(x - t)^(-1), {t, 0, x}]
It will be of a big help to me.
3
votes
1answer
66 views
Evaluating an integral combining a Bessel function with some other functions [closed]
How can I evaluate the integral of $j_1 ^2(x)\exp(-bx)/x$ from 0 to ∞?
5
votes
1answer
70 views
Variable dimensional integrals
I need to Integrate a function, but let the number of dimensions vary.
Integrate[Product[Exp[-( t[[i]] )^2], {i,1,4}], options]
...
2
votes
2answers
136 views
Compute unstable integral with high precision
I want to calculate
$$
\int_{\mathbb{R}^n} e^{-t_1^4-t_2^4- ...-t_n^4-(1-t_1-t_2-...-t_n)^4} \text{d}t_1 \text{d}t_2 ... \text{d}t_n
$$
with the highest possible n.
I tried with
...
3
votes
4answers
433 views
How can I simplify $\lim_{n\to \infty } \, \int_0^n e^{-x} \left(1-\frac{x}{n}\right)^n \, dx$?
How can I simplify $\lim_{n\to \infty } \, \int_0^n e^{-x} \left(1-\frac{x}{n}\right)^n \, dx$ ?
Limit[Integrate[(1 - x/n)^n/E^x,
{x, 0, n}], n -> Infinity]
...
5
votes
3answers
483 views
How do I find and plot the intersection of these three surfaces?
I know how to find the region of intersection of these cylinders using integration, but how would I plot that in mathematica? This is what I have
...
2
votes
1answer
91 views
Calculation of sum $\begin{aligned}\sum_{k = 1}^{n - 1}\end{aligned}\left(1+\cos\left(\frac{k\,\pi}{n}\right)\right)^n$
Having established that Mathematica cannot calculate the following summation:
sum = Sum[(1 + Cos[k Pi/n])^n, {k, 1, n - 1}]
I implemented the classic "plan B", ...
0
votes
1answer
58 views
Is there any way to do this integral without expanding the exponential power?
Integrate[
x^(1/2) Exp[-10.7*
c/x] Sqrt[(1 - (125/1000)^2) (1 - (2*4.18/125)^2) - (1 - (2 x/
1000))^2], {x, 5.03, 994.97}]
or can i do this indefinite ...
1
vote
1answer
68 views
Problem with minimization of $x^z$
I have plugged in the following problem
Minimize[{x^z, z >= 1, z <= 10, x >= 1}, {x, z}]
Can anyone explain please why the software does not provide the analytical solution? Clearly NMinimize is ...
0
votes
1answer
35 views
How to deal with the derivative of an variable upper limit integration by the symbolic form?
I need a rusult with the following form
D[Integrate[f[t], {t, low[x], upper[x]}], x]
(*Out[]=-f[low[x]] (low^\[Prime])[x]+f[upper[x]] (upper^\[Prime])[x]*)
My ...
1
vote
0answers
39 views
Integration methods used by Mathematica [migrated]
Writing:
Integrate[ArcTan[(E Cos[x] + Pi Sin[x])^2], {x, 0, 2 Pi}]
I get:
2 Pi ArcTan[Sqrt[1/2 (-1 + Sqrt[1 + (E^2 + Pi^2)^2])]]
I was wondering if anyone ...
0
votes
1answer
35 views
Interesting output in Integrate [closed]
I found interesting behaviour, best illustrated in the following example. When trying to evaluate
Integrate[1/(d^2 - 1), d]
Mathematica 11.3 gives the following ...
1
vote
0answers
39 views
Fitting a Curie-Weiss law function with a diverging point to a data
I would like to fit a function to some experimental data such that the extracted value of Curie temperature (Tc) describes the temperature at which the response flattens out to a constant value. ...
0
votes
1answer
64 views
Integrate a function of NDSolve
I'm having this problem of integrating a function that's the solution of a NDSolve multiplied by another function.
I basically compute a cycle wich add at each step the quantity
...
0
votes
1answer
55 views
Coupled PDEs: Wave and String Equations
I need to solve a system of mixed string and wave equations. Omitting some constants it looks like this:
$$u_ {\text {yy}} (y, t) - u_ {\text {tt}} (y, t) = \varphi _{t}(x, y, t)$$
$$\nabla _{\{x,y\}}...
1
vote
0answers
12 views
Eingenvalues and Listplot take a long time optimize
I try to calculate the following eigenvalues of a matrix of N=5 10^3 X5 10^3 and representing it using Listplot as follow
...
1
vote
3answers
77 views
Summation and product term expressed in Mathematica [duplicate]
I have the following term which I would like to express correctly in Mathematica:
$$\sum_{i=1}^{m}\frac{1}{\prod_{j=1,j\neq i}^m(\rho_i-\rho_j)}$$
This means that for $m=3$ for example you should ...
0
votes
1answer
61 views
Evaluate integral over quadratic form [duplicate]
What would be the most kosher way to evaluate an integral like this in Mathematica:
$$\int_\mathbb{R^3} \exp\left((x-b)^tA(x-b)\right)dx \ ?$$
Here $A$ is a real symmetric matrix and $b$ some vector ...
1
vote
1answer
63 views
0
votes
0answers
21 views
Assign a Derivate to a new fuction variable
I'm having a new problem. I'm trying to create a new function from a Derivate of another function, but i'm can't getting it. Someone could help me? I'm thinking that is something in the syntax of the ...
-1
votes
1answer
60 views
How to find the maximum of a function with several variables symbolically?
I want to maximize a function with respect to tau1, tau2, and alpha, and it is like the following:
V + β1 τ1 - α τ1^2 - β2 τ1^2 + \τ2 - 2 τ2^2 + α τ2^2
I know I ...
2
votes
1answer
75 views
Why is Mathematica returning the original function from FindMaximum? [closed]
I am trying to find the maximum of the following expression:
I can not figure out why Mathematica returns me the original expression.
I am new to Mathematica and tired to search around but could not ...
1
vote
1answer
52 views
Area of a parametric plot with interpolating functions
I am looking for the area of a parametric plot defined by two solutions (defined as: VVLt & PVLt) of Numerically solved differential equations. The problem is when i'm trying to integrate the two, ...
1
vote
1answer
86 views
How to integrate exponent of sine?
I am trying to integrate Abs|E^ $\sin \theta$| (the absolute value of Euler's constant to the power of sin(theta), and tried the code Integrate[Abs[E^Sin[x]], x] ...
