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.
1
vote
1answer
38 views
How do I divide an integral into parts and sum the total?
I have an exercise looking like this: Analyze the integral $4\int_{0}^{1} \sqrt{1-x^2}\ \tex$ numerically by ng the interval (2,1) into three
equal parts, then summarize the integral parts.
I just ...
4
votes
2answers
39 views
Identifying total derivatives
I would like to identify which terms in my expression are a total derivative and which ones are not.
For expressions in which all terms are a total derivative, one can simply use integrate to find ...
5
votes
2answers
200 views
Derivative of an interpolated function
I am trying to take the derivative of an interpolated function. I am given the function values and the derivatives at some irregular points. Here is my minimal working example to reproduce the error:
...
0
votes
1answer
74 views
Second Derivative in MMA
I would like to ask, why the following second derivative is wrong in MMA
...
1
vote
0answers
40 views
4
votes
1answer
95 views
Concerning the origin of four terms in DSolve's hyperbolic solution to an ODE
Typing
DSolve[{g''[y] - k^2/Gy g[y] == 0}, g[y], y]
where $k$ and $G_y$ are constants gives
$$\left\{\left\{g(y)\to C_1 \exp\left(\frac{k y}{\sqrt{G_y}}\right)+...
-1
votes
2answers
118 views
Direct and easy calculation of the N-th derivative of a function at 0?
Does Mathematica have an easy way to calculate the n-th derivative of a function at zero?
I was looking for something along the way D[e^x,{x,n,0}].
How do I do ...
0
votes
0answers
73 views
Why does Mathematica not integrate my function? [on hold]
Why doesn't Mathematica integrate my function?
Srry I am at the first time at this forum. So i have no idea how it works. I hope the following input is enough.
...
0
votes
0answers
24 views
Why is an integration error appearing during this code?
I'm really confused about this to be honest. If i evaluate approx[x,k] for some k, and then input the list into the Plot function, I get a plot. But if I try to use approx[x,k] directly, I get an ...
0
votes
0answers
20 views
Acquire a single-variable function by slicing a 3d graph along a given line
Consider my Calculus 2 math problem: Given a function f[x,y]=x^2-3xy and two vectors a = {1,2} and b = {1,1}. Spanning a linear ...
2
votes
3answers
149 views
Compute a certain “separability probability” via a constrained 4D integration over $[-1,1]^4$
I have employed the GenericCylindricalDecomposition command (applied to $6 \times 6$ "density matrix" positivity constraints) to compute the following expression
...
0
votes
1answer
44 views
Replacing a derivative [on hold]
If I want to calculate e.g. $\frac{dr(t)}{dt}$ then in Mathematica we can write e.g. ,
answer = D[r[t],t]
This outputs r'[t] ...
3
votes
0answers
36 views
On using ConditionalExpression inside Integrate vs. getting the result, then applying Assuming on result
I thought at first that
sol = Integrate[ ConditionalExpression[ integrand, conditions],....] (1)
Is equivalent to and should give the same answer as
...
1
vote
0answers
22 views
Define an operator without chain rule
I am trying to define a linear operator myOp[f[x, t]], which commutes with linear differential operators without involving the chain rule, i.e., ...
0
votes
1answer
37 views
Help with recurrence relation
I try to solve the following recursion for $n \in \mathbb{N}$.
$r_i = r_{i-1} - \frac{1}{2} \cdot \sqrt{1 - \frac{4\pi^2\cdot r_{i-1}^2}{n^2} \cdot \cos^2 \left(\frac{\pi}{n}\right)}$
$r_0 = \...
1
vote
1answer
31 views
Midpoint Riemann Sum Code [duplicate]
I am trying to write a code in order to find the Riemann sum for the midpoint but can't seem to get the correct answer. I double checked what the answer should be but I haven't been able to find what'...
-3
votes
0answers
40 views
How can I take the derivative wrt to qx-px? [closed]
I HAVE TO TAKE A DERIVATIVE WRT TO qx-px
d/d(qx-px)
ie: d/d(qx-px) F(qx-px, py, T)
but Mathematica will not allow this
even if I try to "trick it " by defining rx=qx-px
ANY IDEA HOW i MIGHT TAKE ...
2
votes
3answers
87 views
Can the indefinite integral of this trigonometric function be written in closed form?
Mathematica isn't evaluating this integral and I'm wondering if there's a bug in how I'm asking:
the function I want to integrate is
$$f(\theta) = \cos^2(\theta)\sqrt{A \cos^2{\theta} + B \sin^2{\...
0
votes
0answers
39 views
Finding tangents and curvature of a parametric curve in 3D [duplicate]
I am in a calculus 3 class and cannot figure out how to get Mathematica to solve for unit tangent, normal tangent, binormal tangent, and curvature without getting a supper messy result. This is the ...
2
votes
0answers
43 views
Symbolic perturbation expansion for quantum mechanics using Hellmann-Feynman derivaties
I am interested in some quantum mechanical perturbation expansion for energies. Actually I want to implement these terms $E_n^{(k)}$. As is stated below one can do that using CAS. I would be ...
5
votes
1answer
100 views
Integrating function with /; in its definition
why
f[x_ /; x>=0]:=x;
Integrate[f[x] ,{x,0,2 Pi}]
return unevaluated? Notice that the above definition of f[x] works OK ...
7
votes
2answers
519 views
Forcing Mathematica's Integrate to give more general answers
I have a simple gaussian integral: $\int^{\infty}_{-\infty}dx\:e^{i\alpha x^2}$.
If $\alpha \in \mathbb{R}$, then:
$$
\int^\infty_{-\infty} dx\; e^{i \, \alpha x^2} = \sqrt{\frac \pi {-i \alpha}} \...
0
votes
1answer
92 views
ImplicitRegion vs DiracDelta [closed]
I've been having troubles with numerical integration over implicit regions, so I checked in a simple example if the result coincides with integrating with a DiracDelta function, and found this rather ...
2
votes
0answers
81 views
Can this integral equation problem $\int_\Gamma \frac{e^{ik|x-y|}}{4\pi|x-y|}\varphi(y) \, \mathrm{d} y = u_{x_0}^{in}(x)$ be solved?
I am not sure if Mathematica is capable of solving integral equations in 2D/3D. I found this page in the documentation, but this is just for 1D.
The following is what I would like to solve, it can ...
2
votes
0answers
46 views
Make Mathematica evaluate an integral symbolically
The following integral has a symbolic solution, which MMA does not return.
$\kappa(x) \begin{cases}
\text{mod}(x,1)\le0.5: & 1\\
\text{mod}(x,1)>0.5: & 0
\end{cases}$
$f(z)=\int_0^1 \...
1
vote
1answer
90 views
Multivariate series for approximating implicit system
I'm trying to approximate the solution of an implicit set of equations by means of a Taylor series. I have managed to do so for a solution expressed in terms of a single independent variable, by using ...
0
votes
0answers
27 views
Linear dependency of vectors of functions
I would like to find a symbolic way to obtain a Q-function (Boolean output) for the following problem.
I have the following inputs:
...
0
votes
1answer
65 views
Infinite sum + integral
I have the following sum + integral to solve:
$\int_0^{\infty}dt\, t^{\frac{m-3}{2}} e^{-t(a^2+b^2)} \sum_{n=-\infty}^{\infty} e^{-t(c^2 n^2 +2bcn)}$
My try:
...
4
votes
0answers
50 views
How to calculate specific area on surface of sphere?
I am trying to calculate the solid angle subtended by arbitrary-shaped loops on a sphere's surface.
First, I parametrize circular loops by:
$$\theta(t,k_{x0},r) = k_{x_0} + r \cos(t);$$
$$\phi(t,k_{...
5
votes
4answers
261 views
Computing surface integral on the unit sphere
I have a function $f$ which takes three unit vectors of $\mathbb R^3$ and returns a number, so $f \colon \mathbb R^3 \times \mathbb R^3 \times \mathbb R^3 \to \mathbb R$ which is defined as $f(\...
0
votes
0answers
48 views
How to find D[F[a(x,y,t),b(x,y,t)],t] in terms of D[F[a(x,y,t),b(x,y,t)],x] and D[F[a(x,y,t),b(x,y,t)],y]?
I have a function $F(a(x,y,t),b(x,y,t))$ whose derivative I would like to be displayed as
$\frac{\partial F}{\partial t} = f(x,y,t)\frac{\partial F}{\partial x} + g(x,y,t)\frac{\partial F}{\partial y}...
10
votes
3answers
650 views
How to use Mathematica to do a complex integrate with poles in real axis?
I want to use Mathematica to compute the following complex integral:
Integrate[Exp[I z ] 1/z, {z, -Infinity, Infinity}]
Mathematica reports that it does not ...
1
vote
2answers
87 views
How to calculate arbitrary area on surface of sphere?
I am trying to calculate the solid angle subtended by arbitrary-shaped loops on a sphere's surface.
First, I parametrize circular loops by:
$$\theta(t,k_{x0},r) = k_{x_0} + r \cos(t);$$
$$\phi(t,k_{...
1
vote
2answers
59 views
Verifying equality
I am trying to test equalities involving integrals. What are the possible causes for not being able to test the following:
...
0
votes
1answer
47 views
0
votes
1answer
113 views
Definite double integral in Mathematica
I am a beginner in Mathematica. I want to perform the following definite double integral equation in Mathematica and want to compare the solution with mine.
$$
A_{P-A} = R^2\int_{\theta=\frac\pi2-\...
0
votes
1answer
58 views
Find area under real part of complex curve without numerically integrating
I have a function defined within the domain 0<=x<=2*Pi.
f = Cos[x] * (a - 1 - r*Cos[x])^1.5 on the domain: ArcCos[(a-1)/r] < x < 2*Pi-ArcCos[(a-1)/r]
f = 0 elsewhere
'a' and 'r' are ...
1
vote
1answer
88 views
How do I draw a function along with a particular tangent line at a specific point? [duplicate]
I would like to draw a function (x/x^2+1), and at a certain point, i'd like to draw a dot along with the tangent line at that specific point.
what I have tried is the following:
...
1
vote
0answers
44 views
10
votes
1answer
240 views
Linearity Assumption [duplicate]
I'd like to define a set of variables of this sort:
{K1[t],K2[t],...}
Adding a general assumption: all such variables are linear.
I'd like to do this to avoid ...
5
votes
1answer
225 views
Neglect higher order derivatives in expression
Say I have an expression of this sort:
Derivative[5][x1][t]+Derivative[3][x2][t]+x1[t]+Derivative[2][x1][t]
and so on...
I would like to ask Mathematica to ...
3
votes
1answer
30 views
Defining a two variables function by Hessian Matrix
I need to define a function h by the Hessian matrix:
f[x_, y_] := x^3 + y^2
h[x_, y_] := D[f[x, y], {{x, y}, 2}]
and ...
5
votes
1answer
69 views
Computing a functional derivative using the VariationalMethods package
To test the VariationalMethods` toolbox, I tried to compute $\frac{\delta I[y]}{\delta y(x)}$ for the following functional, assuming $w(x)$ is some well-behaved ...
2
votes
1answer
67 views
Specifying $0^+$ as the lower bound of an integral [closed]
Consider the following evaluation.
Integrate[1/Sqrt[x], {x, 0, 1}]
(* 2 *)
The integral is right only if the first extreme of integration is $0^+$.
I want to ...
-3
votes
1answer
88 views
Solving an integral equation analytically [closed]
I have an inegral equation like this
$\qquad n(\phi)=\int_0^\sqrt{\phi} f(w)\sqrt{2w+\phi}dw$.
I need to find $f(w)$ analytically. Here $n(\phi)$ is known.
Here $n(\phi) = -\frac{2}{\delta ^2 \phi ...
2
votes
2answers
79 views
Derivative of integrated noise Gaussian likelihood
In a Bayesian problem with Gaussian likelihood with mean $\mu$ and a uniform prior on the standard deviation $\sigma$, it is possible to derive the marginal posterior (where $\sigma$ has been ...
0
votes
1answer
47 views
0
votes
1answer
55 views
How can I calculate the time-series of a vector?
I need to calculate a time-series of the vector $\|\theta(t)\|=2.2^{-t}\|v(t)\|$ where $t$ for 10 time-steps.
I was thinking of expressing the vector as a list of functions, and then the functions ...
0
votes
1answer
44 views
Complex Infinity of Hypergeometric+Gamma function
I am solving some integral which gives an hypergeometric function+gamma function . The point is that my values of n (see the code below) are integers, so ...
1
vote
2answers
129 views
Analytic or numerical integral calculation
I would like to integrate the following,
$$f(x,y) = \frac{1}{(a - x)^2 + b (y-x^2)^2},$$
where $a>0$ and $b>0$ between definite bounds to obtain an analytic result ideally. I've tried the ...
