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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
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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

how to linearize an equation? [migrated]

How does one linearize this equation?
4
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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
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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
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

Piecewise function inside Integrate - Wrong result

I tried to evaluate the following integral: ...
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
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 ...