Questions related to the calculus and analysis branches of Mathematica, including, but not limited to, limits, derivatives, integrals, series, and residues.

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3
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0answers
107 views

Function for the Second Derivative Test

I wrote the following function. It is based on Mathematica for Rogawski's Calculus, 2nd Ed, 2007, Based on Mathematica 7. See: http://users.rowan.edu/~hassen/Math_Rogawski_Calc.htm, Chapter 14. I made ...
-1
votes
1answer
65 views

How can I prove the following equality [closed]

I have the following equality : $$\int_0^\pi \frac{\cos(2t)}{a^2\sin^2(t)+b^2\cos^2(t)}dt=\frac{a}{a+b}-\frac{1}{2}$$ where $0 <b \leq a$. I used the residues but I could not prove this equality
1
vote
1answer
76 views

Multivariable Calculus Chain Rule

I am getting ready to write a notebook for the chain rule in Multivariable Calculus for my students. I do know: ...
0
votes
1answer
45 views

Setting trigonometric polynomial terms to zero

Consider an expansion $$\eta = a\cos (kx-\omega t) +\frac{1}{2} a^2 k \cos 2 (kx-\omega t).$$ I would like to compute expansions like $$ V = \frac{1}{2\pi} \int _0^{2\pi} \eta^2 \ d\theta ,$$ ...
1
vote
0answers
45 views

Numerical Derivative after numerical integration [closed]

I am trying to find the numerical derivative of a function whose argument defines the bounds of a numerical integral. ...
1
vote
1answer
74 views

Manipulate to determine $\delta$ given $\epsilon$ in continuity question

This is a continuation of a question I posed at: Examining the function $f(x,y)=xy(x^2-y^2)/(x^2+y^2)$. The quest is to analyze the partial derivative $$ f_x(x,y)=\begin{cases} ...
6
votes
1answer
81 views

Examining the function $f(x,y)=xy(x^2-y^2)/(x^2+y^2)$

Consider the function $$f(x,y)=\begin{cases} xy\dfrac{x^2-y^2}{x^2+y^2},&(x,y)\ne(0,0)\\ 0,&(x,y)=(0,0) \end{cases}$$ I can show that $f_x(0,0)=f_y(0,0)=0$. ...
2
votes
1answer
149 views

Evaluating a surface integral

I'm trying to compute the integral $$\int_S x^2+z^2\,{\rm d}S,$$where $S$ is the surface $$S\colon~ \frac{x^2}{2} + \frac{y^2}{3} + \frac{z^2}{2} = 1, \quad y \geq 0.$$ One possible parametrization ...
0
votes
0answers
64 views

Clairaut's Theorem

Clairaut's Theorem: Suppose $f$ is defined on a disk D that contains the point $(a,b)$. If the functions $f_{xy}$ and $f_{yx}$ are both continuous on D, then $$f_{xy}(a,b)=f_{yx}(a,b).$$ Now, check ...
1
vote
0answers
79 views

Meaning of D[f[x,y],x,y]

In calculus, we show our students: $$\frac{\partial^2 f}{\partial x\partial y} =\frac{\partial}{\partial x}\left(\frac{\partial f}{\partial y}\right) $$ Now, in Mathematica, which of the following ...
4
votes
1answer
104 views

Difficulty with total derivative

I am trying to introduce myself to the total derivative Dt in the documentation tutorial/TotalDerivatives. But I ran into what I think is a problem. There seems to be a serious problem in the ...
3
votes
3answers
174 views

Partial derivatives of an implicit equation

Looking for an easy way to find partial derivatives of an implicit equation. For example: Find $\partial z/\partial x$ and $\partial z/\partial y$ if $z$ is defined implicitly as a function of $x$ ...
0
votes
1answer
86 views

Why is the indefinite integral $e^\frac{x}{2}x^\frac{g}{2}dx$ the upper Gamma function in Mathematica? [closed]

I have an indefinite integral $e^\frac{x}{2}x^\frac{g}{2}dx$, I try to solve in Mathematica. Solve[Int[e^{x/2}x^{g/2-1}dx ,x]] (* Gamma[g/2,x/2] *) How does ...
0
votes
1answer
63 views

Divergent integral error

I'm integrating a product of two functions which is bounded on the interval from -Inf to 0, and diverges from 0 to +Inf. The error that I get says that the integral diverges on {0,+Inf}... How did ...
0
votes
1answer
64 views
0
votes
1answer
119 views

Existence of an analytical form for integral

Can the following integration be performed with an analytical output? ...
5
votes
1answer
114 views

ParametricPlot3D with tube and arrow

I have this: ...
3
votes
1answer
64 views
6
votes
1answer
105 views

Why is RegionMeasure so slow when calculating intersection area of a 2D and a 3D object?

If you think this description is too long, you can read the problem directly I know normally when one wants to calculate an region, this guide is useful. However, when it comes to calculating an ...
10
votes
1answer
183 views

Dirichlet coefficients as limits: wrong

Perhaps I should have included the word "bug" in my question. Here we go There is a bug in this Limit Limit[3^s (-1 - 2^-s + Zeta[s]), s -> ∞] (* 0 *) which ...
1
vote
0answers
47 views

Partial Derivative of f with respect to x1 [duplicate]

How can I calculate the partial derivative of a series like e.g. $$ a = \sum _{i=1}^n \left(2 x_i^2+z\right) $$ with respect to $x_{1}, x_{2},... x_{n}$ using Wolfram Mathematica? I tried it with: ...
2
votes
1answer
85 views

Implicit derivatives in one line. How do I do it?

I have this little script to derive implicitly $PrePrint = # /. {D[y_, x_, NonConstants -> {y_}] :> y'[x]} &; Example: ...
2
votes
0answers
68 views

Minimax for conditioned UnitStep functions

I made a simplified version of my problem, and I have now: ...
3
votes
1answer
75 views

How do I color Riemann integral rectangles using conditionals?

I'm builind a function to visualize the Riemann Integration process to any function using rectangles. Everything is fine, except that I need the color of the rectangles to change if they're under the ...
2
votes
2answers
127 views

MaxValue of a periodic function

I want to find the maximum of a periodic function on the interval [-Pi,Pi]. What I tried is the following: ...
6
votes
3answers
195 views

Limit of Nested Radical $\sqrt{1+\sqrt[2!]{2^2+\sqrt[3!]{3^3+…}}}$

I want to find limit of infinite nested radical $\quad \quad \sqrt{1+\sqrt[2!]{2^2+\sqrt[3!]{3^3+...}}}$ but I don't know how to define this expression in Mathematica. How can I define it and ...
5
votes
1answer
261 views

Need help with the Lagrange multiplier method

Well I am new to Mathematica and got really stuck solving this problem. In class, I was ask to use the Lagrange multiplier method to find the maximum and minimum value of $f(x,y) = x^2+y^2$ which ...
1
vote
2answers
83 views

Ask about the variable and constants in partial derivative in a loop

EDITED I have a function in a list like : func={2(x^2)+x*y+3(y^2)}; I want to have a UI that ask me which parameter (x or y) is a variable in a ...
6
votes
2answers
212 views

Curvature Application

I found these amazing animations at https://courses.engr.illinois.edu/tam212/avt.xhtml#avt which I think will be wonderfully useful when teaching multivariable calculus next semester. I've started to ...
1
vote
2answers
62 views

How to tell Mathematica that a limit converges to a product of two variables

I have looked through the manual and Im simply just stuck trying to make Mathematica solve this: ...
4
votes
3answers
134 views

How to get multiplication of the elements of a specific list/Matrix

Assume there is a list: List1={a,b,c,d}; Then I want to get these calculation: ...
6
votes
1answer
158 views

Defining a function implicitly and calculating its derivatives

I'm interested in using Mathematica's symbolic manipulation to obtain, for a particular function $f$, derivatives of arbitrary order evaluated at zero. Normally I'd use the ...
3
votes
2answers
346 views

How I can integrate $\int_0^{a} x e^{-\frac{b^2 x^2}{2 c}} J_0(n x) dx$?

How can I get a solution to the integral given below ? $\quad \quad \int_0^{a} x e^{-\frac{b^2 x^2}{2 c}} J_0(n x) dx$ where $a,\,b,\,c$ and $n$ are constants and $J_0$ is a Bessel function of the ...
0
votes
1answer
85 views

FullSimplify not giving me a physically useful answer for Mott scattering

I am using Mathematica to find the cross-section for Mott scattering over an extended target. I evaluated ...
3
votes
3answers
135 views

Calculating $\lim_{x\to 1} \, \int -\frac{i \text{Li}_2\left(x-x^2\right)}{\sqrt{3} \left(x-\frac{1}{2}-\frac{i \sqrt{3}}{2}\right)} \, dx$

How can we force Mathematica to compute this limit? $$\lim_{x\to 1} \, \int -\frac{i \text{Li}_2\left(x-x^2\right)}{\sqrt{3} \left(x-\frac{1}{2}-\frac{i \sqrt{3}}{2}\right)} \, dx$$ It seems it ...
2
votes
0answers
87 views

Converting integral equations to differential equations

I am trying to use Mathematica to convert integrals to differential equations, of any order. An example of an integral equation is given below, in Mathematica code. Could you please advise as to the ...
7
votes
1answer
91 views

How to calculate residues using different branches of the logarithm

My goal is to be able to compute the residues of something like $z^{\alpha} R(z)$ where $0 < \alpha < 1$, $R(z)$ is rational, and the exponential can be chosen with any branch. I found a way to ...
0
votes
0answers
47 views

Bounds of a closed curve

how to find the rectangular bounds of a curve which is of the form f(x,y) and forms a closed loop. The sides of the rectangle are parallel to the axes and are tangents to the curve. The time taken for ...
4
votes
2answers
158 views

Analytic expression for a Complex Hilbert Transform in Mathematica

I need to solve the following integral equations for a problem I'm working on - $\frac{-i}{2 \pi}$ $\int_{-a}^{a} \mathrm{dt}\,\, \frac{e^{i k t}}{t + i \tau}$ and $\frac{-i}{2 \pi}$ ...
0
votes
1answer
85 views

Strange behavior of `FourierSinCoefficient`

FourierSinCoefficient works normally like this and the result is checked with Integrate. ...
4
votes
1answer
76 views

Formal Differentiation

I'm trying to formally differentiate an expression. (I am aware of how to differentiate formally?, but am unable to generalize the idea) For example: ...
2
votes
1answer
118 views

Integration doesn't work

I'm trying to obtain the following integral: ...
2
votes
1answer
91 views

How to simplify and evaluate Integral with lots of DiracDelta dists

I have a question about an Integrals which I can do by hand but I want to implement this Integral in Mathematica. Actually it is the definition of a 2 Particle Phase Space. The Expression is the ...
5
votes
1answer
166 views

Confirming conservation laws for complex valued functions

Consider the nonlinear Schrödinger equation (I would like to do this for a more complicated set of equations, but to gain understanding I'll consider this simplified case) $$A_t+iA_{xx}+i|A|^2A =0,$$ ...
0
votes
0answers
61 views

PolyLog does not simplify

How come mathematica can not simplify a simple expression involving the PolyLog function? An example is: You know that ...
0
votes
0answers
74 views

Plotting Real and Imaginary parts of Riemann Surfaces

I'm working on a project where I need to take curves like, for example, $y^{2}=(x^{2}-1)(x^{2}-3)$ and plot their real and imaginary parts. More specifically, let $x$ just be real and then plot ...
10
votes
2answers
249 views

Asymptotics of $\frac{\sum _{i=0}^{\lfloor n/2 \rfloor} {2(n-2i) \choose n-2i} {n \choose 2i} {4i \choose 2i}}{2^{3n - 1}}$

I am fairly sure that asymptotically $$\frac{\sum _{i=0}^{\lfloor n/2 \rfloor} {2(n-2i) \choose n-2i} {n \choose 2i} {4i \choose 2i}}{2^{3n - 1}} \sim \frac{2}{\pi n}.$$ I tried ...
2
votes
2answers
117 views

Dealing with Norm, Complex situation not desired

I am working on a notebook for my calculus students and am dealing frequently with the norm of vector valued functions. I always seem to run into this kind of situation. ...
1
vote
1answer
85 views

Absolute uncertainty and error propagation with lists [closed]

I am trying to do some error propagation using mathematica. This is my second day using the program. I am trying to take the derivative of a numerical function that is made up of the following lists. ...