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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0
votes
1answer
143 views

Taylor Series vs. Series Function [closed]

Could someone please explain why the these two functions give two different results ...
4
votes
2answers
142 views

A Chain rule proof using Mathematica

If $z=f(x,y)$, where $x=r \cos\theta$ and $y=r\sin\theta$, how can I use Mathematica to prove that: $$\frac{\partial^2z}{\partial x^2}+\frac{\partial^2z}{\partial y^2}=\frac{\partial^2z}{\partial ...
14
votes
2answers
382 views

Symbolic area calculation for a parametric self-intersecting closed curve

The parametric equation of the curve is: $$\begin{cases} x &= -9 \sin (2 t)-5 \sin (3 t) \\[6pt] y & = 9 \cos (2 t)-5 \cos (3 t) \end{cases}\quad t\in[0,2\pi]$$ which can be easily ...
2
votes
0answers
69 views

How can I get constant of integration? [closed]

When using Integrate, Mathematica sets a constant of integration automatically. But sometimes we are given that constant already and it should be used in order to ...
3
votes
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
77 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
82 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
150 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
65 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
109 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
188 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
265 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
159 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
92 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
86 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 ...