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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5
votes
1answer
160 views

Visualize transformation used in multiple integration

In class, we will sketch the region $D$ in $R^2$ bounded by the curves $xy=1$, $xy=4$, $y=x$, and $y=x+2$. Then we will use the change of variables given by $u=xy$ and $v=y-x$. We need to show that ...
7
votes
1answer
157 views

Old fashioned region method?

Just stumbled across this idea in http://users.rowan.edu/~hassen/Mathematica/Volume%20III/Chapter%2015.pdf. ...
0
votes
0answers
59 views

The system can not be solved

I'm trying to solve this equation: f[a_][r_]=((a*r/12)*(1+(r/12))^n)/(((1+(r/12))^n) - 1) When I type Solve[f[a][r]==p, a] ...
12
votes
1answer
894 views

1/x integration giving 0 instead of infinity [duplicate]

I stumbled across the fact this integral: Integrate[1/y, {y, 0, 1}, GenerateConditions -> False] returns 0 Anybody ...
3
votes
1answer
180 views

Derivative of conjugate multivariate function

I have a problem with Mathematica, taking the derivative of the conjugate of some function. I know that a similar question has been posed before here, but the solution did not work for multivariate ...
-1
votes
1answer
83 views

Wrong condition for the convergence of the integral $\int_0^\infty t^a \exp(-i t^b) dt$

The integral $\int_0^\infty t^a \exp(-i t^b) dt$ converges for $-1 < a < b - 1$ or $b - 1 < a < -1$. However, ...
4
votes
0answers
58 views

How to take the second derivative of a complicated function using limits [closed]

I have a very complicated function of one variable consisting of exponential and error functions combined in various ways. Mathematica can't take the derivative of this function in the usual way ...
0
votes
0answers
24 views

Show[cp1,cp2, …] but PlotLabel in cp2 does not show up [duplicate]

I am visualizing the extrema of $f(x,y)=x^3+3y^2$ constrained to the level curve $g(x,y)=xy=-4$, found by using the Lagrange Multiplier Method. ...
2
votes
1answer
109 views

Why can't Mathematica do this limit?

(Glancing past the fact that one can do this by hand in a second...) Are there any extra conditions I can give Mathematica to make it evaluate limits like this? I'm not sure why it's getting choked ...
2
votes
1answer
166 views

Calculate the limit of an integral

How can I use Mathematica to find the limit:$$\lim_{x\rightarrow 0^{+}}\frac{\int_{1}^{+\infty}\frac{e^{-xy}\quad-1}{y^3}dy}{\ln(1+x)}=?$$ I tried this ...
0
votes
2answers
121 views

How does one define a function that takes vectors as arguments but also derivatives?

Summary: tldr; I just want to implement a version of the following mathematics function: $$ f(a) = c_1 e^{-(a - t_1)^2} + c_2 e^{-(a - t_2)^2} + c_3 e^{-(a - t_3)^2}$$ in mathematica, and be able ...
7
votes
1answer
195 views

Why should the spatial derivative order of the ODE *not* exceed two?

Following this question I came across this strange behaviour. Let me define a 1 D interval implicitely ...
1
vote
1answer
131 views

How does one verify the derivative of $ f(x) = \sum^{K_2}_{k_2=1} c_{k_2} \exp\left(- \big(a^{(2)} - t^{(2)}_{k_2}\big)^2\right)$

I wanted to verify the derivative of the following equation: $$ f(x) = \sum^{K_2}_{k_2=1} c_{k_2} \exp\left(- \big(a^{(2)} - t^{(2)}_{k_2}\big)^2\right)$$ for that I wanted to write a symbolic ...
1
vote
1answer
71 views

ND and D yielding strange results

Both of the following codes, using ND and D, differentiates my function Flux[] incorrectly. If the derivatives are plotted, it can be seen that both are the original function multiplied by some ...
0
votes
0answers
23 views

Plotting a dispersion curve with complex wavevectors [duplicate]

I want to reproduce a dispersion curve from P. Zou's thesis found here: http://www.ipd.anl.gov/anlpubs/2001/06/39620.pdf on Pg. 31, Eq. 2.13. The problem is, the code below only gives values for the ...
5
votes
3answers
271 views

Extrema of a function of three variables

To identify and classify the critical points of the function $f(x,y,z)=x^3+xz^2-3x^2+y^2+2z^2$, I used the Hessian matrix method. ...
2
votes
0answers
116 views

Evaluating Real and Imaginary parts in a dispersion equation via ContourPlot [closed]

I want to reproduce a dispersion curve from P. Zou's thesis found here: http://www.ipd.anl.gov/anlpubs/2001/06/39620.pdf on Pg. 31, Eq. 2.13. The problem is, the code below only gives values for the ...
2
votes
1answer
64 views

Integrate complains about incompatible quantities in integration limits

When I run the following code, Integrate throws ...
1
vote
1answer
75 views
1
vote
1answer
73 views

Issue with Differentiation over a tensor

I am new user in Mathematica. I am working with liquid crystal theory, and one of the notation there is the tensor: $ \Pi_{ij} = \frac {\partial {F_{el}}} { \partial ( \partial {n_i} / \partial { ...
1
vote
2answers
105 views

How to get the desired (equivalent) answer from Integrate?

This is the integrand: 1/beta/(2*z)/(1 - beta) // TraditionalForm Integrating with respect to z, gives ...
0
votes
1answer
56 views

Why can not the vector be transposed in integrating over a region? [duplicate]

Clear[x, A]; A = {{2, -1, 0}, {-1, 2, -1}, {0, -1, 1}}; Integrate[Exp[-x.A.{x}\[Transpose]/2], x \[Element] FullRegion[3]] This code raises an error ...
3
votes
2answers
142 views

Help with Integration

I am new to Mathematica, and I need help with integration of the following Reliability expression (from a well-known Reliability Model). $$R(t)=e^{-Ne^{-bt_i}(1-e^{-bt})}, t \geq 0$$ I have ...
2
votes
0answers
75 views

Derivative of vector dot product with respect to a vector

I have the expression: Transpose[gvecI, {2, 1}].x (m[1] + m[2]) gvecI and x are [3x1] ...
5
votes
2answers
275 views

Module inside of Manipulate Example

Here is a very nice example from The Student's Introduction to MATHEMATICA. ...
5
votes
2answers
591 views

Order of a pole

Is there a simple way to determine the order of the poles of a rational function? I have a difficult function where I need Mathematica to find the poles. It would be interesting to also know what the ...
1
vote
1answer
60 views

How to apply Simplify only to elements of list that cause specific Warning

I have a set of replacement rules I use to analytically compute nested integrals of very long symbolic expressions since Integrate literally takes forever. To do ...
7
votes
1answer
94 views

Integrate yielding a ConditionalExpression but I don't think the condition is necessary

Suppose I take the PDF of the LogNormal distribution with parameters m and s evaluated at x. I obviously get an expression involving m. I now want to integrate that expression not with respect to x ...
1
vote
0answers
51 views

Discrepancy between laplace and fourier transform of gaussian

The function in question is Exp[x^2/2] for x>0 Laplace transform should be the same as the fourier transform since the function is absolutely integrable. Fourier: They are indeed the same ...
0
votes
1answer
84 views

Partial derivatives with respect to multiple variables

For any natural number $N$: $$_t k_{mn}^1 = \frac{1}{m!n!}\frac{\partial^{n+m} k_1 (0, 0)}{\partial x^m \partial t^n}, \qquad (m,n=0,1,\ldots, N)$$ where $k_1(x,t)$ is a known function, for example ...
12
votes
2answers
422 views

Why doesn't Mathematica evaluate this simple limit?

I want to evaluate $$\displaystyle\lim_{n\to\infty}\left(n-\sqrt{\sin(n)+10n+n^2}\right)^2$$ I used this code ...
5
votes
2answers
179 views

Finding the equation of a level curve

Is there a way in Mathematica to determine a parametric equation of a level curve in Mathematica. For example, consider: ...
1
vote
2answers
100 views

Easy way to add another parameter to Manipulate

Vector definitions: I'm sorry that I didn't include the Vector definitions. I forgot because I had them in initialization cells. Here they are: ...
2
votes
1answer
83 views

Limit problem calculating directional derivative

Given $f(x,y)=1-x^2-y^2$, find the directional derivative at the point $(x_0,y_0)$ in the direction of the unit vector $\vec u$. I am experiencing some strange behavior with the Limit function. I ...
0
votes
1answer
165 views

Taylor Series vs. Series Function [closed]

Could someone please explain why the these two functions give two different results ...
4
votes
2answers
176 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
438 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
79 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
133 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
70 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
3
votes
1answer
159 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
55 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
68 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
93 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
113 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
284 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
82 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 ...
3
votes
2answers
130 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
146 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
243 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 $x$ ...