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.

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32 votes
4 answers
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Finding unit tangent, normal, and binormal vectors for a given r(t)

For my Calc III class, I need to find $T(t), N(t)$, and $B(t)$ for $t=1, 2$, and $-1$, given $r(t)=\{t,t^2,t^3\}$. I've got Mathematica, but I've never used it before and I'm not sure how to coerce ...
a98's user avatar
  • 423
36 votes
4 answers
11k views

Usage of Assuming for Integration

For some reason, when I enter the following integration in Mathematica Assuming[{k ∈ Integers}, Integrate[ Exp[ I k t], {t, -π, π}]] the result turns out to be 0....
TriSSSe's user avatar
  • 533
76 votes
5 answers
27k views

Multivariable Taylor expansion does not work as expected

The basic multivariable Taylor expansion formula around a point is as follows: $$ f(\mathbf r + \mathbf a) = f(\mathbf r) + (\mathbf a \cdot \nabla )f(\mathbf r) + \frac{1}{2!}(\mathbf a \cdot \nabla)...
matheorem's user avatar
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59 votes
6 answers
7k views

Generating evenly spaced points on a curve

In the KnotData package a simple command such as points = Table[KnotData[{3, 1}, "SpaceCurve"][t], {t, 0, 2 Pi, 0.1}]; will ...
scallionpancake's user avatar
20 votes
2 answers
8k views

How to make traditional output for derivatives

Hello again after some pause. i have a problem how to present partial derivatives in traditional form, not as Mathematica gave it to me in its notation. So I want to present this ...
Pipe's user avatar
  • 1,099
45 votes
2 answers
23k views

How to calculate contour integrals with Mathematica?

How to calculate the integral of $\frac{1}{\sqrt{4 z^2 + 4 z + 3}}$ over the unit circle counterclockwise for each branch of the integrand?
user64494's user avatar
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14 votes
3 answers
2k views

Why does Integrate declare a convergent integral divergent?

When I try this command Integrate[1/Sqrt[(s^2 - u)^2 - 1], {s, m, Infinity}, Assumptions -> u > 2 && m > 10] Mathematica declares that the ...
Davide's user avatar
  • 141
30 votes
3 answers
29k views

Quick Hessian matrix and gradient calculation?

I am absolutely new to Mathematica and I actually want to try implementing a little optimization method . Long story short assuming I have a predefined two-variable function f(x,y) I want to ...
Pranas's user avatar
  • 409
11 votes
3 answers
6k views

Using D to find a symbolic derivative

I need to do the following: Define a function Take the derivative of this function and have a look at the symoblic representation Substitute in some values With the bonus that I want to use the ...
C. E.'s user avatar
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7 votes
5 answers
4k views

sliding a tangent line along a curve

I am confused by Dynamic. I am trying to slide a tangent line along a curve, without using DynamicModule, so that I can include ...
jamie's user avatar
  • 107
34 votes
2 answers
1k views

Symbolic derivatives are being calculated numerically

Update: (1) By V11, not sure of the exact version, the derivative IntegerPart' has been given a symbolic definition. (2) The numeric derivative computed has changed ...
Dr. belisarius's user avatar
10 votes
3 answers
805 views

How to do algebra on unevaluated integrals?

I am working with functions calculated from a set of general basis functions. ...
Jerry Guern's user avatar
  • 4,602
21 votes
3 answers
10k views

How can I implement the method of Lagrange multipliers to find constrained extrema?

I want to form the function $h=f-\lambda_{1}g_{1}-\lambda_{2}g_{2}$ where $f$ is the function to optimize subject to the constraints $g_{1}=0$ and $g_{2}=0$ so that I can solve the first partial ...
Logan's user avatar
  • 517
16 votes
2 answers
17k views

How to specify assumptions before evaluation?

If I request mathematica evaluate an integral for me, I'll often get a more general ConditionalExpression than I want. Example : ...
Peeter Joot's user avatar
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18 votes
2 answers
10k views

Paths integrals in the complex plane

I can't find how to calculate path integrals of complex functions in the complex plane. For example: $$\oint_{\mid z \mid =2}\frac{1-e^z+z}{z^3 (z-1)^2}dz$$
Agustin Santiago's user avatar
10 votes
2 answers
2k views

Simplifying the derivative of $|x|$

Context In[855]:= D[Abs[x], x] /. x -> 1 Out[855]= Derivative[1][Abs][1] In[856]:= D[x, x] /. x -> 1 Out[856]= 1 Question Why is ...
user avatar
30 votes
4 answers
13k views

Implementing discrete and continuous Hilbert transforms

What is an efficient and accurate Mathematica implementation of the Hilbert transform, for both continuous and especially discretely sampled functions? This transform relates phase and amplitude in ...
Mr.Wizard's user avatar
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21 votes
1 answer
6k views

How does Mathematica integrate?

Basically, this question can be considered to be an extenstion to my other question. What I wanted to do was this integral as homework (it is indefinite BTW so no approximations using Simpson's Rule ...
The-Ever-Kid's user avatar
  • 1,129
17 votes
2 answers
3k views

Getting an InterpolatingFunction from a ContourPlot

I have a function, say minimizeme[Ω_][β_][ϵ_] = ϵ^2 Ω - Log[2 (Cosh[2 β] + Cosh[2 β ϵ])]/(2 β); I want to find its critical points in $\epsilon$ for a given ...
Editortoise-Composerpent's user avatar
15 votes
2 answers
3k views

Plot showing discontinuity where it shouldn't

I was trying to integrate a continuous function with a kink and I did it two ways and both ways the plot of the result shows a discontinuity. I also later want to differentiate the Integrated function....
Amatya's user avatar
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13 votes
5 answers
1k views

Double series over primes

I'm very curious if the following double series over primes has a closed form: $$\sum_{k \in \mathcal{P}}\sum_{n \in \mathcal{P}}\frac{1}{k\;n(k+n)^2}$$ where $\mathcal{P}$ denotes the set of all ...
user 1357113's user avatar
  • 1,415
10 votes
2 answers
3k 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 ...
camzor00's user avatar
  • 101
8 votes
4 answers
2k views

Higher-order partial derivatives w.r.t. variable raised to some power

This is a follow-up question to Differentiate w.r.t. squared variable. I want to be able to compute partial derivatives of the form $$\partial_{x^n}^m \, f(x) = \frac{\partial^m}{\partial^m x^n} \, f(...
Janosh's user avatar
  • 1,281
5 votes
3 answers
587 views

Integrating a real function I get a complex value, while after variable transformation the result is real. Bug?

I have the following integral: Integrate[1/Sqrt[0.7 + 0.3*(1 + z)^3], {z, 0, ∞}, Assumptions -> z ∈ Reals] >> -3.36354 - 3.85013 I The output is complex, ...
Santiago's user avatar
  • 1,201
19 votes
3 answers
6k views

how to simplify symbolic integration

The following is the code: ...
howard's user avatar
  • 403
11 votes
1 answer
438 views

What exactly does GenerateConditions do?

Consider for example this strange behavior: Integrate[1/x, {x, 0, Infinity}, GenerateConditions -> False] (*0*) I'd also like to know the difference between <...
Emerson's user avatar
  • 1,207
9 votes
2 answers
2k views

The time-like geodesics (orbits) in the Schwarzschild spacetime

I am trying to plot Schwarzschild's orbit without invoking the geodesic equation. As a reference I am using Chandrasekhar's Book (The Mathematical theory of Black Holes, Oxford University Press). In ...
ricci1729's user avatar
  • 196
32 votes
3 answers
6k views

How to deal with complicated Gaussian integrals in Mathematica?

As we know, for most Gaussian integrals, we can get the analytical result. Now I have many Gaussian integrals to treat, which have the following general form, ...
simpler's user avatar
  • 321
29 votes
2 answers
6k views

How does Mathematica understand branchcuts of the complex logarithm?

Say I have the function $f(x) = x \tanh(\pi x) \log (x^2 +a^2)$ where $a$ is some positive real number. Then it seems to be me that Mathematica when given such a ...
user6818's user avatar
  • 1,171
17 votes
5 answers
3k views

How to make a Line[] with no end?

I'm trying to do this: In this graph, the secant points are aproximated in order to become the tangent, it seems I need some kind of function which plots a line based on two points and it's points ...
Red Banana's user avatar
  • 5,203
15 votes
5 answers
3k views

Graphically approximating the area under a curve as a sum of rectangular regions

I am completely new to Mathematica. Basically I was trying to write a code to plot a function and draw the approximate area by rectangles. To be more precise, plot a function ...
Gio's user avatar
  • 201
4 votes
3 answers
9k views

Step-by-step definite integration

I am interested in obtaining the intermediate steps for rather straightforward symbolic definite integrations (e.g., from an elementary calculus text). As an example, for $\int_0^1 x\,dx$ I'd like ...
JohnD's user avatar
  • 3,301
40 votes
2 answers
2k views

Bug in mathematica analytic integration?

Bug introduced in 9.0 or earlier and fixed in 13.0 or earlier I found Mathematica provides me a wrong answer for a relatively simple analytically solvable integral: ...
Collector's user avatar
  • 503
29 votes
3 answers
5k views

Can we teach Mathematica about functional differentiation?

The key relation for functional differentiation is $$\frac{\delta}{\delta f(y)}f(x)=\delta(x-y), $$ where $\delta(x-y)$ is the Dirac delta function, and the usual properties of differentiation (e.g. ...
Emerson's user avatar
  • 1,207
15 votes
3 answers
13k views

How to find the nth derivative?

This question is not the same as my last one. How do you find the $n$-th derivative where $n$ is a variable? For example, you can find the nth derivative for a specific $n = 3$ ...
T. Webster's user avatar
9 votes
1 answer
304 views

When to use GenerateConditions -> True

Many functions, usuallly those involving integration, take a GenerateConditions option which often defaults to False, or at ...
Daniel Mahler's user avatar
9 votes
3 answers
1k views

Inverting a function in a certain region

InverseFunction works well for globally invertible functions, like f = 2*# + 2 &; InverseFunction[f] ...
David's user avatar
  • 14.9k
8 votes
3 answers
5k views

Creating the Nabla operator (also known as Del operator) as an operator

How can I define the nabla operator (also known as Del operator) as a an operator, acting on everything to the right of the operator! Also taking \[Del]^2 would ...
TehHO's user avatar
  • 109
27 votes
3 answers
7k views

Visualizing a Complex Vector Field near Poles

I've been playing around with a visualization technique for complex functions where one views the function $f: \mathbb{C} \rightarrow \mathbb{C}$ as the vector field $f: \mathbb{R^2} \rightarrow \...
Jackson Walters's user avatar
20 votes
3 answers
9k views

Derivative of real functions including Re and Im

When deriving functions using Re, Im or Arg (and probably some other functions as well), ...
celtschk's user avatar
  • 19.1k
15 votes
2 answers
957 views

What is the best way to define Wirtinger derivatives

Wirtinger derivatives ( also called Cauchy operators) in complex analysis are widely used tools. They are defined in the case of one dimensional complex plane as follows $$\frac{\partial}{\partial z}=...
Aymane Fihadi's user avatar
14 votes
4 answers
9k views

How to plot and find the volume of a solid?

How to plot and find the volume of the solid enclosed between the paraboloid z=5(x^2+y^2) and z=6-7x^2-y^2 And the answer of ...
user11253's user avatar
  • 141
14 votes
2 answers
7k views

Derivative of the Absolute Value Function in Mathematica looks very different from the actual answer

Derivative of the Abs Function in Mathematica looks very different than the actual answer. Why? How to correct this? For example I entered ...
user14056's user avatar
  • 241
11 votes
2 answers
9k views

Equation of a line that is tangent to a curve at point

A common problem in the derivative section of calculus texts is "find the equation of the line that is tangent to the curve $y = \ldots$ at the point $P$." To find the line that is tangent to $y = 2 ...
dharmatech's user avatar
11 votes
2 answers
2k views

Why aren't these additions of integrals and summations equal?

I have the following code: Simplify[Integrate[f[x] + g[x], x] == Integrate[f[x], x] + Integrate[g[x], x]] To test: $$\int{\left(f(x) + g(x)\right)dx}=\int{f(x)...
Matt Groff's user avatar
  • 1,111
5 votes
5 answers
6k views

Using implicit differentiation to find a line that is tangent to a curve at a point

A few days ago I asked about using differentiation to find a line that is tangent to a curve at a given point. J.M. provided a very elegant way to solve these kinds of problems in Mathematica. Now I'...
dharmatech's user avatar
3 votes
4 answers
3k views

How do I evaluate several n-th derivatives of a function at one point?

I have a question where I have to compute a Table containing $f^{(n)}(0)$ for n = 1, ..., 5, where $f^{(n)}$ denotes the $n$th ...
ppkjref's user avatar
  • 51
3 votes
3 answers
426 views

Evaluate a certain three-dimensional constrained integral

The result of the three-dimensional integration ...
Paul B. Slater's user avatar
23 votes
2 answers
5k views

Differentiating functions of vectors/matrices?

I'm dealing with derivatives of scalar functions of matrices and wondering if Mathematica can help me here. The standard approach of expanding it in terms of components is cumbersome. As an ...
Yaroslav Bulatov's user avatar
22 votes
2 answers
2k views

Incorrect results for elementary integrals when using Integrate

Bug introduced in 8.0 or earlier and persisting through 13.2 or later There is a rather simple integral ($K_0$ is the 0-th order MacDonald function) $$\int_0^\infty e^{-x \cosh\xi}\, d\xi = K_0(x)$$ ...
Fabian's user avatar
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