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.
825
questions
32
votes
4
answers
26k
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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 ...
- 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....
- 533
77
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)...
- 17.1k
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 ...
- 711
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
...
- 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?
- 26.3k
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 ...
- 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 ...
- 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 ...
- 70.6k
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 ...
- 107
34
votes
2
answers
1k
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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 ...
- 116k
10
votes
3
answers
812
views
How to do algebra on unevaluated integrals?
I am working with functions calculated from a set of general basis functions.
...
- 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 ...
- 517
16
votes
2
answers
17k
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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 :
...
- 6,418
31
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 ...
- 272k
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$$
- 181
18
votes
2
answers
3k
views
Getting an InterpolatingFunction from a ContourPlot
I have a function, say
minimizeme[Ω_][β_][ϵ_] = ϵ^2 Ω - Log[2 (Cosh[2 β] + Cosh[2 β ϵ])]/(2 β);
I want to find its critical points in $\epsilon$ for a given ...
- 1,186
10
votes
2
answers
2k
views
Simplifying the derivative of $|x|$
Context
In[855]:= D[Abs[x], x] /. x -> 1
Out[855]= Derivative[1][Abs][1]
In[856]:= D[x, x] /. x -> 1
Out[856]= 1
Question
Why is ...
user1602
21
votes
1
answer
6k
views
How does Mathematica integrate?
Basically, this question can be considered to be an extenstion to my other question.
What I wanted to do was this integral as homework (it is indefinite BTW so no approximations using Simpson's Rule ...
- 1,129
15
votes
2
answers
3k
views
Plot showing discontinuity where it shouldn't
I was trying to integrate a continuous function with a kink and I did it two ways and both ways the plot of the result shows a discontinuity. I also later want to differentiate the Integrated function....
- 6,888
13
votes
5
answers
1k
views
Double series over primes
I'm very curious if the following double series over primes has a closed form:
$$\sum_{k \in \mathcal{P}}\sum_{n \in \mathcal{P}}\frac{1}{k\;n(k+n)^2}$$
where $\mathcal{P}$ denotes the set of all ...
- 1,415
10
votes
2
answers
3k
views
Derivative of conjugate multivariate function
I have a problem with Mathematica, taking the derivative of the conjugate of some function. I know that a similar question has been posed before here, but the solution did not work for multivariate ...
- 101
8
votes
4
answers
2k
views
Higher-order partial derivatives w.r.t. variable raised to some power
This is a follow-up question to Differentiate w.r.t. squared variable.
I want to be able to compute partial derivatives of the form $$\partial_{x^n}^m \, f(x) = \frac{\partial^m}{\partial^m x^n} \, f(...
- 1,281
5
votes
3
answers
594
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, ...
- 1,201
19
votes
3
answers
6k
views
11
votes
1
answer
443
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 <...
- 1,217
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 ...
- 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,
...
- 321
30
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 ...
- 1,181
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 ...
- 5,329
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 ...
- 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 ...
- 3,311
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:
...
- 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. ...
- 1,217
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 $n = 3$
...
- 886
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]
...
- 14.9k
9
votes
1
answer
307
views
When to use GenerateConditions -> True
Many functions, usuallly those involving integration, take a GenerateConditions option which often defaults to False, or at ...
- 1,095
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 ...
- 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 \...
- 655
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), ...
- 19.1k
15
votes
2
answers
978
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
$$\frac{\partial}{\partial z}=...
- 363
14
votes
4
answers
9k
views
How to plot and find the volume of a solid?
How to plot and find the volume of the solid enclosed between the paraboloid z=5(x^2+y^2) and z=6-7x^2-y^2
And the answer of ...
- 141
14
votes
2
answers
7k
views
Derivative of the Absolute Value Function in Mathematica looks very different from the actual answer
Derivative of the Abs Function in Mathematica looks very different than the actual answer. Why? How to correct this?
For example I entered ...
- 241
11
votes
2
answers
2k
views
Why aren't these additions of integrals and summations equal?
I have the following code:
Simplify[Integrate[f[x] + g[x], x] == Integrate[f[x], x] + Integrate[g[x], x]]
To test:
$$\int{\left(f(x) + g(x)\right)dx}=\int{f(x)...
- 1,141
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 ...
- 919
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'...
- 919
3
votes
3
answers
426
views
Evaluate a certain three-dimensional constrained integral
The result of the three-dimensional integration
...
- 2,335
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 ...
- 51
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 ...
- 7,793
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)
$$\int_0^\infty e^{-x \cosh\xi}\, d\xi = K_0(x)$$
...
- 1,424