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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16
votes
4answers
7k views

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 ...
11
votes
2answers
5k 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 : ...
7
votes
3answers
1k 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 ...
32
votes
5answers
2k 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 ...
13
votes
2answers
4k 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?
13
votes
1answer
934 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 ...
8
votes
2answers
525 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 ...
12
votes
1answer
1k 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 ...
7
votes
4answers
1k 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 ...
12
votes
3answers
3k 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 ...
17
votes
2answers
315 views

Symbolic derivatives are being calculated numerically

Just found the following while debugging a problem. Mathematica is calculating the derivative of IntegerPart[x] in some odd way: ...
12
votes
2answers
1k 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 ...
5
votes
1answer
929 views

Calculating volume between two surfaces of revolution

Once again, I'm very new to Mathematica and winging it as I go. I probably don't have the math education at this point to even justify buying it in the first place, but that's off topic. I wanted to ...
-1
votes
3answers
1k views

How do I determine the maximum value for a polynomial, given a range of x values?

I need to determine the maximum value for y = a x^2 + b x + c, where I know the coefficients and the upper and lower x values. ...
11
votes
4answers
548 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 ...
8
votes
2answers
560 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) + ...
9
votes
3answers
618 views

Inverting a function in a certain region

InverseFunction works well for globally invertible functions, like f = 2*# + 2 &; InverseFunction[f] ...
4
votes
1answer
867 views

Symbolic integration in the complex plane

Context While answering this question, I defined (symbolic and numerical) path integrations as follows ...
16
votes
3answers
4k 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 ...
2
votes
1answer
709 views

Finding residues of multi-dimensional complex functions

Say I have a function $f$ of $n$ complex variables, $\{ z_i \}_{i=1}^{i=Nc}$. And then I want to contour integrate the expression such that for each $z_i$ its an integration on an unit circle about ...
10
votes
5answers
549 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 ...
9
votes
4answers
1k views

Plot a 2D vector path onto a surface

In my calculus 3 course, we're studying gradients and have a project that takes a combination of 3D Gaussian radial surfaces and a basic parametric path $r(t) = \{x(t),y(t)\}$ to see how the gradient ...
21
votes
3answers
1k 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. ...
9
votes
2answers
3k 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 ...
11
votes
6answers
4k views

Finding the centroid of the area between two curves

When I have an area bounded by curves, is there a built-in way to find the center of the area? Or do I have to plot it first and then use ComponentMeasurements on ...
22
votes
2answers
967 views

Bug in mathematica analytic integration?

I found Mathematica provides me a wrong answer for a relatively simple analytically solvable integral: ...
4
votes
2answers
1k 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: $$\oint_{\mid z \mid =2}\frac{1-e^z+z}{z^3 (z-1)^2}dz$$
6
votes
1answer
143 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 ...
9
votes
3answers
351 views

Maximize violating constraints

I have Maximize[{(h*10)/(300*(100 - (l^.5 + d^.4 + H^.6))), (l + d + H + h) == 669, l > 0, d > 0, H > 0, h > 0}, {h, l, d, H}] I believe ...
8
votes
2answers
853 views

Bug in Integrate for Mathematica

Consider the following: ...
7
votes
1answer
403 views

Mathematica 9 can't integrate this function but earlier versions could

Integrate[ ArcTan[x]/(1 + x) Log[(1 + x^2)/2], {x, -1, 1}] I used Mathematica 9.0.1 on Windows7 32bit, Mathematica 9 cannot compute this, but Mathematica 8 gives ...
7
votes
2answers
278 views

How to simplify the integration result of the following program?

In the following program, the fec[x] is a function depending on $x$ while q3[t] and q5[t] ...
7
votes
1answer
641 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 ...
29
votes
6answers
1k views

Finding length of intersection of two surfaces

I would like to know how we find the length of the intersection of two surfaces. For instance, in the following example,a surface intersects with a plane: How do we find the length of intersection ...
9
votes
3answers
3k views

How can I differentiate Numerically?

Mathematica has two ways to integrate: Integrate and NIntegrate. But what about D? ...
12
votes
5answers
2k views

How can you compute Itō Integrals with Mathematica?

How can you compute Itō Integrals with Mathematica? I tried searching through the documentations but I didn't find anything. P.S. I was not at all sure how to tag this question. I had to put in at ...
27
votes
4answers
3k views

How to find the period of an arbitrary mathematical function?

Is there a function to find the period of an arbitrary (possibly complex) function in Mathematica?
10
votes
3answers
2k views

Derivative of real functions including Re and Im

When deriving functions using Re, Im or Arg (and probably some other functions as well), ...
9
votes
2answers
609 views

Find closed form expression for series expansion coefficients [duplicate]

Is there a built-in function that will find a general expression for the coefficient of the series expansion of a function? Series will only give the explicit ...
9
votes
3answers
1k views

how to simplify symbolic integration

The following is the code: ...
6
votes
1answer
909 views

How to expand a function into a power series with negative powers?

Is there any way to expand this expression a+b(1-Exp[-T/(b c)]/(z-Exp[-T/ (b c)]) (where a, ...
4
votes
2answers
3k views

How to find (numerical) value of a derivative at point?

I have the following function: f[0, 0] = 0 f[x_, y_] := Exp[-(x^2 + y^2)^(-1)] How do I find its partial derivatives at any given point, including $(0,0)$? This ...
1
vote
3answers
629 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 ...
8
votes
1answer
324 views

ReplaceAll[] and Limit[] don't give correct results for this expression under extreme variables [duplicate]

Possible Duplicate: Funny behaviour when plotting a polynomial of high degree and large coefficients 1/x^2 + (3 + x)/(6 (1 - Exp[x] + x)) ——This is a ...
4
votes
1answer
95 views

When to use GenerateConditions -> True

Many functions, usuallly those involving integration, take a GenerateConditions option which often defaults to False, or at ...
4
votes
1answer
502 views

Order of integration changes output of indefinite multiple integral in Mathematica 7

I'm trying to integrate a form-factor used in the calculation of radiation between two rectangles in perpendicular planes. While the integral is usually done over fixed limits, I am trying to do the ...
-4
votes
1answer
426 views

Symbolic Definite Integration

How do I perform Integrate[Sin[x]^2/(z^2 + R^2 - 2 zRCos[x])^(3/2), {x, 0, Pi}] where z and ...
16
votes
4answers
6k views

Finding Limits in several variables

Is there a way to find a limit of a multivariable function, like $$\lim_{(x,y)\to (0,0)} f(x,y)$$ with Mathematica? When $f$ is continuous, we can use $$\lim_{(x,y)\to (0,0)} ...
15
votes
1answer
374 views

How to represent a continuous monotonic phase of Airy functions?

Note: In this question I am concerned only with real-valued variables and functions. DLMF, §9.8 Airy Functions, Modulus and Phase, formula $9.8.4$ defines the phase of Airy functions: ...
13
votes
2answers
800 views

Numerical partial derivative

For a one-variable numerical function, it's simple to calculate the derivative at a point with Derivative as Szabolcs has pointed out before: ...