Questions tagged [vector-calculus]
Questions on dealing with vector calculus functions of Mathematica such as Grad, Div, Curl, Laplacian and their representations in various coordinate systems.
299
questions
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How to make Jacobian automatically in Mathematica
If we have two vectors, $a$ and $b$, how can I make Jacobian matrix automatically in Mathematica?
$$ a=\left(
\begin{array}{c}
x_1^3+2x_2^2 \\
3x_1^4+7x_2
\end{array}
\right);b=\left(
\begin{array}{...
- 923
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 ...
- 423
25
votes
4
answers
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views
What is the definition of Curl in Mathematica?
I have a usual mathematical background in vector and tensor calculus. I was trying to use the differential operators of Mathematica, namely Grad, ...
- 1,697
19
votes
4
answers
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views
How do I plot the unit normal field for a surface?
The question is pretty much in the title; I'm about to teach my multivariable calculus students about orientations on surfaces, and I would like to be able to show them pictures. Any ideas?
- 291
19
votes
2
answers
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Is it possible to do vector calculus in Mathematica?
I am trying to rearrange and manipulate some vector differential equations in Mathematica. As far as I understand you have to tell Mathematica that a variable is a vector by specifying the components ...
- 358
17
votes
1
answer
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views
How can I define or use a new coordinate system?
I want to use the dipole coordinate system as defined in this paper: http://arxiv.org/abs/physics/0606044
I see that Mathematica can do all kinds of vector analysis using different kinds of ...
- 417
15
votes
2
answers
6k
views
How to declare a 3D vector variable?
How can I do vector calculations without telling Mathematica the vector entries?
I have very many arbitrary linear combinations in $\mathbb{R}^3$ which I want to perform some general calculations on (...
- 335
15
votes
2
answers
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views
How do I plot a proper streamline plot, including spacings and line endings?
Mathematica includes two nice built-in tools to visualize vector fields, VectorPlot and StreamPlot. The latter is a useful tool, ...
- 10.2k
12
votes
2
answers
1k
views
Creating randomly oriented planes
I would like to create randomly oriented planes.
This is how I'm attempting to do that:
I create a 2 random unit vectors, $\mathbf{v}_1$, and $\mathbf{v}_2$, in the $x$-$y$ plane
I assume that if I ...
- 2,785
11
votes
1
answer
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views
Why I get the "Set::write: "Tag Times in is Protected." error?
Why I get the error below with this code:
...
- 361
11
votes
3
answers
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How to expand dot product by applying properties
I would like to expand a dot product which includes vectors $ \vec{v_1}, \vec{v_2}, \dots $ and constants $ c_1, c_2, \dots $
So that:
$$ c_1 \vec{v_1} \cdot \left(c_2 \vec{v_2}+ c_3 \vec{v_3}+\dots ...
- 277
10
votes
3
answers
988
views
Creating random configurations of spherocylinders or cylinders
About the setting:
We have a 3D simulation box with side $l$ and our catesian coordinate system is set with its origin at the centre of the box.
We have a number $N$ of spherocylinders of aspect ...
user52181
10
votes
2
answers
4k
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Singularities using VectorPlot
I am trying to plot a vector function of a fluid flow given by
$\vec{V} = (\frac{-\cos(\theta)}{r^2},-\frac{\sin(\theta)}{r^2})$
I am trying to plot it in Mathematica using the command below, I ...
- 715
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votes
1
answer
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Dipolar magnetic field lines inside a cylinder
I'm drawing the magnetic field lines of a rotating dipole (magnetic field distorded by emission of radiation). Currently, my Mathematica 7.0 code is fully working, but I'm having a constraint which I ...
- 3,949
10
votes
0
answers
405
views
matrix calculus with types (similar to matrixcalculus.org) [duplicate]
Is it possible to do matrix calculus in Mathematica, like in http://www.matrixcalculus.org/ ?
One of the main limitations I've found in Mathematica is that symbols are assumed to be scalars by ...
- 317
9
votes
1
answer
931
views
Einstein summation convention for symbolic vector calculus
I am trying to do some vector calculus in Mathematica in index notation form because it gives a clear result that can be compared to pen and paper calculations. Since there is no built in Einstein ...
- 765
9
votes
0
answers
178
views
Symbolically evaluating gradients/Hessians
I'm taking a machine learning course, which involves taking a lot of analytical gradients and Hessians. It would be ideal if I could perform these calculations in Mathematica. However, I am only aware ...
- 91
8
votes
2
answers
299
views
Confused about the solution obtained from vector linearization
I am trying to linearize a vector expression
$$\frac{(\mathbf{u}+d\mathbf{u})\times(\mathbf{v}+d\mathbf{v})}{\vert(\mathbf{u}+d\mathbf{u})\times(\mathbf{v}+d\mathbf{v})\vert}$$
Here is my code
...
- 193
8
votes
2
answers
684
views
Line Integral Difficulty
I am having difficulty with the following question:
Compute the line integral of
$$f(x,y)=\frac{xy}{1+x+2y},$$
along the unit quarter-circle in the first quadrant from (1,0) to (0,1).
My problem ...
- 14.8k
8
votes
3
answers
492
views
How is Grad defined for array particularly in non-Cartesian coordinates?
This question can be viewed as a follow-up of
What is the definition of Curl in Mathematica?
First argument of Grad can be an array, but what definition does ...
- 62.8k
8
votes
4
answers
1k
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Finding surface normal for 3D region at a specific point
I would like to find the surface normal for a point on a 3D filled shape in Mathematica.
I know how to calculate the normal of a parametric surface using the cross product but this method will not ...
- 4,090
8
votes
1
answer
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Smoothing/Averaging 2D Vector Fields
I have a list of 2D vectors defined by {{x,y},{u,v}} and would like to smooth or average the vectors. For example here are 2 vector fields, the second has noise ...
- 103
8
votes
0
answers
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Visualizing the Lagrange Multiplier Solution
This question is based on work in Susan Colley's Vector Calculus, 4th ed.
The question is to find and classify critical points of $f(x,y,z)=3xy-4yz+5xz$ subject to the constraint $g(x,y,z)=3x+y+2z=13$...
- 14.8k
7
votes
2
answers
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views
How to change coordinates of a differential operator?
I'm doing a basic quantum mechanics problem and am trying to learn how to do it in Mathematica. Any help would be much appreciated.
$\vec{L} = \vec{x} \times \vec{p}$
where $\vec{x}$ has components:...
- 81
7
votes
1
answer
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views
Why $u\cdot \operatorname{grad}(u)$ is not equal to $\operatorname{div}(u u)$?
The Navier-Stokes equation contains term
$\vec{u} \cdot \nabla \vec{u}$
which should be equal to
$\nabla \cdot \left(\vec{u}\vec{u} \right)$
provided
$\nabla \cdot \vec{u}=0$.
However this ...
7
votes
1
answer
2k
views
Plotting a parametrically defined vector field
Let's say that you have a three dimensional surface defined by two parameters. E.g.
$$ S(\theta,\phi) = \{\cos{\theta}\sin{\phi},\sin{\theta}\sin{\phi},\cos{\phi}\} $$
Additionally, you have some ...
- 173
6
votes
3
answers
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How to calculate the surface integral of a vector field?
Suppose the oriented surface is described as, the outside of an upper hemisphere $S:x^2+y^2+z^2=1$ inside the cylinder $x^2-x+y^2=0$
The vector field is : ${\vec F}=<x^2,y^2,z^2>$
How to ...
- 2,997
6
votes
2
answers
467
views
How do I verify a vector identity using Mathematica?
I am trying to verify well known Frenet–Serret formulas in general setting using Mathematica.
I need to consider a general space curve $r(s)=(x(s),y(s),z(s))$ in $\mathbb{R}^3$ and define its unit ...
- 359
6
votes
1
answer
3k
views
Lie-Bracket of two vector fields
I'm new to Mathematica (installed couple of hours ago) and I need to compute a few Lie brackets between two vector fields $f$ and $g$.
$$ f\left(\mathbf{x}\right) =
\left(
\begin{array}{c}
...
- 63
6
votes
3
answers
1k
views
Working with abstract vectors
I often need to compute derivatives or integrals involving N-dimensional vectors (where the dimension could be equal to 2 or 3 but is not particularly relevant for the sake of the derivation). The ...
- 163
6
votes
1
answer
206
views
Comparing unit normal definition in calculus with FrenetSerretSystem
I'm trying to compare the unit normal definition in calculus texts (i.e., $\vec N=\vec T'/\|\vec T'\|$) where $\vec T$ is the unit tangent vector, with the unit normal vector returned by ...
- 14.8k
6
votes
1
answer
320
views
Is there a way to add my own coordinate chart?
You all may have seen something like this:
U = Laplacian[Phi, {r, theta, phi}, "Spherical"]
What I want is to add my own Chart with its own coordinates, metric ...
- 61
5
votes
2
answers
724
views
'gradient' function in MMA
How to calculate Numerical gradient of 2D arrays using the "gradient function" ("Matlab-like")?
"[___] = gradient(F,hx,hy,...,hN) specifies N spacing parameters for
the ...
- 1,816
5
votes
3
answers
4k
views
Extrema of a function of three variables
To identify and classify the critical points of the function $f(x,y,z)=x^3+xz^2-3x^2+y^2+2z^2$, I used the Hessian matrix method.
...
- 14.8k
5
votes
1
answer
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Plotting a complicated vector field
If we consider the vector $\left ( A \cdot \nabla \right) \: B$, we have in Cartesian coordinates
$$\left ( A \cdot \nabla \right) \: B = \left ( A \cdot \nabla B_x \right ) e_x + \left ( A \cdot \...
- 389
5
votes
1
answer
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views
Finding unit tangent, normal, and binormal vectors for interpolated function
As an extension of this question, is it possible to find the unit tangent, normal, and binormal vectors for an interpolated function? eg
...
- 8,102
5
votes
1
answer
2k
views
Get the vector Norm without absolute values? [duplicate]
I want to get the Norm of a vector which involves sines and cosines (So I really need to replace it by the Pythagorean theorem). Since I know that all my results are going to be real and positive I ...
- 673
5
votes
2
answers
351
views
Finding vector of same direction with smallest integer coordinates
To determine Miller Indices of crystal lattice planes I would need a stable algorithm which determines the smallest set of integer coordinates of a vector which has same direction as a given vector (e....
- 2,841
5
votes
2
answers
363
views
How do I get actual values from a Jacobian matrix?
I have a Jacobian function:
D[{Sqrt[x^2 + y^2], ArcTan[x, y]}, {{x, y}}]
It gives me a matrix with the formulas I need for my transposition matrix:
...
- 1,550
5
votes
1
answer
1k
views
5
votes
1
answer
2k
views
Vector calculus integration identity problem
This is a follow up from another post. I was using the integration symbols available in the Basic Math Assistant palette.
I am new to vector calculus operations. There is a known identity found in my ...
- 1,408
5
votes
1
answer
309
views
Extending D for four-derivatives
I'm currently trying to consistently define rules for extending D[] to four-derivatives. As 'backend' I'm using the package TRACER (http://library.wolfram.com/infocenter/MathSource/2987/), which can ...
5
votes
3
answers
794
views
Explicitly construct tensor quantities with given symmetries
I am trying to use mathematica to generate explicitely a tensor. I know multiple thing about it. Let's call it $C_{\mu\nu\lambda\sigma}$. This guy has to be symmetric in the two first indices ${\mu,\...
- 349
5
votes
1
answer
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views
Finding the attractors of the vector field constructed as the gradient of an interpolated 3D scalar field
I have a list of data existing as {{{x, y, z}, f},...} from which I construct a 3D interpolation function using Interpolation[ ]. I am able to construct a vector field from the gradient of this scalar ...
- 143
5
votes
1
answer
157
views
How to extend a root finding method from 1D to multi dimensions
The number (0.25 Pi) is a third order root of f[x] below.
Clear[f];
f[x_] := Sin[x] - Cos[x] - Sqrt[2] (x - 0.25 Pi);
Chop@Normal@Series[f[x], {x, 0.25 Pi, 4}]
-...
- 6,954
5
votes
1
answer
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views
Integration over a convex combination of a region: $\int_{\Omega} (w_1 z_1 + w_2 z_2)^{1-\sigma} d (z_1, z_2)$ where $\Omega = \{ z_1 + z_2 = 1\}$
Take $w,z\in R^{n}$. I am interested in integrating (as generically if possible)
$$\int_{\Omega}(w \cdot z)^{1-\sigma} d z$$
Where the domain of $\Omega$ is $1$ dimension, and includes the convex ...
- 957
5
votes
0
answers
351
views
How to verify the convexity of a function?
I have an optimization problem with the following objective function in $(x,y)$
$$
A\log \left(\sum_{i=1}^n x_i\right)+\log\left(1-\frac{f}{n}\left(\sum_{i=1}^n\frac{x_i}{y_i}\right)\right)
$$
where $...
- 173
4
votes
3
answers
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How do you show a cone inside of a sphere?
I have a sphere of radius 5 and a cone (i.e. z=√(x^2+y^2)) inside of it. I can show them separately, but I'd like to show them together. I've tried a few tricks with opacity, implicit specifications, ...
- 41
4
votes
2
answers
2k
views
Finding a vector field from a set of points
I have a data set of the type $\{x,y,z\}$ where $(x,y)$ is a point and $z$ is the "value" or magnitude at that point. This gives me a triangular sort of shape since $(x,y,z)$ is not defined along the ...
- 1,725
4
votes
1
answer
248
views
Summation of Kronecker deltas should give the dimension
I have a really simple problem but I don't know how to solve it. Basically, I am doing vecor manipulation and I am summing a lot of Kronecker delta here and there. How can I teach Mathematica that for ...
- 349