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.
52
questions
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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
42
votes
3
answers
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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
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
7
votes
2
answers
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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
25
votes
4
answers
1k
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, ...
- 1,707
5
votes
2
answers
727
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
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
4
votes
4
answers
1k
views
Hypergeometric function with a matrix argument
I am looking for the evaluation of a Hypergeometric function with a matrix argument as for example in Koev and Edelman or as showcased in this Wikipedia article.
From what I understand from ...
- 483
19
votes
4
answers
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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
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votes
1
answer
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Why I get the "Set::write: "Tag Times in is Protected." error?
Why I get the error below with this code:
...
- 361
10
votes
3
answers
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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
3
votes
1
answer
619
views
SphericalPlot3D of an OblateSpheroid via coordinate transformation
This is an effort to reproduce an ellipse and a hyperbola of revolution from OblateSpheroidal coordinates with constant $\eta$ and $\theta$ .
My approach consisted in getting a Coordinate ...
- 1,418
3
votes
2
answers
113
views
How to invoke JacobianDeterminant in MMA >9.0?
According to the JacobianDeterminant help for my MMA 11.0 (W7, 64-bit)
As of Version 9.0, vector analysis functionality is built into the Wolfram Language
but the only Jacobian headwords are for ...
- 2,560
0
votes
1
answer
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How do you manipulate this in 2D
Use Manipulate to create an interactive plot that shows the parametrically-defined curve described by {x[t], y[t]} = {t Sin[t], t^2/15} for ...
15
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2
answers
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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
10
votes
2
answers
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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
322
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
3
votes
1
answer
3k
views
Plot a function based on Derivative - Gradient field
I'm trying to plot the gradient field of a function in such a way that is possible to change it easily, only editing the function.
Consider the code:
...
- 693
3
votes
2
answers
401
views
Reconstructing a function from its gradients
I have a list of the components of the gradients, $\partial f/\partial x_i$, of a function $f(x_1,x_2,\cdots)$. Is there some neat way to reconstruct the function $f$?
One approach to doing this ...
- 168
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votes
1
answer
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How can I use a unit vector notation found in physic texts?
In physics (I'm thinking of Taylor's Classical Mechanics or Griffith's Classical Electrodynamics) it is common to use "hat" vectors to denote unit vectors like $\hat{x}$ or $\hat{r}$. If I wanted to ...
- 529
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votes
2
answers
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Finding scalar potential function [duplicate]
Consider the vector field defined by:
$$\vec F(x,y)=\langle 2xy-\sin x,x^2+e^{3y}\rangle$$
We can check to see if the vector field is conservative with the following calculations:
$$\begin{align*}
\...
- 14.8k
2
votes
0
answers
785
views
Symbolic derivatives with vectors and matrices
I'm currently trying to solve some problems using symbolic vectors and matrices of arbitrary size. However, I have some problems with understanding and verifying the results:
I defined the vectors as ...
- 167
1
vote
1
answer
182
views
Quadratic form derivative in Mathematica
How to correctly differentiate quadratic form by vector in Mathematica, i.e.:
$Q=\omega^T I_{p} \omega$
$\frac{dQ}{d\omega}= ??? $
...
- 2,344
0
votes
1
answer
274
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Visualizing combined Vectors
I'm trying to visualize a cross-section of a finite continuous solenoid's B-Field (cylindrical coordinate equations from Wikipedia). I believe I have calculated the radial and z-axis fields, but I'm ...
- 7,485
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
8
votes
3
answers
495
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 ...
- 62.9k
8
votes
4
answers
1k
views
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 ...
- 4,100
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 ...
- 3,007
6
votes
3
answers
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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
5
votes
2
answers
354
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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
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
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
1
answer
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4
votes
1
answer
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How does one plot a three-dimensional electric field in spherical coordinates?
I have the following three-dimensional electric field:
e[r_, θ_, ϕ_, t_] := (Sin[θ]/r)*(Cos[r - t] - Sin[r - t]/r)*{0, 0, 1}
where the {0,0,1} vector is the unit ...
- 1,627
4
votes
1
answer
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Manipulations with tensors that keep so(3,1) symmetry manifest
I am doing some manipulations with tensors in curved background. There are, however, coordinates, where Lorentz symmetry is manifest. So, on one hand, I am dealing with particular coordinates and, on ...
- 143
4
votes
2
answers
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How are Symbolic Vectors Defined?
It is my understanding that undefined symbols are considered scalars, so I can't index variables that aren't defined.
It is possible to define a symbol as a vector of symbols as follows, allowing ...
4
votes
2
answers
1k
views
Plot a level curve and its gradient
Suppose that I have the following ellipse function,
$f(x,y)=4x^2+y^2-5$.
The gradient of this ellipse is calculated as
$\nabla f(x,y)=[8x,2y]$.
I know how to plot and join them. It is easy. I do ...
- 1,243
4
votes
3
answers
1k
views
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
1
answer
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views
Using Vector Operations In Mathematica
I would like to calculate
$$ \mathbf T = (\mathbf B \bullet \nabla) \mathbf B$$
Where nabla (the upside down triangle) is the grad operator $(\partial/\partial x,\partial/\partial y,\partial/\...
- 4,100
3
votes
2
answers
201
views
Why is TensorExpand so slow for vector operations?
I would like to expand the following tensor expression:
...
- 1,121
3
votes
2
answers
124
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Solving an integral vectorial equation by iterations
I'm triying to solve a problem like the following:
$$
\binom{f(t)}{g(t)}=\binom{f_0(t)}{g_0(t)}+\lambda\int_0^t ds\hat{A}(s)\binom{f(s)}{g(s)},
$$
where $f_0(t),g_0(t)$ and $\hat{A}(t)$ (a $2\times2$ ...
- 491
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votes
5
answers
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Computing the intersection area of two disks in 3D
Closely related to this question about highlighting intersection of two disks, I am trying to figure out if one can do so similarly for disks embedded in $3D$ (e.g. in a bounding box). The difference ...
user52181
3
votes
5
answers
2k
views
How to obtain the gradient of a function as a function?
The Grad function allows me to get the gradient of a function like this:
...
- 1,867
2
votes
0
answers
341
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] ...
- 831
2
votes
1
answer
376
views
Generating random vectors using Mathematica and an inequality
Consider $n$-dimensional vectors $c^i = \left(c^i_1, c^i_2 ...c^i_n \right)$ , whose entries are complex and generated at random. These vectors are normalized such that
$$|c^i|^2 = 1.$$
Using ...
- 247
2
votes
0
answers
79
views
12.1 VectorPlot - How to show differing magnitudes in the new implementation
I'm graphing vector fields for a class. Magnitude is important at least initially. In 12.0 and earlier VectorPlot plotted the different magnitudes as length. In 12.1 VectorPlot represents magnitude by ...
2
votes
0
answers
261
views
Computing quadratic differential trajectories with Mathematica
There was a question about a particular case of this, Quadratic differentials; seemingly it contained too little information, so let me try again. This will be also a second take on my previous ...
- 1,770
2
votes
0
answers
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Draw Vector Tangent to Point on Parameterized Curve [duplicate]
I define the vector function $\vec{r}(t)$ by
r[t_] = {t, Cos[t], Sin[t]}
The command r'[t] gives the vector tangent to the ...
- 185
1
vote
1
answer
481
views
how to define unit vectors in mathematica
I'm struggling a little bit trying to understand how to address this problem, I would like to do this in mathematica: is just that I don't know how to do the dot product between $\mathbf{J}$ and $\...
- 13
1
vote
1
answer
875
views
How to use a convective derivative
I am trying to figure out how to do a convective derivative in Mathematica in the context of the Navier-Stokes equation. The issue is, I have no clue how dot product a vector with the gradient ...
- 11