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.
311
questions
1
vote
1
answer
54
views
How to Solve Partial Derivative with Summation (for Linear Regression)?
I'm demonstrating linear regression so I would like to calculate derivatives 1 & 2 using Mathematica, something like this
This is based on the textbook notation in Johnston's Econometric Methods ...
- 31k
0
votes
1
answer
67
views
ArcTan action in more than two dimensional space [closed]
There is a four-dimensional vector, (w, x, y, z).
Can ArcTan, ArcTan[w, x, y, z] be applied? Or what is the alternative to it?
- 575
1
vote
0
answers
69
views
Integration by parts for Green's first identity
How should one implement
$$\int _{\Omega }\nabla u\cdot \nabla v\,d\Omega \ =\ \int _{\Gamma }v\,\nabla u\cdot {\hat {\mathbf {n} }}\,d\Gamma -\int _{\Omega }v\,\nabla ^{2}u\,d\Omega\,,$$
Symbolically ...
- 141
3
votes
2
answers
131
views
Plotting a vector field on a cylinder
Assume a parametric equation for a cylinder
$$\mathrm{cyl} (\theta, z) = (r \cos\theta, r \sin\theta,z)$$
and a vector field given by
$$\mathrm{vecField} (\theta, z)=\frac{\sin(\alpha)}{r}\partial_\...
- 31
5
votes
3
answers
392
views
Creating a 3D Gradient Vector Field Plot in Mathematica
I am new to Mathematica and currently learning how to visualize mathematical functions and their gradients. I am trying to reproduce a specific image that illustrates the gradient of a two-variable ...
- 51
0
votes
0
answers
41
views
Perturbing a tensorial expression
I am new to Mathematica. I am trying to simplify an expression of the some form like:
$$
n_i \sigma_{ij} n_j - \gamma n_i \hat{\sigma_{ij}} n_j = 2 + v_i x_i + \kappa E_{ij} \chi_{ji}
$$
There are ...
0
votes
0
answers
76
views
Symbolic vector calculus rookie mistakes
I'm trying to learn how to do vector calculus in Mathematica, just seem to be missing some points along the learning curve.
With assumptions
...
- 115
2
votes
1
answer
93
views
Vector calculus in generalised orthogonal frame of reference
I want to write Navier-Stokes equations in generalised orthogonal frame of reference in Mathematica. I therefore want to expand gradient and other vector calculus operations using metric factors such ...
- 55
1
vote
1
answer
87
views
Recursive application of Grad in scalars
I am looking for an efficient way to apply Grad recursively to a scalar function. The code I have so far is in the structure below. I define the scalar function outside the loop, take its derivatives ...
- 669
2
votes
0
answers
131
views
Cavity resonator modeling with Wolfram Mathematica
I am attempting to determine eigenfrequencies and the corresponding electric field distribution in a rectangular cavity resonator with perfectly conducting walls. In the simplest case of a rectangular ...
- 41
0
votes
0
answers
26
views
Simplify vector calculus based on component splitting
I am interested in verifying a few Vector Calculus expressions using Mathematica.
These expressions are based on splitting a vector field into its transverse components (in the xy- plane) and ...
2
votes
1
answer
92
views
How to create a vector as a tensor object for different euclidean bases?
The components of a tensor are always displayed with respect to one or multiple basis vectors.
For a tensor of rank 1, a vector, in 3D-euclidean space, we resort to three orthonormal basis vectors.
...
- 45
2
votes
0
answers
85
views
Defining my own set of orthogonal curvilinear coordinates
I want to use Vector Analysis functions (Grad, Div,Curl etc) with the following set of simple Toroidal coordinates $(r, \theta, \phi)$:
$x = (R+r \cos\theta)\sin\phi$, $y = (R+r \cos\theta)\cos\phi$, $...
- 21
0
votes
0
answers
111
views
Why won't this magnetic field plot?
I have been having some longstanding trouble in getting a 2D vector plot of a magnetic field to show up. For context, I am working with Lienard-Wiechert potentials and as such trying to plot the curl ...
- 55
1
vote
1
answer
172
views
PlotLegends` and VectorAnalysis is now obsolete on Mathematica v 12.0.0.0 Home edition [closed]
My code was working a few months ago. Now I receive the messages:
...
- 175
3
votes
2
answers
118
views
Eigenvalues and classification of critical points [closed]
I started with a function (x,y) and tried to write the code to work out the eigenvalues and classify the critical points.
The output it all good up until I try to use Which[] to classify the critical ...
- 31
2
votes
1
answer
86
views
Differentiating unit tangent vector does not work
I am new to Mathematica and I want to visualize the tangent and normal vector field of a particular embedding of the circle. In my code it is [CapitalGamma][s]. When I put the unit tangent vector (uT[...
- 35
1
vote
1
answer
101
views
Plotting the electric field of the Lienard-Wiechert potentials
I am having a very rough time trying to properly define and then vectorplot the expression for the electric field of the Lienard-Wiechert potentials I have been trying to code. Up to this point, the ...
- 55
0
votes
0
answers
78
views
How to get $ (\vec{A} \cdot \nabla) \vec{b}) $ to evaluate in Wolfram language [duplicate]
I am quite new to Wolfram language, and I am trying to use it to prove the equality $$ \nabla (A \cdot B) = A \times (\nabla \times B) + B \times (\nabla \times A) + (A \cdot \nabla)B + (B \cdot \...
1
vote
2
answers
228
views
Map water runoff flows near Ellicott City, Maryland
Ellicott City, Maryland was the site of a flash flood several years ago due to the local topography. How can we use the Wolfram Language to show the direction water flows at many points near that city?...
- 969
4
votes
2
answers
846
views
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 ...
0
votes
0
answers
33
views
How do I prevent, or get rid of the <code>Abs^\[Prime]</code> in the output of a partial of a Normalize[vector]? 12.0.0 for Microsoft Windows [duplicate]
12.0.0 for Microsoft Windows
How do I prevent, or get rid of the Abs^[Prime] in the output of a partial of a Normalize[vector]?
The code below shows the problematic ...
- 1,567
1
vote
1
answer
86
views
'not a variable' error when applying D function [closed]
I am trying to apply differentiation, D, operator in Mathematica to a vector. Importantly, I want to carry out the operation with respect to another vector. Basically, I am computing a Jacobian matrix....
- 55
1
vote
0
answers
72
views
Vector Potential Function of Vector Field with DSolve?
(This is a redux of this.)
Why doesn't this work? I have a vector field F1 whose div is 0 with a known vector potential function A2, and I try to get DSolve to solve the differential equation for A2 ...
- 263
2
votes
1
answer
123
views
Preventing Mathematica from applying product rule
I am programming viscous terms of compressible Navier-Stokes equations
...
- 55
1
vote
2
answers
133
views
Showing each component of velocity in a helix
In this animation, I want to show each component of velocity (V_x, V_y and V_z) with arrows ...
- 2,688
2
votes
1
answer
73
views
Simplifying expressions with dot or non-commutative product (without the help of NCAlgebra)
I have two expressions (with dot and non-commutative product):
$f=J^T.x+J.x$
$g=J^T**x+J**x$
I want to simplify them and get the following expressions:
$f=(J^T+J).x$
$g=(J^T+J)**x$
In an attempt to ...
- 2,464
4
votes
3
answers
493
views
Calculating Matrix Derivatives
I'm sorry if this is an easy question, but I'm trying to refresh my memory on calculating derivatives of matrix products and have been reading the Matrix Cookbook. I'd like to rebuild my intuition by ...
- 215
1
vote
0
answers
118
views
How to define a vector operator?
I know how to define a single-variable differential operator like
Clear[Op]
Op[x_] := a D[#, {x, 1}] &
and I can do operations like
...
- 199
3
votes
5
answers
1k
views
Calculating length of curve based on data points?
I had around $1000$ data points $(x,y,z)$ and when I plot them they seem to lie on a closed curve. I carefully checked the data points and the points do not repeat rather they are somewhat close to ...
- 199
3
votes
1
answer
221
views
Why isn't the vector field being plotted over the entire torus?
I am trying to plot a vector field over a torus. Here is my code and the image it produces:
...
- 215
2
votes
1
answer
66
views
Solving a vector diferential equation lead to ComplexInfinity Encountered non-numerical value for a derivative at t == 0
I'm trying to solve the following Vector PDE:
I implemented the following code:
...
- 763
0
votes
0
answers
148
views
Implementing the 3D Radon transform
I am wondering how to implement the Radon transform, the 3D Radon transform, that is, given a 'density' function $f: \mathbb{R}^3\to \mathbb{R}$ The Radon transform of $f$ is $$Rf(s,w)= \int_{x\cdot w=...
- 109
2
votes
0
answers
54
views
Using `GradientFilter[]` and `GradientOrientationFilter[]` to reproduce Matlab's `gradient()` function
In 'gradient' function in MMA, two methods are given to reproduce Matlab's gradient() function. I thought that ...
- 1,210
0
votes
0
answers
62
views
How to express elemental operations in matrix form?
I ran the following code:
X = Array[x, {5, 4}];
W = Array[w, {4, 3}];
Y = Array[y, {5, 3}];
D[Norm[X.W-Y, "Frobenius"]^2, {W}]//MatrixForm
However, mma ...
- 1
0
votes
0
answers
54
views
How to input a specific three-variable function into this definition?
I have the following expression:
...
- 101
0
votes
0
answers
48
views
Looking to compute $e^{\phi}\Delta^2 f e^{-\phi}$ in a coordinate-free way
Let $f, \phi:\mathbb{R}^n\to\mathbb{R}$ with $\phi(x)$ looking like $\phi(x)=(x+a)^2$ for some $a\in\mathbb{R}^n$. I am interested in computing $$L_\phi(f)=e^{\phi}\Delta^2 \left(e^{-\phi}\,f\right)$$ ...
- 101
0
votes
0
answers
63
views
Calculation of SolidAngle not recognizing region?
Basically I'm trying to calculate the solid angle of a square seen from the origin. I defined the region (in two different ways) and tried running the function but it does not give any output.
Why is ...
- 58
3
votes
2
answers
130
views
Solving an integral vectorial equation by iterations
I'm triying to solve a problem like the following:
$$
\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
0
votes
0
answers
46
views
Output of expressions unable to have operations be performed on the expression
I am currently trying to use the code linked in the catalog of spacetimes pdf to calculate chirstoffel symbols for a metric which follows as
n := 4
...
- 119
0
votes
0
answers
65
views
Algorithm: Permutations & Signature
Theoretical side
Simple example: If I have two sets $A_1=\{1,3\} ,A_2=\{2,3\}$
and Permutations[{1, 2, 3}]= $\left(
\begin{array}{ccc}
1 & 2 & 3 \\
1 &...
- 1,014
1
vote
1
answer
90
views
Derivative of quadratic form in NCAlgebra
I installed the NCAlgebra package in the hope that it contains enough rules for symbolic work with most branch-matrix expressions.
...
- 2,464
2
votes
1
answer
170
views
Inactive form for PDE and symmetry
I would like to solve the following non-linear Poisson equation (as a toy problem for a more complicated problem)
...
3
votes
2
answers
176
views
Set of quadratic forms and linear algebra
I have a set of quadratic forms.
$L_{1}=u_1^TJ_{1}u_1$
$L_{2}=u_2^TJ_{2}u_2$
$L_{3}=u_3^TJ_{3}u_3$
where $u_{i=1,2,3}$ - 3$\times$1 vector;
where $J_{i=1,2,3}$ - 3$\times$3 matrix;
I need to pack ...
- 2,464
0
votes
1
answer
243
views
How do I find the unit vector in the direction of the minimal rate of change?
A ball falls on the surface $x^4 z+x^9 y^2+y^2 z^3=65$ at the point (-1,-3,2). How do I find a three-dimensional unit vector in the direction that the ball will begin to roll. I know that the ball ...
- 115
1
vote
1
answer
177
views
How to compute the divergence of a four-vector?
I have a quadri-vector which is given by
u = {(E^(-φ0[r]))*(1 - ε δφ[t,
r]), (E^(-φ0[r])) D[ε ξ[t, r], t], 0,
0}
and a quantity n which is given by
<...
0
votes
0
answers
69
views
Difference in covariant derivative and divergence?
defining polar Minkowski metric (with indecies down in polar coordinates) as
gdd = {{-1, 0, 0, 0},{0, 1, 0, 0},{0, 0, r^2, 0},{0, 0, 0, r^2 Sin[\[Theta]]^2}}
...
- 790
0
votes
1
answer
94
views
Finding the Curvature of the path of a particle
How do I find the curvature of the vector when t=5?
The function is:
v[t_] := {2 t^4 + 2, 6 t^3 + 1, 3 t^2 + 4}
- 115
1
vote
1
answer
400
views
The square of Abs of a complex number
We all know that
4^2 + 22^2 = 500
Is there a natural way to generates 500 from the complex number ...
- 4,325
3
votes
1
answer
232
views
Pattern matching with two variables - derivatives on 4-vectors
I am interested in pattern matching with two variables and not just one, and more specifically implementing derivatives on 4-vectors. In the case of interest the 4-vectors have two indices; one is in ...
user49048