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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.

2
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1answer
117 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 ...
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2answers
60 views

How can I make an arrow's shaft more visible?

{Graphics[{Arrowheads[.1], Arrow[{{0, 0}, {2, 1}}]}], Graphics[{Arrowheads[.1], Arrow[BezierCurve[{{0, 0}, {1, 1}, {2, 0}}]]}]} say I have the above code. Can I ...
0
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1answer
41 views

Make Arrows Smaller

I have the following program that describes a 3D curve in space with an osculating circle and TNB vectors moving with the curve. The arrows indicating the TNB vectors are way too large making it seem ...
0
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1answer
70 views

Triple product in spherical coordinates

Given six real numbers $a,b,c,d,e,f$ (say between $0$ and $\pi$) I would like to express the following determinant in a compact and "reasonable" way: $$ \det \begin{bmatrix} \sin a \cos b & \sin ...
16
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3answers
646 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, ...
0
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2answers
82 views

Verifying equality $ (\vec a \times \vec b)^2 = \left| \vec a \times \vec b \right| ^2 $

I'm trying to verify the equality $ (\vec a \times \vec b)^2 = \left| \vec a \times \vec b \right| ^2 $ in Mathematica. How can I do it? Thank you for your time.
1
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1answer
41 views

Hessian matrix product in Taylor expansion of vector function

I am trying to get the 2nd order coefficient of the Taylor expansion at $\pmb{x}=\pmb{0}$ of ...
0
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0answers
33 views

Methods for generating dense packings of capsules

Are there methods inherent to Mathematica that offer efficient ways of filling a finite cubical box by as many capsules as possible (or given a target number) without any overlaps between them? ...
0
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0answers
45 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 ...
4
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2answers
88 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,\...
4
votes
1answer
146 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 ...
0
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1answer
47 views

Recombining vectors after taking derivatives

I want to map two expansions order by order and to solve to get unknowns coefficient. We define $X_{\mu}= \frac{(x-y)_{\mu}}{(x-y)^2}$. The quantity I need to expand in the end is $t_{\mu\nu}= \frac{...
6
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2answers
279 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 ...
4
votes
1answer
75 views

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 ...
3
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2answers
153 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
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1answer
164 views

DiracDelta convergence in 3D - Cartesian vs. spherical coordinates

Integrating DiracDelta in 3D in Cartesian coordinates works just fine i.e. gives vecf[{x, y, z}] ...
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0answers
64 views

Plotting or Visualizing a Higher dimensional vector field

I am trying to visualize a higher dimensional vector field using Mathematica. Is there a way to do this. As an example, one can use $$\begin{eqnarray} \dot{x}&=& \left(J-R\right)x \end{...
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0answers
21 views

keep variables in the correct position when TensorExpand

I would like to expand a tensor expression of 3-d vectors. However, the code runs really slow, the problem is asked here: Why is TensorExpand so slow for vector operations? However, I found that by ...
3
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2answers
117 views

Why is TensorExpand so slow for vector operations?

I would like to expand the following tensor expression: ...
0
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1answer
33 views

Integrating a vector according to elements of another vector

I have two vectors, $\bar{x},\bar{v}$ and I want to produce a third vector such that:$$ u_i = \int_{x_i}^{x_i+\delta} f(v_i,t)dt \ .$$ I tried (with a simple function for example): ...
8
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2answers
273 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 ...
2
votes
1answer
106 views

How to take the curl of a vector function involving hypergeometric functions?

I have a vector function involving a hypergeometric function as its inner constituent. I need to take the curl of this vector and when I do, Mathematica prompts this array of errors: ...
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0answers
33 views

Del as a Differential Operator: (Matrix times Del) cross vector [duplicate]

I tried to reply to this answer, but don't have enough reputation points yet. Basically the poster constructed Del (i.e. $\nabla = \left( \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \...
3
votes
2answers
195 views

How to solve equations combined with vector variables with unknown length

I want to solve the following equaitions combined with vector variables and scalar variables. How can I use it in MMA? \begin{aligned} \mathbf{x}+w\mathbf{a}-\mathbf{v}&=0\\ \mathbf{v}&\ge 0\\...
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0answers
39 views

How do you treat gradient as a vector? [duplicate]

I need to express $\nabla = (\partial_x,\partial_y,\partial_z)$ as a vector, so that I can compute the differential operator given by $A\cdot\nabla$ (where here $A$ is a 3x3 matrix) using Mathematica. ...
5
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1answer
70 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 ...
7
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0answers
94 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 ...
0
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0answers
62 views

Integration involving vectors, in 3D

Since I'm new to Mathematica I was looking for ways to learn vector integration and how to do them efficiently, so I came across this article where they did calculation for field in a point above a "...
0
votes
1answer
68 views

Plot the curve into the xz plane with time interval

At time t ≥ 0, a new laser rocket is at the position: ...
1
vote
1answer
61 views

Plotting cross-sections of $z=4x^2+y^2$ parallel to the $yz$-plane

I want to plot the cross-sections of $z=4x^2+y^2$ parallel to the $yz$-plane. I have tried the below code ...
0
votes
1answer
279 views

How to plot circle, with it centered at a point in 3D in a plane of xyz equation? [duplicate]

I can figure out how to plot this for calc 3. Question: a) Plot the circle of radius 3 centered at the point $\{-1, 1, 1\}$ in the plane whose xyz-equation is $2(x + 1) + 3(y - 1) + (z - 1) = 0$. ...
2
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1answer
83 views

Passing a function into another function defined with Module and using it there

I'm trying to write a generic module that will take a vector-valued function of a space curve and show the animation of a unit tangent vector traversing the curve. I found a really nice example from ...
3
votes
1answer
73 views

How to get a list of 3D coordinates using point and rotation

I have one point: p1={82,80,0} I want to rotate it around the Y axis. The start time is 0 seconds ($0s$) and the end time is 5 seconds ($5s$). ...
2
votes
1answer
391 views

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 ...
2
votes
2answers
397 views

Does Mathematica 11 have spherical coordinate unit vectors?

I have a spherical vector wave for an electric field: e[r_, θ_, ϕ_, t_] := (Sin[θ]/r) [Cos[r - t] - Sin[r - t]/r] which points in the ϕ direction. In mathematics,...
1
vote
2answers
43 views

Create matrix from combinational arrangement of vectors

I am just trying to produce a short hand code which shall do the following: I have a function of 2 vectors $f(\vec{v_1},\vec{v_2})$ f=#1^2+#2^2-#3-#4&; and ...
0
votes
1answer
48 views

How to solve for points in a region of a plane? [closed]

I have three points in 3D Cartesian space: A = {-0.154, -0.246, -0.439}; B ={-0.0055, -0.3945, -0.3895}; C= {-0.154, -0.444, -0.241}; that all lie on the ...
1
vote
0answers
217 views

numerical gradient of an array

I have 2D arrays of numeric data and I would like to calculate the vector gradient of these arrays. I would like to achieve something like the 'gradient' function of MatLab: https://www.mathworks....
3
votes
2answers
105 views

Symbolic representation of vector function

I want to symbolically represent a function $p: \mathbb R^n \to \mathbb R^n$, where the eventual goal is to compute an exact partial derivative. The function in question is given by $$ p_i(z) = \...
1
vote
3answers
63 views

Unexpected behavior when using Cross

I'm trying to calculate the equation of the plane that is formed from the position of the Earth, Moon and Sun using precise coordinates from NASA's SPICE toolkit. I found this extremely clear ...
5
votes
3answers
236 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 ...
9
votes
3answers
609 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 ...
2
votes
2answers
157 views

How can I do integration with the Green theorem?

I have an integral $$\int_C xy^2dx-4x\sin y\,dy$$ where $C$ bounded with some constrains, for instance inside $x^2+y^2=1$ and below $y=x^2$. I can integrate of one variable and also with some ...
7
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0answers
157 views

How to locate a stream line starting from a saddle point?

I read in a table of gradient vectors and plot using the ListStreamPlot. I want to locate and color the line that represents the "boundary" of the flux. A flux line that 1)starts from a saddle point 2)...
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2answers
185 views

Trying to define the Lie bracket of two vector fields

I am trying to define in the simplest possible way (only one coordinate system, no checking that variables are vectors, etc.) the Lie bracket of two vector fields in 3-space. What is wrong with the ...
6
votes
1answer
187 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 ...
1
vote
1answer
174 views

Vector analysis in curvilinear coordinates

It is known that vector calculus in 3D takes quite a simple form when one uses orthonormal curvilinear coordinates (check out, for example, here, in section "Differentiation"). Is there any ...
2
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1answer
475 views

How to plot a vector field on a circle?

I have two vector fields (electric fields) for inside and outside an sphere of radius $R$ (lets suppose $R=1$), expressed as: ...
0
votes
1answer
164 views

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 ...
1
vote
1answer
297 views

Plotting vector field in cylindrical co-ordinates

I am trying to plot the following. Let $\Gamma^\top \Gamma=1$, and $e_3=\{0,0,1\}$ (unit vector along Z-axis). I am trying to plot $\dot \Gamma=e_3\times \Gamma$. Tried the following: $\Gamma =\{\...