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.

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Referencing the Component of a vector

If I were to define a vector r = {2, 0} + t*({5, 4} - {2, 0}) >>{2 + 3 t, 4 t} Sqrt[(D[r[1]])^2 + (D[r[2]])^2] How do I properly reference a single ...
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Line Integral with Vector Plot Parametric Plot and Show

How can I make different plottings of this line integral where $C$ is the square bounded by lines $x = \pm 1$ and $y = \pm 1$. $$ \begin{equation} \int_C 4y^3dx-2x^2dy \end{equation} $...
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58 views

How to reproduce `JacobianMatrix` function

I want to use the JacobianMatrix function in package VectorAnalysis, but I get the following warning message after running the ...
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26 views

Plot trajectory and angle of object using multivariable vector plot

I want to plot a projectile motion graph, with and without drag. So I want the x-axis to be the horizontal displacement. y-axis is the change in angle. z-axis is the vertical displacement. ...
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77 views

Help with simplification

Could anyone please help me simply the following expression ...
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2answers
36 views

About composition of functions (I don't want to use a vector variable function)

f2[x_, y_] := x^2 + y^3; g[t_] := {t^2, 3*t+1} $g$ returns a list, but $f_2$ is a two variable function. So, we cannot compose these functions: ...
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1answer
105 views

Efficiently finding spheres furthers apart along and perpendicular to a given direction

Setting I am dealing with a box of spheres, all having the same radius and inserted into a finite box size without any overlaps. Along a given direction $\mathbf v$, I am trying to find: (a) The two ...
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1answer
112 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 ...
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40 views

Obtain the vector potential and magnetic vector of infinite wire carrying charge

I'm working on the question above, but I'm really stuck for days. my goal is to implement the following Problem in Mathematica. Any help would be appreciated. I want to evaluate the following ...
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0answers
53 views

Find vectors describing two interconnected pyramids in Euclidean space

I need the help of competent linear algebra specialists in Mathematica. In the Euclidean space, two tetrahedrons are turned at different angles. Their vertices coincide with the beginning of the base ...
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1answer
53 views

Using Grad and NDSolve with vector variables

A simple orbital model might be developed like this: ...
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1answer
172 views

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 ...
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Error when trying to solve elastic PDEs using FEM: “Compile::argcompten: The comparison, LessEqual, is invalid for tensor arguments.”

I'm trying to solve the following linear elastic problem: The pairs of numbers beside each node is the coordinates in meters. The loads are in Newtons. The part of codes that probably have no ...
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2answers
99 views

Summations on Variables with subscripts

One of my equations uses a set of variables which ultimately for a set of certain values gives a list of 8 numbers: c=13.9506 (31 - k) following cc(sub k) = Round[Table[13.950621339931203...
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5answers
502 views

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 ...
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2answers
65 views

How can I concisely define a matrix-form partial derivative operator?

I have a vector $\boldsymbol{u}(x,y) = \begin{bmatrix} u_x(x,y) \\ u_y(x,y) \\ \end{bmatrix}.$ I would like to get $\boldsymbol{\sigma}(x,y) = \begin{bmatrix} \sigma_x \\ \sigma_y \\ \tau_{xy}...
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48 views

How can I take the divergence of a symbolic vector? [duplicate]

I would like to compute expressions that have the following form: $$\left(\partial_{p_1} - \partial_{p_2}\right)^2 \left(\partial_{p_1} - \partial_{p_2}\right)^2 \frac{(p_1-p_2)\cdot(p_3-p_4)}{p_1^...
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2answers
244 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: ...
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50 views

How do you get a partial derivative of a vector to another vector?

I have a translation function between Cartesian and Polar coordinates: xi[xPrime_] := {xPrime[[1]]*Cos[xPrime[[2]]], xPrime[[1]]*Sin[xPrime[[2]]]} I now want to ...
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1answer
60 views

Computing the average curvature of a curve [closed]

Compute the average curvature of the curve $\qquad r(t) = ⟨t \cos t,\, t \sin t⟩\ \mathrm{for}\ 0 ≤ t ≤ 2π$. Use Mathematica to arrive at an exact value and its numerical approximation. ...
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1answer
127 views

Visualising and detecting stacked oblate particles

Context: I am trying to learn how to visualize a box filled with oblate-shaped particles. These are in contrast to prolates which can be drawn using for example ...
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1answer
40 views

RecurrenceTable: numercial exception

When I added vector b into expression, it returned with RecurrenceTable::excptn: Value {2/3,-(1/3),2/3,-(1/3)} is a numerical exception. ...
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49 views

Implementing a generic product rule

I'd like to implement a generic rule for expanding differential forms by the product rule. For instance, I'd like to transform the following PDE: $\phi \frac{\partial \phi}{\partial t} = \phi \nabla^...
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1answer
198 views

Solve the vector-matrix equation. Minimize the length of the desired n-dimensional vector

There is the following vector-matrix equation: $$\mathbf x^\top\mathbf M\mathbf x=\begin{bmatrix}x_1&x_2&x_3\end{bmatrix}\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}...
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Mathematica Scalar Field on 3D Space

Fellow Mathematicai, is there any way to create a visual plot of a Scalar Field on 3D space in Mathematica? With Gratitude, - SDH
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1answer
121 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 ...
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1answer
39 views

Vector shape and Dimensions

I think I have not understood clearly the difference between dimension and shape. I want to construct a vector remembering recurrence of the type : ...
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56 views

How to get results in terms of vector operations?

Is there a simple way to simplify an expression in terms of vector operations? For example, when I evaluate this; ...
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74 views

Vector calculus expression formatting

I have the following example vector calculus code ...
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2answers
136 views

Solving equation with Curl in angular velocity

In 3 dimensions, angular velocity is given as: $$\boldsymbol{\omega} = \frac{1}{2} \nabla \times \boldsymbol{v}$$ How can I use mathematica to solve for velocity $\boldsymbol{v}$, if I give a value ...
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129 views

Does Laplacian use first argument “theta” for polar angle or azimuthal angle?

Only example I could find in the help for Laplacian is this My question is, is $\theta$ meant to be the polar or the azimuthal angle? Using $\theta$ for polar is ...
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1answer
57 views

Problems encountered in plotting direction fields

I learned how to plot direction fields according to the answer of @Robert Jacobson to the question How can I plot the direction field for a differential equation?. However, when I tried to plot the ...
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1answer
150 views

Laplacian operator examples

I was reading up in the help about the Laplacian operator. and tried to understand the one example I saw: Laplacian[Sin[r^2],{r,\[Theta]},"Polar"] results in: ...
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1answer
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 ...
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1answer
595 views

Solving Laplace equation in spherical coordinates

I was trying to solve Laplace's equation for a spherical capacitor, which is not difficult by hand, just to figure out the commands so I can eventually try something more complicated. Then, I ran into ...
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1answer
117 views

Where to find order of arguments for default functions

Lets take for example the Laplacian. So I want to apply it in spherical coordinates, so I go the the associated documentation page (https://reference.wolfram.com/language/ref/Laplacian.html?view=all) ...
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105 views

Curl in Polar Coordinates

I was trying to figure out Curl in other coordinates. The Curl documentation (https://reference.wolfram.com/language/ref/Curl.html ) didn't really say how Mathematica interpreted the input. Anyway, I ...
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2answers
344 views

How to add a vector variable to a vector value?

If we have a three dimensional vector variable: $Assumptions = a ∈ Vectors[3]; and a three dimensional vector value for example: ...
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52 views

Curl of a scalar in Mathematica [duplicate]

What is the reason for Mathematica to produce expressions with opposite signs? ...
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2answers
116 views

Vector calculus with index notation

How would you calculate this in Mathematica? $$\sum_i\partial_i\partial_i G_{jk}$$ $$G_{jk}=\bigg(\frac{\delta_{jk}}{r}+\frac{(x_j-y_j)(x_k-y_k)}{r^3}\bigg)$$ where $r=|\boldsymbol{x}- \boldsymbol{...
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1answer
392 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
80 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 ...
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1answer
116 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 ...
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1answer
96 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 ...
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4answers
935 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, ...
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2answers
96 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.
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2answers
120 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 ...
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0answers
115 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 ...
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2answers
231 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,\...

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