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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In simple terms, what is a Tensor Space? [closed]

I've heard of tensor spaces, and they sound intriguing to my non-mathematically educated mind. I was wondering if anyone could give a, "Tensor Space for Dummies," rundown. If not possible to ...
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49 views

How can I make NDSolve aware of vectorial nature of an ODE?

I am trying to solve the equilibrium equation $\text{DIV } \mathbf P(\textbf{u}(x,y)) = \mathbf{f}$, where $\mathbf P$ is the stress tensor defined by $$ \mathbf P = \mathbf F + (1-\det\mathbf F)^...
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33 views

How to calculate the gradient in cylindrical coordinates in more efficient way?

I would like to obtain the gradient for the following function in cylindrical Coordinates ...
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Cannot derive `Norm` or `Normalize` when recreating Frenet Serret equations

I'm trying to calculate the torsion of a curve at a point using the following code: ...
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45 views

Eliminate doesn't work

I tried to define r by the polynomials a,b,c only. The result which I want is r=4/3 a. But this eliminate doesn't work! It is continue running without stop and without any output. Eliminate[{a == x[1] ...
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Why is ParametricPlot3D not producing a graph

I have the vector V = {2, y z^2, 3 y + z} I wanted to get a 3d parametric plot of V using: x[t] = Sin[t^2]; y[t] = t^2 - Cos[t]; z[t] = Sinh[t] - Cos[t]. So I did: <...
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Ket representation in mathematica

I have a simple case: I tried using ket representation for vectors in $\mathbb{C}^{2^2}$. I tired assing for Ket[] notatation from mathematica like this Ket[3] = $\{0,0,0,1\} $. We can assign it in a ...
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Functions of several variables composition

If we have two functions $G,F:\mathbb R^2\to\mathbb R^2$ then the mechanism $F\circ G(x,y)=F(G(x,y))$ gives us a new map $F\circ G:\mathbb R^2\to\mathbb R^2$ called its composition. In a "...
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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 ...
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1answer
56 views

Find Solution to Exact ODE with 4 Variables?

OK, I give up. I was trying to use the solution in this post to solve an exact ODE with 4 independent variables. This works ...
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1answer
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Aesthetic Constant Vectorial Field 3D-Plot

I am trying to 3D-Plot a constant Vectorial Field: ...
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1answer
37 views

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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64 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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28 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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89 views

Help with simplification

Could anyone please help me simply the following expression ...
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2answers
41 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
110 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
137 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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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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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
56 views

Using Grad and NDSolve with vector variables

A simple orbital model might be developed like this: ...
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1answer
205 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
135 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
542 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
67 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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1answer
54 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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253 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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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
63 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. Please include ...
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1answer
132 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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57 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
244 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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1answer
135 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
43 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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0answers
57 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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78 views

Vector calculus expression formatting

I have the following example vector calculus code ...
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2answers
146 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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154 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
58 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
179 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
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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 ...
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1answer
664 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
126 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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116 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
387 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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