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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.

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34 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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44 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
46 views

Using Grad and NDSolve with vector variables

A simple orbital model might be developed like this: ...
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1answer
126 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
55 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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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
62 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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42 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
233 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
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
117 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
38 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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36 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
143 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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38 views

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
107 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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33 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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53 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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68 views

Vector calculus expression formatting

I have the following example vector calculus code ...
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2answers
117 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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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
46 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
104 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
469 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
109 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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90 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
209 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
111 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
306 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
70 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
99 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
88 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
893 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
93 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
89 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
93 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
163 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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1answer
159 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 ...
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1answer
69 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{...
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2answers
305 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 ...
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1answer
107 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 ...
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2answers
515 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 ...
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1answer
179 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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25 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 ...
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2answers
133 views

Why is TensorExpand so slow for vector operations?

I would like to expand the following tensor expression: ...