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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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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
70 views

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

There is the following vector-matrix equation. where C - nonzero constant. [x1 x2 ... xn] - vector of real numbers. M - matrix of real numbers of dimension n x n It is required to find a vector $x$...
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3answers
27k views

How to make Jacobian automatically in Mathematica

If we have two vectors, $a$ and $b$, how can I make Jacobian matrix automatically in Mathematica? $$ a=\left( \begin{array}{c} x_1^3+2x_2^2 \\ 3x_1^4+7x_2 \end{array} \right);b=\left( \begin{array}{...
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1answer
68 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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34 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
99 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
29 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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1answer
83 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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0answers
51 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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1answer
218 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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0answers
65 views

Vector calculus expression formatting

I have the following example vector calculus code ...
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2answers
96 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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1answer
93 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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4answers
871 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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1answer
42 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
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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2answers
101 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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2answers
575 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,...
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1answer
177 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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77 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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1answer
128 views

Curl and other functions of functions

Where can one find the source code of the Mathematica function Curl? Alternatively, how can one define a function the argument of which is restricted to be a 3D vector function of three variables ...
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2answers
148 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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0answers
51 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
76 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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2answers
68 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
87 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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2answers
92 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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0answers
78 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
139 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,\...
2
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1answer
121 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: ...
4
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1answer
153 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 ...
6
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2answers
300 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
177 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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1answer
98 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
384 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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2answers
131 views

Why is TensorExpand so slow for vector operations?

I would like to expand the following tensor expression: ...
8
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2answers
281 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 ...
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0answers
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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1answer
36 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): ...
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1answer
82 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 ...
3
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2answers
214 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
37 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}, \...
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0answers
40 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. ...
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0answers
171 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 ...
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1answer
102 views

Plot the curve into the xz plane with time interval

At time t ≥ 0, a new laser rocket is at the position: ...
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1answer
82 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 ...
3
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2answers
122 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) = \...
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1answer
566 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 $2(x + 1) + 3(y - 1) + (z - 1) = 0$. ...