# 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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### 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 ...
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### Singularities using VectorPlot

I am trying to plot a vector function of a fluid flow given by $\vec{V} = (\frac{-\cos(\theta)}{r^2},-\frac{\sin(\theta)}{r^2})$ I am trying to plot it in Mathematica using the command below, I ...
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### Why I get the “Set::write: ”Tag Times in is Protected." error?

Why I get the error below with this code: ...
118 views

### Symbolically evaluating gradients/Hessians

I'm taking a machine learning course, which involves taking a lot of analytical gradients and Hessians. It would be ideal if I could perform these calculations in Mathematica. However, I am only aware ...
1k views

### Dipolar magnetic field lines inside a cylinder

I'm drawing the magnetic field lines of a rotating dipole (magnetic field distorded by emission of radiation). Currently, my Mathematica 7.0 code is fully working, but I'm having a constraint which I ...
280 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 ...
182 views

### Line Integral Difficulty

I am having difficulty with the following question: Compute the line integral of $$f(x,y)=\frac{xy}{1+x+2y},$$ along the unit quarter-circle in the first quadrant from (1,0) to (0,1). My problem ...
196 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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### Smoothing/Averaging 2D Vector Fields

I have a list of 2D vectors defined by {{x,y},{u,v}} and would like to smooth or average the vectors. For example here are 2 vector fields, the second has noise ...
157 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 ...
185 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)...
789 views

### Visualizing the Lagrange Multiplier Solution

This question is based on work in Susan Colley's Vector Calculus, 4th ed. The question is to find and classify critical points of $f(x,y,z)=3xy-4yz+5xz$ subject to the constraint $g(x,y,z)=3x+y+2z=13$...
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### How to change coordinates of a differential operator?

I'm doing a basic quantum mechanics problem and am trying to learn how to do it in Mathematica. Any help would be much appreciated. $\vec{L} = \vec{x} \times \vec{p}$ where $\vec{x}$ has components:...
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### 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 ...
294 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 ...
639 views

### Working with abstract vectors

I often need to compute derivatives or integrals involving N-dimensional vectors (where the dimension could be equal to 2 or 3 but is not particularly relevant for the sake of the derivation). The ...
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### Extrema of a function of three variables

To identify and classify the critical points of the function $f(x,y,z)=x^3+xz^2-3x^2+y^2+2z^2$, I used the Hessian matrix method. ...
571 views

### Plotting a parametrically defined vector field

Let's say that you have a three dimensional surface defined by two parameters. E.g. $$S(\theta,\phi) = \{\cos{\theta}\sin{\phi},\sin{\theta}\sin{\phi},\cos{\phi}\}$$ Additionally, you have some ...
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### Finding vector of same direction with smallest integer coordinates

To determine Miller Indices of crystal lattice planes I would need a stable algorithm which determines the smallest set of integer coordinates of a vector which has same direction as a given vector (e....
1k 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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### Comparing unit normal definition in calculus with FrenetSerretSystem

I'm trying to compare the unit normal definition in calculus texts (i.e., $\vec N=\vec T'/||\vec T'||$) where $\vec T$ is the unit tangent vector, with the unit normal vector returned by the ...
342 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 ...
357 views

### Finding the attractors of the vector field constructed as the gradient of an interpolated 3D scalar field

I have a list of data existing as {{{x, y, z}, f},...} from which I construct a 3D interpolation function using Interpolation[ ]. I am able to construct a vector field from the gradient of this scalar ...
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### 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 ...
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### Integration over a convex combination of a region: $\int_{\Omega} (w_1 z_1 + w_2 z_2)^{1-\sigma} d (z_1, z_2)$ where $\Omega = \{ z_1 + z_2 = 1\}$

Take $w,z\in R^{n}$. I am interested in integrating (as generically if possible) $$\int_{\Omega}(w \cdot z)^{1-\sigma} d z$$ Where the domain of $\Omega$ is $1$ dimension, and includes the convex ...
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### Integrating a gradient and matching terms automatically

I am trying to integrate a function $\nabla w(x,y)$ indefinitely knowing both components of $\nabla w(x,y)$. I have a function dw[x_,y_] that I have defined. I then ...
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### How to extend a root finding method from 1D to multi dimensions

The number (0.25 Pi) is a third order root of f[x] below. Clear[f]; f[x_] := Sin[x] - Cos[x] - Sqrt (x - 0.25 Pi); Chop@Normal@Series[f[x], {x, 0.25 Pi, 4}] -...