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Questions tagged [scalar-fields]

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Proof that $\| u \|=\| v\|\iff\langle u+v,u-v\rangle=0$

Let $(\cdot,\cdot):\mathbb{R}^n\times \mathbb{R}^n\to\mathbb{R}^n$ be any map that fulfills the properties $u,v,w\in\mathbb{R}^n;\; \lambda\in\mathbb{R};\; \langle u;v\rangle:=(u_1v_1)+\cdots+(...
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0answers
16 views

What are useful application of that “non-commutativity of quaternion multiplication prevents the transition of changing ij = +k to ji = −k”?

I read here I don't understand well this conclusion The non-commutativity of quaternion multiplication prevents the transition of changing ij = +k to ji = −k. I understand that this is not a real ...
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2answers
21 views

The conjugate of a scalar is the same scalar in a matrix-scalar multiplication?

I am reading the book, Applied Linear Algebra and Matrix Analysis. When I was doing the exercise of Section 2.4, Page 97, I thought the equation is right as followed: $(cA)^{*} = cA^{*}$ But answer ...
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0answers
23 views

Under what conditions are partial derivatives continuous?

For any continuous scalar and vector fields which doesn't contain any singular points, under what conditions are their partial derivatives continuous?
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1answer
27 views

Why is this notation equal to its transpose?

In my econometrics textbook, I have this step which is not clear to me: \begin{align} S &= e'e \\ &= (y-W\beta)'(y-W\beta) \\ &= \underbrace{y'}_{1\times T} \ \underbrace{y}_{T\times 1} ...
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2answers
45 views

If $\vec{\nabla} \times \langle P,Q,Q \rangle=\vec{0}$ iff $Pdx+Qdy+Rdz$ is exact differential form.

If $\vec{\nabla} \times \langle P,Q,R\rangle=\vec{0}$ iff $Pdx+Qdy+Rdz$ is an exact differential form. My attempt:- If $Pdx+Qdy+Rdz$ is an exact differential form. Then there exists $U(x,y,z): dU=...
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0answers
41 views

2 types of Line integrals in scalar field

Consider the line integral, $\int _ c$f(x,y)$\vec dr$ , where f(x,y) is a scalar field, and it is evaluvated on a curve c . After integration we get a vector let it be $\vec I$ . $\int _ c$f(x,y)$\...
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1answer
43 views

Expressing Vectors, Finding the Angle and Finding the Area

Part 1 According to the Rectangular solid in the following image, express the vectors $\overrightarrow {AF}$, $\overrightarrow {GD}$, $\overrightarrow {FC}$, $\overrightarrow {EC}$ in terms ...
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2answers
39 views

Can eigenvalues be outside of the scalar field of a vector space?

Suppose V is a vector space over the scalar field F, and T is a linear operator on V. Can T have an eigenvalue l such that l is not an element of F? I am working through Linear Algebra Done Right, 3 ...
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0answers
73 views

Method of images using Green functions with Dirichlets and Neumann conditions for diffraction problems

In the solution of the wave equation $$[\nabla^2-\frac{1}{v^2} \frac{\partial^2}{\partial t^2}]U=0$$ and consequently the Helmholtz equation $$[\nabla^2+k^2]G(R,\tau)=\delta(R)=\delta(\vec{x}-\vec{...
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1answer
243 views

Visualising the difference between scalar and vector fields

I know the definition that scalar fields output numbers (magnitudes) for a particular point in space (in its domain) while vector fields output vectors but I am having a hard time visualizing the ...
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1answer
327 views

Fields of characteristic 2

Can someone please explain what are the filds with characteristic 2 with examples? In the book " Lectures in Abstract Algebra" in the section 5, it is written that it is convenient to treat separately ...
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0answers
38 views

New parameter in ODE seems to not affect solution at a point but then contributes a divergence at that point

I'm studying the following ODE (for those interested, it is a Klein-Gordon eq with $\partial_t =0$ for a test field around a black hole). $$ \partial_x (x(x+2)\partial_x \phi) = x(x+2)\ddot{\phi} + 2(...
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1answer
13 views

Does it mean “ scalar field changes in a particular direction” by that the curl of the gradient of a scalar field is zero in practice?

What does mean that the curl of the gradient of a scalar field is zero in practice? My effort: the gradient means the direction at which the magnitude of vector change maximally and the curl says ...
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1answer
9 views

Does a figure which has a contour plot have a systematic function to define it?

Does a figure which has a contour plot have a systematic function to define it? I mean I am plotting a contour plot (the contour plot which shows z slices in x-y graph) of a figure like (suppose) ...
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0answers
29 views

Curl And Potential

Every vector field which has a potential, therefore it can be expressed as the gradient of a certain potential, doesn't have any curl. Thus : $\vec{\nabla} \times \vec{v} = 0$, where $\vec{v}$ is ...
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34 views

Vector as a part of function

I need to calculate directional derivative for such scalar field: $$u=y\ln(l+x^2)-\arctan z\;\text{ where }\;\vec \ell=2\vec i-3\vec j-2\vec k, \;\text{ and }\;M(0,1,1).$$ The question is - what ...
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39 views

Examples of generalisations in which $k$-ring might be, for example, a monad ? What are $\textit{scalars}$, really?

I reflect on scalar nature so my question is very simple: what are $\textit{scalars}$, really ? I read about ground ring There are also generalisations in which k might be, for example, a monad. ...
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1answer
46 views

What is $\mathbb{F}^{2, 2}$?

If $\mathbb{F}$ is a scalar field, what is meant by $\mathbb{F}^{2,2}$? Is this the same as $\mathbb{F}^2 \times \mathbb{F}^2$? Or is it $\mathbb{F}^2 \otimes \mathbb{F}^2$?
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1answer
50 views

What is meant by 'scalars in modules'?

I want to understand better when from here they say When the requirement that the set of scalars form a field is relaxed so that it need only form a ring [..] In this case the "scalars" may be ...
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2answers
125 views

What are the Effects of Applying the $\nabla$ Operator to a Scalar Field? [closed]

For a scalar field $f(\mathbf{r})$, where $\mathbf{r} = (x, y, z)$, what kind of mathematical object results from each of the operations below: $$ \nabla f(\textbf{r}) $$ $$ \nabla \cdot f(\textbf{r}...
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2answers
52 views

What kind of projection does a specific map (3D -> 2D) correspond to?

Suppose I have a 3D scattered data $$(A_1,B_1,Z_1)$$$$(A_2,B_3,Z_3)$$$$(A_3,B_3,Z_3)$$$$...$$$$(A_n,B_n,Z_n),$$ shown schematically in the picture below and there is no order in the data. Now, I ...
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1answer
46 views

Visualizing a Scalar Field: $T(x,y,z)=10e^{-(x^2+y^2+z^2)}$

For a scalar field of temperature, for example, let $$T(x,y,z)=10e^{-(x^2+y^2+z^2)}$$ where $T(x,y,z)$ is a function of temperature in terms of position variables $x, y$ and $z$. How can we ...
2
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1answer
101 views

Recover scalar field from gradient

In one dimension, if I have a Riemann-integrable derivative $f'$ of a function $f$ which I don't know, I can (almost) recover $f$ from integrating $f'$. A simple example would be $f'(x)=2x$, then by ...
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2answers
237 views

Index notation for vector calculus proof

I’d like to prove that $\nabla v \cdot \nabla w = \frac{1}{2} \Big(\nabla^2(vw) - v\nabla^2 w -w\nabla^2 v\Big)$. I’ve attempted to use index notation, but I am unsure of how to rely on the chain rule ...
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0answers
187 views

Gradient of a scalar field

I know that the gradient points toward the maximum of a scalar field. If a scalar field has many maximum points, the gradient calculated in a point aims always toward the nearest maximum ? If yes, why ...
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1answer
264 views

Finding the gradient of a scalar field in cylindrical coordinates

How do I find the gradient of the following scalar field in cylindrical polar coordinates? $\ f(x,y,z)=2z-3x^2-4xy+3y^2$ Should I express it in polar form first, then take the partial derivatives? ...
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0answers
42 views

Two scalar fields whose gradients are orthogonal

Given a sufficiently smooth scalar field $\phi$, how to get another scalar field $\psi$ so that their gradients are orthogonal. In 2D case, this is equivalent to:$$\nabla \phi \cdot \nabla \psi=\frac{...
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38 views

Can someone explain to me scalar fields?

I would like to gain a deeper understanding on scalar fields, especially those which are derived out of vectors. Take for an example a multi-variabled scalar function. The function itself would be a ...
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2answers
111 views

Are angles in a Cartesian coordinate system vectors?

For any given vector, denoting a point, (x,y), on a Cartesian coordinate system, if we consider the angle, phi, of this vector to the horizontal axes, x, is this angle, phi, considered a vector? I ...
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1answer
209 views

Compute gradient of scalar field defined by trilinear interpolation of sample grid

I'm trying to figure out how to compute something for a game I'm creating, and I'm having trouble finding a solution in language I can understand. The extent of my math education is that I'm just ...
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1answer
224 views

Vector and scalar field Laplacian

My vector field is $$F=(-6x^2yz^2+ye^x)i + (-2x^3z^2 + e^x)j - (4x^3yz)k$$How do i find the scalar field $$f,\quad where\quad F=del\,f$$Am i meant to integrate $F$? Also i need to find the laplacian ...
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1answer
411 views

Chain rule with normal directional derivative and radial derivative of a scalar field

Consider a volume $V$ in $\mathbb{R}^3$ enclosed by a closed surface (orientable) $S$ with outgoing normal unit vector $\hat{n}$. Consider also a scalar field $f:\mathbb{R}^3 \to \mathbb{R}$ that ...
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1answer
85 views

Is the elementary definition of a scalar misleading?

An elementary definition of a scalar is that it is ''a quantity having only magnitude''. I've been trying to reconcile this statement that I've always accepted with (1) the more exact definition of a ...
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1answer
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Conclude the “MVT for scalar fields”- proof [closed]

So, e.g $f=a^2+b^2$ (red function in the picture). For two points $x,y$ $\in$ $(-3,3)\times(-3,3)$ we define $g$ as $g: f(ty+(1-t)x)$, and that will be the gray curve. MVT says, there is a $t \in(0,1)$...
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2answers
674 views

why the curl of the gradient of a scalar field is zero? geometric interpretation

This is probably a very silly question, but am I correct in saying that a vector field has non zero curl at some point when the direction of transformation changes? If so I can think of plenty of two ...
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1answer
23 views

where to find good exercise about Eigenvalue and determinant and scalar product

where to find good exercise about Eigenvalue and determinant and scalar product , exercise in which i use all the possible varities?
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1answer
467 views

Derive expression for the gradient of an arbitrary scalar function

I'm trying to do the following problem but I have no idea how to start and was wondering if someone could point me in the right direction? Start from the expression for the metric (the square of the ...
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0answers
66 views

Initializing wave propagation from an arbitrary curve on a triangular mesh

I have a torus with a contour that cuts through its triangles. this contour represents the levelset $f=0.5$ for a given scalar field $f$ which is defined on mesh vertices. I want to initialize a new ...
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1answer
207 views

Direction of Gradient of a scalar function

So I have always read that taking the gradient of a scalar function gives the direction of maximum rate of increase in that scalar field (for example Wikipedia). For the problem at hand I have a field ...
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0answers
27 views

Laplacian definition as difference of scalar function value at point and its average at neighbouring points

Morse and Feshbach in their classic book, pp. 6$-$7, give a definition of the second derivative $d^{2}\psi/dx^{2}$ in one-dimension for a scalar field $\psi(x)$ as being related to the difference ...
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190 views

How is the gradient of a radial scalar field, radial?

I'm not sure if I'm mistaken: but my notes say the following. A scalar field $f$ is radial if $f({\textbf{x}}) = \phi ( ||{\textbf{x}}||)$ for some $\phi : [0,\infty) \rightarrow \mathbb{R}$. I ...
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1answer
258 views

Show that no scalar field exists

Q) Consider the vector field $F(x,y)=(y,-x)$. Show that there exists no scalar field $f:R^2 \rightarrow R$ whose gradient is $F$. Find a closed path C so that $\int_{0}^{1}F . dr \neq 0$. A) So far I ...
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1answer
263 views

Is every vector field the gradient of a scalar field?

I was wondering whether every vector field is the gradient of a certain scalar field. If not, could you provide a counterexample? Thanks.
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79 views

Vector Field Verification

I'm going through past exam questions and I came across part of a question I wasn't sure how to tackle. I don't think I've ever seen something like this: For $(x,y) \neq (0,0)$, define a vector field ...
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0answers
30 views

Some insight about the directional derivative

Here is the definition of directional derivative given; $$\lim_{\vec{c} \to \vec{u}} \frac{f(\vec{c} + h\vec{u}) - f(\vec{c})}{h}$$ which gives for a scalar field $f: R^n \to R$, the derivative we ...
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0answers
159 views

Scalar line integrals vs. line integrals when to calculate each?

I have read some discussion of the difference between scalar line integrals and line integrals, but I am still confused. For the problem: 1) Evaluate the scalar line integral $\int_{H}^{}$ $(x^2 + y^...
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1answer
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Meaning of this line integral

I have seen expressions for 4 different line integrals: $\int f\text{d}s $ $\int f\text{d}\textbf{s} $ $\int \textbf{F} \boldsymbol{\cdot} \text{d}\textbf{s}$ $\int \textbf{F} \times \text{d} \...
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1answer
80 views

Difference between the notions of Real multivariate function and Scalar field and also between Vector multivariate function and Vector field

I think I already know the definitions of "Real (multivariate) function" and "Vector (multivariate) function", but correct me if I'm wrong: A Real function: A function which takes some real numbers ...
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2answers
185 views

How to prove this is an harmonic function?

I need some help with the following: If $f$ and $g$ are two harmonic functions in $R^2$, i.e., $\Delta f = 0$ and $\Delta g = 0$. Can we say that the scalar function $\nabla f \cdot \nabla g$ is ...