Scalar Definition and 829 Threads

In mathematics and physics, a scalar field or scalar-valued function associates a scalar value to every point in a space – possibly physical space. The scalar may either be a (dimensionless) mathematical number or a physical quantity. In a physical context, scalar fields are required to be independent of the choice of reference frame, meaning that any two observers using the same units will agree on the value of the scalar field at the same absolute point in space (or spacetime) regardless of their respective points of origin. Examples used in physics include the temperature distribution throughout space, the pressure distribution in a fluid, and spin-zero quantum fields, such as the Higgs field. These fields are the subject of scalar field theory.

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  1. P

    Dot, Scalar, Inner Product Question

    I have been searching for a way to relate known concepts (known to me) to the computation of the dot product in an effort to understand why it takes the form it does. I ran into a little snippet in a classical dynamics book that seems like it just may be the ticket. Here is what it says...
  2. P

    Killing Vector and Ricci curvature scalar

    Homework Statement I'm currently self-studying Carroll's GR book and get stuck by proving the following identity: K^\lambda \nabla _\lambda R = 0 where K is Killing vector and R is the Ricci ScalarHomework Equations Mr.Carroll said that it is suffice to show this by knowing: \nabla _\mu...
  3. S

    Questions about the Kretschmann curvature scalar

    The Kretschmann curvature scalar is defined to be K = RabcdRabcd where Rabcd is the Riemann tensor. I believe I heard in class that this scalar can be used to demonstrate the existence of a curvature singularity. Can somebody tell me why this is so? Also, I heard that it is better (easier I...
  4. T

    Scalar multiplication of two equal vectors

    i have vector "a" why a*a=1?? it make no sense the formula says |a|*|a|*cos0=a^2 (not 1)
  5. S

    Finding Scalar Function G: Step-by-Step Guide

    Hi, I got a doubt question for this. Given this general expression for the scalar function, G such that del G = F(x,y,z)=2xyi + (x^2 - Z^3)j + (-3yz^2 + 1)k How do i go about finding G?
  6. D

    Why is the Higgs Field a Scalar Field? Exploring Its Nature

    Why is the Higgs field a scalar field? I understand if it is one, it will have no spin and no angular momentum. But understanding that a particle is a scalar seems to me a leap of faith. What am I not getting?
  7. D

    Scalar triple product, volume, and ordering

    Greetings all, I'm reading about a way to solve for the volume of a "parallelepiped" in 3 space, which is determined by vectors u, v and w. The volume is apparently the absolute value of the determinant given by the matrix u1 u2 u3 v1 v2 v3 w1 w2 w3 which is the same as the scalar triple...
  8. L

    Scalar factors of parabolic cylindrical coords

    Homework Statement I have a question to find Scalar factors of parabolic cylindrical coords and element dV with provided tranformation equations. I know the values for both of them and that the product of the scalar factors is the dV, but how do i derive those scalar factors? I don't even know...
  9. K

    Understanding Negative Scalars in Physics

    I don't understand how a scalar can be negative. Like work for example, how can this be - or + yet still be a scalar. I've read that scalars only have magnitude, while vectors have both magnitude and direction. As vague as these definitions seem, I reason that scalars wouldn't have any +/-...
  10. Saladsamurai

    Difference between Scalar Function and Vector Function?

    Okay I know the definition of a Vector and of a scalar... but I am getting a little confused for some reason. Wolfram.com gives this definition of a scalar function: A function f(x_1,x_2,...,x_n) of one or more variables whose range is one-dimensional, as compared to a vector function...
  11. P

    Deriving the Poincare algebra in scalar field theory

    Homework Statement Find the commutators [P^\sigma,J^{\mu \nu}] The answer is part of the Poincare algebra [P^\sigma,J^{\mu \nu}]=i(g^{\mu \sigma}P^\nu-g^{\nu \sigma}P^\mu) If someone can convince me that \partial_i T^{0\mu} = 0, (i.e. the energy-momentum tensor has no explicit spatial...
  12. W

    Lorentz Transformation of Scalar Fields

    Homework Statement Working on an exercise from Srednicki's QFT and something is not clear. Show that [\varphi(x), M^{uv}] = \mathcal{L}^{uv} \varphi(x) where \mathcal{L}^{uv} = \frac{\hbar}{i} (x^u \partial^v - x^v \partial^u ) Homework Equations (1) U(\Lambda)^{-1} \varphi(x)...
  13. W

    Question: Spin particles in scalar gravitational field

    The action of non-spinning particles in scalar gravity is S=-\int{\sqrt{-g}(\frac{1}{8\pi}g^{\mu\nu}\Phi_{,\mu}\Phi_{,\nu}-\rho e^{\Phi})d^4x} where \rho presents the comoving density. Now, I want to know the formula when particles with classical spin. Thank you!
  14. R

    Scalar Product of a diffrential.

    Hey, in my textbook they keep doing this and I can't follow for example r.\ddot{}r = 1/2 \ddot{}r^{}^2{} and r.\dot{}r = 1/2 \dot{}r^2{}. Can anyone explain this to me? I know I should probably know it. P.S Can't quite get the dot product to look right apologies.
  15. S

    Power series of a scalar function with a vector

    I was trying to expand a scalar function with a power series but it accepts a vector argument. Do I simply use the multivariable power series expansion with the components of the vector acting as the argument OR do I use the single variable power series and take the vector's magnitude in the...
  16. M

    How Do You Convert Cardinal Directions to I+J Vector Form?

    I have a question on an assignment that expects the I+J form of a vector but is only giving the direction(cardinal) or angle of the vector. See example below: Homework Statement Unit vectors and are directed east and north, respectively. Calculate the unit vector (in terms of I and J) in...
  17. H

    Proof - the derivative of a scalar multiple

    Homework Statement I am confused how the scalar multiple is divided out of the proof of this rule without taking an h with it in the denominator, which would get very tiny meaning the entire thing would go to infinity or negative infinity or zero, you can't tell. Start with: f(x) = k...
  18. S

    Which of these are vector and which are scalar?

    a) displacement b) velocity c) average velocity d) distance e) speed f) acceleration g) position which of these are vector and which are scalar?
  19. I

    Variation with scalar field coupled to gravity in 2D

    The two dimensional action is: S_k = \int d^2\sigma\sqrt{h}\left(\partial_\alpha\phi\partial^\alpha\phi - \frac{i}{2}kR^{(2)}\phi\right) where k is a constant, R^{(2)} is the two dimensional scalar curvature. I'm trying to derive the following energy momentum tensor: T_{\alpha\beta}^k =...
  20. I

    Conformal dimension of a scalar function?

    Actually, I'm not fully understand what the meaning of conformal dimension is. But I know how to read off the conformal dimension of a tensor, say, t^{++}{}_+, then the conformal dimension is -2 + 1= -1, where the lower index carries conformal dimension 1 and upper index carries conformal...
  21. e2m2a

    Invariance of scalar dot product across inertial and non-inertial frames

    I have a question concerning scalar invariance with respect to an accelerating and an inertial reference frame. Here is the problem. Suppose we have a rotating spherical object, which we denote as the rotator, attached to a near-massless wire. The other end of the wire slips loosely over a...
  22. C

    Calculus: I can't understand why curl of gradient of a scalar is zero

    (Sorry, the title should read "...why curl of gradient of a scalar "function" is zero) Of course I know how to compute curl, graident, divergence. Algebrically I know curl of gradient of a scalar function is zero. But I want to know the reason behind this...and also the reason why gradient of...
  23. K

    Understanding Scalar and Vector Projections: A Layman's Guide

    I'm re-visiting calculus again, and I've stumbled onto the concepts of scalar and vector projections in the vector chapter. While keeping in mind which equation to use for what projection is quite doable, I cannot seem to see the purpose of keeping scalar and vector projections in mind. Can...
  24. S

    Proving that the scalar product is invariant

    Is there a general way of proving that the scalar product xuxu = (x0)2 - (x1)2 - (x2)2 - (x3)2 is invariant under a Lorentz transformation that applies no matter the explicit form of the transformation.
  25. H

    Exploring Scalar Electromagnetics: Fact or Fiction?

    Are they real are what? I've read in various places on the internet that there is a vast amount of energy stored in the vacuum of space that is supposedly extractable and extremely efficient. The top link below contains quite a bit on information about the potential weapons (scalar howitzer...
  26. C

    Understanding Scalar Fields: Assigning Values to Space

    "A scalar field assigns every point in space to a scalar value" Would this be a correct definition of a scalar field? Thanks
  27. snoopies622

    Produce Scalar from Tensor: What's the Name?

    As I understand it, for a tensor of any rank I can produce a corresponding scalar in the following way: Create an inverted form of the tensor by lowering its upper indices and raising its lower indices, and then taking the inner product of this tensor and the original one. My only question...
  28. J

    Scalar Projection: Find Distance Point to Line

    Homework Statement Use the scalar projection to show that a distance from a point P(x1, y1) to the line ax + by + c = 0 is \frac{ax1 + by1 + c}{\sqrt{a^2 + b^2}}Homework Equations scalar projection = \frac{a . b}{|a|} The Attempt at a Solution The first thing that I did was to say that b =...
  29. K

    Energy-momentum tensor for a scalar field (sign problem)

    Hi I have a small subtle problem with the sign of the energy-momentum tensor for a scalar field as derived by varying the metric (s.b.). I would appreciate very much if somebody could help me on my specific issue. Let me describe the problem in more detail: I conform to the sign convention...
  30. hxtasy

    Scalar equations, should be EASY but this has been bothering me for years

    Ok this is a pretty simple concept and in the electronics field I use it on a monthly basis, I can do it but I do not understand how it works and It is bugging the crap out of me. EXAMPLE: I have a pressure sensor, it reads pressure from 0 psi to .73 psi, and outputs a corresponding voltage...
  31. Fra

    Logic of E-H action, ricci scalar, cosmological constant?

    Logic of E-H action, ricci scalar, cosmological constant?? This crazy thread is mean to stimulate some reflections on the logic of Einsteins Equations. It would be interesting if those who have any ideas can join. Maybe it could be enlightning? The common way of thinking about GR is that we...
  32. U

    Scalar waves, is this a complete fabrication?

    Many of you will have heard of these. Does anyone knowledgeable on conventional electromagnetics, suspect there may be some truth in it?
  33. A

    Can photon couple to scalar field?

    I have seen in one paper that photon is coupled to dilaton field which is scalar and motivated by string theory. I do not understand this. Photon is carrier of electromagnetic field and so I thought it can only couple to electrically charged fields. Can anyone explain?
  34. J

    Interactions of Fermion & Scalar Fields: Exploring the Difference

    Suppose I couple a fermion field to a scalar field using \mathrm{i} g \bar{\psi}\psi \varphi and \mathrm{i} g \bar{\psi}\gamma_5\psi\varphi. I'm trying to understand what would be the physical difference between these interactions. I know that (1/2)(1\pm \gamma_5) approximately projects out...
  35. M

    Meaning of Curvature Scalar (R) in GR & Its Evolution

    What is the meaning of the curvature scalar (R) in GR? More precisely, what is the meaning of it's evolution? Why when we are concerning the solar system we take R to be small and when we are concerning the cosmological scales the we assume R to be large? Thanks in advance.
  36. M

    Explicit expressions for creation/annihilation operator of the free scalar field

    I've been trying to work my way through some of my lecture notes, and have hit this snag. (n.b. I use k_0 \equiv +\sqrt{\vec{k}^2 + m^2}) We have a(q) = \int d^3 x e^{iqx} \{ q_0 \phi(x) + i \pi(x) \} a^{\dagger}(q) = \int d^3 x e^{-iqx} \{ q_0 \phi(x) - i \pi(x) \} To calculate the...
  37. E

    Finding a Scalar Function on a Bounded Surface

    Hi guys and gals This is a conceptual question. Let's say I have a scalar function, f(x,y,z) defined throughout \mathbb{R}^3. Further I have some bounded surface, S embedded in \mathbb{R}^3. How would I find the function f, defined on the surface S? Would it be the inner product of f...
  38. D

    Exploring the Possibility of Variable Ricci Scalar in Spacetime Calculations

    Does it make sense for the Ricci Scalar to be a function of the spacetime coordinates? In previous calculations I have carried out in the past, everytime the Ricci Scalar has been returned as a constant, rather than being explicitly dependent on the coordinates. Thanks for any replies
  39. T

    A quick question about scalar product of vectors

    Attached is a .jpg of my problem. I know how to find the scalar product of B*C (I think... 5, right?), but I don't really know where the 2 and 3 come into play. I've tried multiplying the values of C by 3 and then finding the scalar product, then multiplying the quantity by two, but that was...
  40. K

    Energy momentum tensor of a scalar field by varying the metric

    Suppose you are given the Lagrangian of a scalar field \Phi(t) \mathcal{L} = \frac{1}{2} \dot{\Phi}- \nabla \Phi - V(\Phi ). By introducing covariant notation with \eta_{\mu \nu} = (1,-1,-1,-1) this reads as \mathcal{L} = \frac{1}{2} \eta^{\mu \nu} \partial_\mu\Phi \;\partial_\nu\Phi-...
  41. M

    Understanding the Difference between Scalar and Dot Product in Tensor Calculus

    Hello all. In a quite easy to follow short piece by Edmond Bertschinger entitled Introductio to Tensor Calculus for General Relativity on page 6 when speaking of the metric tensor he says, referring to the symbol conventions used in the piece :- "" We reserve the dot product notation for...
  42. E

    Working backwards to find scalar equations

    I am given the symmetric equation of the line of intersection of two planes. I also have a point on each plane. Now I have to work backwards to determine the scalar equations of the two planes. I can work to get it by saying for example x + 2y + 3z -6 + k(4x - y -z +4) - these are made up...
  43. E

    Scalar Potentials Conceptual Confusion

    I'm kind of confused as to how to determine when a vector field is conservative. For example, if we consider the following scalar field: \varphi = arctan(\frac{y}{x}) We see that the gradient is: F = \nabla\varphi = \frac{-y i + x j}{x^{2} + y^{2}} However, F is not a conservative...
  44. R

    RS 1, massless scalar field, and sep of vars.

    Hey, this is rather involved but I hope someone can help me out. I am reading http://arxiv.org/abs/0704.3626 ( the casimir force in randall sundrum models) and am trying to get from equation 2.1 : g^{\mu\nu}\partial_\mu\partial_\nu\Phi+e^{2ky}\partial_y(e^{-4ky}\partial_y\Phi)=0 to...
  45. manjuvenamma

    Is current a vector or scalar?

    I see in several physics papers and articles current, current density represented by vectors. But in one book it was mentioned clearly that current is not a vector because it does not obey vector law of addition. Can some one clarify this point with an example and show that current does not...
  46. R

    Question about Scalar Product of Vectors

    Hello there When multiplying two vectors, why do we multiply them with the cosine of the angle between them? why not the sine of the angle? Thanks.
  47. B

    Coupling fermions to a scalar field: Interpretation problem

    Hi all, I have a little problem concerning the coupling of a fermion to CP^N (or better a 2D scalar O(3) model). Its not a mathematical type of problem. I just read on "The coupling of fermions to the three-dimensional noncommutative $CP^{N-1}$ model: minimal and supersymmetric extensions"...
  48. R

    Exploring the Kinetic Term of a Scalar Field

    I've been wondering about terms you typically find in the action of a field theory, for example consider the kinetic term of a scalar field S=\int d^4x(\partial_\mu\phi\partial^\mu\phi). I've read that it can be thought of as the kinetic energy of the field - but this just doesn't sit to...
  49. M

    Doublets Scalar Potential Question

    How can I write the most general real scalar potential in terms of the singlets and doublets corresponding to irreducible representations of S3?
  50. C

    Consider a surface S on which a scalar field f is defined

    "Consider a surface S on which a scalar field f is defined" "Consider a surface S on which a scalar field f is defined" what does "on which is defined" mean phys descript answers appreciated!
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