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. K

    Normals to (hyper)surface must be scalar multiples?

    Homework Statement Let S is a (hyper)surface defined by {x|F(x)=0}. Suppose n1 and n2 are both normal to S at x=a. Then n1 and n2 are scalar multiples of each other. Homework Equations The Attempt at a Solution If S is a surface in R3, then I think it's clear geometrically that the...
  2. F

    Partial Derivative: Finding the vector on a scalar field at point (3,5)

    Homework Statement A scalar field is given by the function: ∅ = 3x2y + 4y2 a) Find del ∅ at the point (3,5) b) Find the component of del ∅ that makes a -60o angle with the axis at the point (3,5) Homework Equations del ∅ = d∅/dx + d∅/dy The Attempt at a Solution I completed part a: del ∅ =...
  3. T

    Scalar projection - finding distance between line and point

    Using a scalar projections how do you show that the distance from a point P(x1,y1) to line ax + by + c = 0 is \frac{|ax1 +b y1 + c|}{\sqrt{a^2 +b^2}} I do not know how to approach this, please provide some guidance.
  4. B

    How will I represent the scalar function?

    Homework Statement show that \nabla \times (f F)= f \nabla \times F+ (\nabla f) \times F The Attempt at a Solution How will I represent the scalar function? Do I write f=\psi(x,y,z) or f=A_x+A_y+A_z I chose F=a_x \vec i +a_y \vec j +a_z \vec k Using f=\psi(x,y,z) I work out...
  5. R

    Whittaker 1904 paper on scalar potential functions

    http://www.hyiq.org/Library/E.T.Whittaker-1904.pdf "On an Expression of the Electromagnetic Field due to Electrons by Means of Two Scalar Potential Functions" by E. T. Whittaker published in Proceedings of the London Mathematical Society, Vol. 1, 1904) In the paper Whittaker 1904...
  6. G

    Temperate a scalar than why negative temperature?

    What's the meaning of negative temperature if temperature can only be a scalar? Why the construction of negative temperature in degrees Fahrenheit?
  7. O

    Variation of scalar kinetic lagrangian

    Homework Statement The goal of the question I'm being asked is to show that the covariant derivatives, D_{\mu}, "integrate by parts" in the same manner that the ordinary partial derivatives, \partial_{\mu} do. More precisely, the covariant derivatives act on the complex scalar field...
  8. U

    EFE's question regarding Ricci scalar

    Quick question about the EFE's. When writing the einstein tensor G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}, and using the definition of the Ricci scalar R=g^{\mu\nu}R_{\mu\nu}, how does this not give you problems when you expand out R so that the second term becomes...
  9. M

    Scalar Field Theory-Vacuum Expectation Value

    Homework Statement I am given an equation for a quantized, neutral scalar field expanded in creation and destruction operators, and need to find the vacuum expectation value of a defined average field operator, squared. See attached pdf. Homework Equations Everything is attached, but I...
  10. H

    Triple Scalar Product and Torque Explained?

    Homework Statement I am working through Boas' Mathematical Methods in the physical sciences book and I don't understand the triple scalar product and torque example. k [dot] (r X F) = 0 0 1 = xF_y - yF_x x y z F_x F_y...
  11. L

    How is the Riemann tensor proportinial to the curvature scalar?

    My professor asks, "Double check a formula that specifies how Riemann tensor is proportional to a curvature scalar." in our homework. The closet thing I can find is the relation between the ricci tensor and the curvature scalar in einstein's field equation for empty space.
  12. R

    Complex scalar field - Feynman integral

    Homework Statement For a real scalar field \phi, the propagator is \frac{i}{(k^2-m_\phi^2)}. If we instead assume a complex scalar field, \phi = \frac{1}{\sqrt{2}} (\phi_1 + i \phi_2), where \phi_1,\phi_2 are real fields with masses m_{\phi 1},m_{\phi 2}, what is the propagator...
  13. O

    Scalar field as quantum operator.

    Hallo, I was wondering what is the physical significance of scalar field \Phi (x) as an quantum operator. \Phi (x) have canonical commutation relation such as [ \Phi (x) , \pi (x) ] so it must be an opertor, thus what are his eigenstates? Thanks, Omri
  14. reddvoid

    Magnetic vector and scalar potential

    I understood that curl H = J H being magnetic field intensity and magnetic flux density B = u H (u being permeability of free space) divergence of B is zero because isolated magnetic charge or pole doesn't exist. but then they define magnetic scalar and vector potentials .i can imagine H...
  15. H

    Usefulness of Kretschmann scalar

    Hello, As I'm sure you are aware the Kretschmann scalar (formed by contracting the contravariant and covariant Riemann tensors) has some use in the identification of gravitational singularities. Specifically, because K is essentially the sum of all permutations of R's components, but is...
  16. D

    Computing the line integral of the scalar function over the curve

    Homework Statement f(x,y) = \sqrt{1+9xy}, y = x^{3} for 0≤x≤1 Homework Equations The Attempt at a Solution I don't even know how to start this problem. I thought about c(t) since that's all I have been doing, but there isn't even c(t). I only recognize domain. Can anyone help me...
  17. DryRun

    Evaluate scalar triple products

    http://s2.ipicture.ru/uploads/20111115/BiYq94IS.jpg Here is the determinant for axb: w x y z 1 -2 3 -4 -1 2 4 -5 Then, how to proceed?? Can someone please help?
  18. S

    Finding the scalar equation for a plane

    Homework Statement Find the scalar equation for the plane containing L1 and L2. Consider the lines: L1 : x = t + 1, y = 2t, z = 3t - 1 L2 : x = s - 1, y = 2s + 1, z = 3s - 1 Homework Equations Scalar equation for a plane: a(x - x0) + b(y - y0) + c(z - z0) = 0 The Attempt at a Solution These...
  19. C

    Show that a line element transforms like a scalar

    Homework Statement Show that a line element of form ds2 = gabdXadXb transforms like a scalar under any general coordinate transform Homework Equations The Attempt at a Solution I think I've actually found the solution here, but I can't make sense of it...
  20. P

    Computing Thermal Average for a Free Real Scalar Field

    Homework Statement Homework EquationsHi! I read a book where a free real scalar field with Hamiltonian H = \int \dfrac{\mathrm{d}^{3}p}{(2 \pi)^{3}} \, E_{\vec{p}} a_{\vec{p}}^{\dagger} a_{\vec{p}} is beeing discussed. Note that: E_{\vec{p}} = \sqrt{\vert \vec{p} \vert^{2} + m^{2}}...
  21. R

    Zero curl and gradient of some scalar potential

    Can someone help me intuitively understand why if a field has zero curl then it must be the gradient of a scalar potential? Thanks!
  22. jfy4

    Magnetic scalar potential and function expansion

    Homework Statement Consider two long, straight wires, parallel to the z-axis, spaced a distance d apart and carrying currents I in opposite directions. Describe the magnetic field \mathbf{H} in terms of the magnetic scalar potential \Phi, with \mathbf{H}=-\nabla \Phi. If the wires are...
  23. M

    Derive Feynman Rules for Complex Scalar Field

    Homework Statement Derive the Feynman rules for for a complex scalar field. Homework Equations L=\partial_\mu\phi^\dagger\partial^\mu\phi +m^2\phi-\lambda/4 |\phi|^4The Attempt at a Solution I wrote the generating functional for the non-interacting theory Z_0[J]=Z_0[0]exp(-\int...
  24. PeterDonis

    Derivation of expansion scalar for FRW spacetime - weird observation

    Derivation of expansion scalar for FRW spacetime -- weird observation In a recent thread... https://www.physicsforums.com/showpost.php?p=3567386&postcount=137 ...I posted a formula for the expansion scalar for the congruence of "comoving" observers in FRW spacetime. When I posted, I...
  25. F

    Why is Work a Scalar? Understanding its Definition

    I am trying to understand why work is a scalar, without knowing ahead of time that work is defined as: W_{ab} = \int ^{\vec{r_{b}}}_{\vec{r_{a}}} \vec{F} \cdot d{\vec{r}} Essentially, I am trying to understand how this definition was derived (based on the one-dimensional work-energy theorem...
  26. A

    Finding a scalar such that vectors p and q are parallel

    Homework Statement Let: p = (2,k) q = (3,5) Find k such that p and q are parallel The Attempt at a Solution Well, I know that for two vectors to be parallel we need to have p = kq. I know the answer will be kind of obvious but I just can't get it lolll, any help please?? Thanks
  27. sweet springs

    Scalar made from electromagnetic four potential

    Hi. What physical meaning does scalar made from inner product of electromagnetic four potential, gαβAαAβ, have? Regards.
  28. N

    Show that T preserves scalar multiplication - Linear Transformations

    Homework Statement Let T:ℝ^{2}→ℝ be defined by T\left(\begin{array}{c} x_{1} \\x_{2}\end{array}\right) = (0 if x_{2} = 0. \frac{x^{3}_{1}}{x^{2}_{2}} otherwise.) Show that T preserves scalar multiplication, i.e T(λx) = λT(x) for all λ \in ℝ and all x \in ℝ^{2} The Attempt at a Solution...
  29. B

    Continuity of Magnetic Scalar Potential

    Hello I have found in some textbooks that the magnetic scalar potential is continuous across a boundary. Now, how can this be explained starting from the two boundary conditions of Maxwell's equations (continuity of normal flux density Bn and tangential field Ht)? Thanks in advance for...
  30. C

    Existence of Scalar Potential for Irrotational Fields

    Hi I know it's easy to prove that if a vectorfield is the gradien of a potential, \vec F = \nabla V, then \nabla \times F = 0. But how about the converse relation? Can I prove that if \nabla \times F = 0, then there exist a salar potential such that \vec F = \nabla V? I get as far as...
  31. X

    Massive Scalar Field in 2+1 Dimensions

    Homework Statement We wish to find, in 2+1 dimensions, the analogue of E = - \frac{1}{4\pi r} e^{-mr} found in 3+1 dimensions. Here r is the spatial distance between two stationary disturbances in the field. Homework Equations In 3+1 we start from E = - \int \frac{ d^3 k }{(2\pi)^3}...
  32. I

    Calculating Boat's Bearing After Changing Direction: Scalar and Vector HELP

    Scalar and Vector HELP please A boat is traveling on a bearing of 25 degrees east of north at a speed of 5 knots ( a knot is 1.852km/hr). After traveling for 3 hours, the boat heading is changed to 180 degrees and it travels for a further 2 hours at 5 knots. What is the boat's bearing from its...
  33. R

    Derivative of a function of a lorentz scalar

    This is probably a dumb question, but I have a book that claims that if you have a function of the momentum squared, f(p2), that: \frac{d}{dp^2}f=\frac{1}{2d}\frac{\partial }{\partial p_\mu} \frac{\partial }{\partial p^\mu}f where the d in the denominator is the number of spacetime...
  34. 3

    Proof for determinant of a scalar multiplied by a vector

    Homework Statement Let A be an n x n matrix and \alpha a scalar. Show that det(\alpha A) = \alpha^{n}det(A) Homework Equations det(A) = a_{11}A_{11} + a_{12}A_{12} + \cdots + a_{1n}A_{1n} where A_{ij} = (-1)^{i+j}det(M_{ij}) The Attempt at a Solution det(A) = a_{11}A_{11}...
  35. A

    Is Pressure a Scalar Despite Acting in All Directions?

    Homework Statement how come pressure have directions, and yet is a scalar quantity? Homework Equations The Attempt at a Solution
  36. A

    Why Do Scalar Fields in Hybrid Inflation Model Diverge with Large Mixing Term?

    I have a puzzle when I study the hybrid inflation model. Suppose we have two scalar fields, \phi_1 and \phi_2 first, let's consider the situation where they are in their independent potentials V(\phi_i)=m_i^2\phi_i^2, i = 1,2 with initial value \phi_i^{ini} We can solve the scalar...
  37. jfy4

    Complex Scalar Field and Probability Field

    Hi, I was looking at the lagrangian and conserved currents for the free complex scalar field and it looks like it has a striking similarity to the conserved current for probability: \frac{\partial \rho}{\partial t}=\nabla\cdot \vec{j} where j_i =-i(\psi^{\ast}\partial_i \psi -...
  38. W

    Complex scalar field and contraction

    Hi guys, If I use the definition of the scalar complex field as the combination of two scalar real fields, I can get \phi (x) = \int \frac{d^3 p}{(2\pi )^3} \frac{1}{\sqrt{2p_0}} [ \hat a _{\vec{p}} e^{-ip.x} + \hat b _{\vec{p}}^{\dagger } e^{ip.x}] which I can rewrite in terms of...
  39. R

    Spectral weight function and the mass shift of a scalar field

    In the Kallen-Lehmann spectral representation (http://en.wikipedia.org/wiki/K%C3%A4ll%C3%A9n%E2%80%93Lehmann_spectral_representation) the interacting propagator is given as a weighted sum over free propagators. The pole of the integracting propagator is, of course, given by p^2=m^2, m being the...
  40. T

    Is mass a scalar or a vector in physics?

    I came across a problem that asked if some things were scalar or a vector. Is acceleration a vector? I thought it just gives you magnitude and not direction at all. For example when velocity and acceleration are put together, velocity gives you the direction and the acceleration gives...
  41. M

    Gradient of scalar function discontinuous on boundary

    suppose g(r) is a scalar function which is constant inside the volume 'v' but discontinuous at the boundaries of 'v'. The magnitude of discontinuity is given by constant 'M' then can we write the following expression \int\nablag(r)dv=M\int\hat{n}\delta(r-rs)dv=M\hat{n}\intd\deltav where...
  42. N

    Induced scalar electric potential

    Hi Forum! I have got a question about the induced scalar potential. I will present the problem from beginning. Lets say we have a Poisson's equation in form: \epsilon \nabla^2 \phi = -4\pi \varrho(r,t) where \epsilon is the dielectric constant. By use of the Fourier transform...
  43. J

    Symbolic replacement of scalar products

    Hello, I have very complicated expressions containing scalar products like a1*b1 + a2*b2 + a3*b3 In order to reduce the complexity, I would like to establish a set of rules like rule={a1*b1 + a2*b2 + a3*b3 -> pAB, ...} in order to replace each time the scalar product by an...
  44. J

    Strategy in solving vector equations involving grad, scalar product operators?

    What is the general strategy in solving vector equations involving grad and the scalar product? In particular, I want to express \Lambda in terms from \mathbf U \cdot \nabla\Lambda = \Phi but it looks impossible, unless there is some vector identity I can use. Thanks in advance.
  45. S

    Stability of the gaussian under addition and scalar multiplication

    If i have the mgf of X and the mgf of Y where X~N(mx,vx) and Y~N(my,vy) and X and Y are independent , then if i want to show that aX +bY ~ N(amx+bmy , a^2vx+b^2vy) how would i do this - need to be able to do the convolutions way and the mgf's way, for the mgfs way is it just, mgf(ax+by) =...
  46. H

    Fun Magnetic Scalar Potential Problem

    "Fun" Magnetic Scalar Potential Problem Homework Statement An infinite cylindrical shell of radius b is placed inside a constant field B which points along the upwards z-axis. A second cylindrical shell of radius a<b is placed inside the first cylindrical shell, and the volume from b>r>a...
  47. E

    Real scalars for complex scalar results

    Hi guys, I'm a bit puzzled. I'm just reading some offline lecture notes where the Feynman rules of real (!) scalars coupled to gluons are given. However, with these rules the amplitude for phi g -> \bar{phi} g is considered. There are no further instructions. I'm just wondering how one can...
  48. A

    What is the Laplacian of the scalar potential with an extra term?

    On page 35 of Jackson's Classical Electrodynamics, he calculates the Laplacian of a scalar potential due to a continuous charge distribution. In the expression for the potential, the operand of the Laplacian is \frac{1}{|r-r'|}, where r is the the point where the potential is to be...
  49. S

    Invariance of scalar products on Lie algebras

    Hi folks, If I have a Lie algebra \mathfrak{g} with an invariant (under the adjoint action ad of the Lie algebra) scalar product, what are the conditions that this scalar product is also invariant under the adjoint action Ad of the group? For instance, the Killing form is invariant under...
  50. B

    Coulomb Law and Vectors - How do you find a scalar answer from the vector form?

    Coulomb Law and Vectors - How do you find a scalar answer from the vector form?? Two small metal spheres carry equal charges q. They are located at positions r1 = (1,1,0) nm and r2 = (0,0,0) nm and feel a repulsive force of magnitude (mod) F = 0.05 N How much charge is on each sphere...
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