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. J

    Sum of Two Vectors: Magnitude & Scalar Product

    Homework Statement If the magnitude of the sum of two vectors is less than the magnitude of either vector, then: -the vectors must be parallel and in the same direction -the scalar product of the vectors must be negative -none of these -the scalar product of the vectors must be...
  2. R

    Exploring Scalar Multiplication: u & v in V

    u and v are contained in V Lets say the scalar multiplication is defined as: ex. ku=k^2 u or ku = (0,ku2) u=(u1,u2) does this mean that this is also the same for different scalar m? mu=m^2 u or mu = (0,mu2) u=(u1,u2) and does this mean the same for any vector v kv=k^2 v or...
  3. R

    Scalar multiplication axiom, quick question

    u and v are contained in V Lets say the scalar multiplication is defined as: ex. ku=k^2 u or ku = (0,ku2) u=(u1,u2) does this mean that this is also the same for different scalar m? mu=m^2 u or mu = (0,mu2) u=(u1,u2) and does this mean the same for any...
  4. N

    Express F as a unit vector and find the Scalar Projection of F onto OA

    Homework Statement Express the 5.2-kN force F as a vector in terms of the unit vectors i, j, and k. Determine the scalar projections of F onto the x-axis and onto the line OA. I have attached an image of the problem. Homework Equations Fx = Fcos(θ) Fy = Fcos(θ) Fz = Fcos(θ)...
  5. Chronos

    Brian Powell's latest paper: Scalar runnings

    Brian Powell [bapowell] has a new paper out - Scalar runnings and a test of slow roll from CMB distortions, http://arxiv.org/abs/1209.2024. This is interesting and challenges some of the more radical ideas that are currently popular. Very nice, Brian!
  6. F

    Which quantities are categorized as vectors or scalars?

    Identify each of the quantities below as Vector or Scalar. Speed of a train. (Vector) Volume of a cube. (Scalar) Acceleration of a rocket. (Vector) Mass of a pint of milk. (Scalar) Velocity of a train. (Vector) Force of gravity. (Scalar) I put my answers in parentheses but I am not...
  7. H

    Solving Scalar Product: Figure Drawing for (AC - AB) * AB = 0

    Hello! I am preparing for an exam, I didn't really had much time for, and it would be nice of you if you could help me! Homework Statement Draw a figure, so that the following is true: (AC - AB) * AB = 0 2. The attempt at a solution Since I had to miss some classes, I don't really have...
  8. L

    (Electric) Scalar and vector potential

    Homework Statement In the problem, the electric scalar and vector potentials are, \phi=0, \vec{A}=A_0 e^{i(k_1 x-2k_2y-wt)}\vec{u_y} I have to find E, B and S. Then, I have to calculate \phi ' that satisfies div\vec{A}+\frac{\partial \phi '}{\partial t}=0 Then calculate E and B...
  9. dexterdev

    What is the difference between zero scalar and zero vector?

    Can anyone clarify the concepts of zero vector and zero scalar? -Devanand T
  10. J

    Solving Scalar Line Integral with Vector {\bf{u}}

    I have the vector: {\bf{u}}(x,y) = \frac{{x{\bf{i}} + y{\bf{j}}}}{{{x^2} + {y^2}}} Where: x = a\cos t y = a\sin t I know I need to use the equation \int\limits_0^{2\pi } {{\bf{u}} \cdot d{\bf{r}}} And the answer is \int\limits_0^{2\pi } {} ((a\cos t/{a^2})( - a\sin t) +...
  11. M

    Finding the geodesic function for scalar * function= scalar

    In this expression the junk on the left is a scalar. The stuff before the integral is another scalar. The integral is a time-like curve between x1 and x2 and at imagine fgf(x1) is a lower left corner of the rectangle and fgf(x2) is the upper right corner and x2-x1 is the length of the base of...
  12. P

    Vector And Scalar Equations Of A Plane

    Homework Statement Find vector and scalar equations that passes through point P(3,7,-1) and is perpindicular to the line of intersection of 2 planes. P1:x-y-2z+3=0 P2: 3x-2y+z+5=0 So initially i started by finding the line between the two planes. N1 Does Not Equal N2 so they are not...
  13. A

    Mass dimension of a scalar field in two dimensions?

    Which is the mass dimension of a scalar filed in 2 dimensions? In 4 dim I know that a scalar field has mass dimension 1, by imposing that the action has dim 0: S=\int d^4 x \partial_{\mu} A \partial^{\mu} A where \left[S\right]=0 \left[d^4 x \right] =-4 \left[ \partial_{\mu} \right]=1...
  14. R

    Scalar Boson and Quantum Non-locality

    It's impossible that the higgs boson was the only scalar boson in nature. Could quantum-nonlocality be mediated by scalar boson or connected with scalar field? How do you discount or refute this?
  15. A

    Is Vacuum Energy of a Free Scalar Field Zero?

    Homework Statement I have the following task: In quantum free scalar field theory find commutators of creation and anihilation operators with total four-momentum operator, starting with commutators for fields and canonical momenta. Show that vacuum energy is zero. Homework Equations...
  16. R

    Electric potential vector or scalar sum?

    I'm having trouble fully understanding what electrical potential means. If there are two point charges of opposite signs and a point charge somewhere around them, we simply add the two voltages separately? Not as a vector sum? Also the concept of negative potential, does this mean that the...
  17. P

    Integral in Commutator of Scalar fields

    So, in the calculation of D(t,r) = \left[ \phi(x) , \phi(y) \right] , where t= x^0 - y^0,~ \vec{r} = \vec{x} - \vec{y} you need to calculate the following integral D(t,r) = \frac{1}{2\pi^2 r} \int\limits_0^\infty dp \frac{ p \sin(p r) \sin \left[(p^2 + m^2)^{1/2} t \right]} { (p^2 + m^2...
  18. D

    Check if a scalar is an eigenvalue of a matrix

    Homework Statement We have a matrix Anxn (different than the identity matrix I) and a scalar λ=1. We want to check if λ is an eigenvalue of A. Homework Equations As we know, in order for λ to be an eigenvalue of A, there has to be a non-zero vector v, such that Av=λv The Attempt at a Solution...
  19. O

    Creating 2D Irradiance Distribution with Exact Scalar Method

    I am trying to create (in matlab) an irradiance distribution in a 2-D array, using the exact scalar method. As of now I have a 2-D array defined as my aperture function and I also have a 2-D array defined as a Gaussian function (light source). I believe that I know how to do this with a...
  20. K

    How to Type Matrices in a Forum to Proper Syntax and Formatting

    Homework Statement Find the Scalar Equation of a Plane containing the points P(1,1,-1) Q(0,1,1) Homework Equations ax + by+ cz = d The Attempt at a Solution PQ = [-1,0,2]T [x,y,z]T = [1,1,-1] + s[-1,0,2]T + t[a,b,c]T ^ This is the vector equation.
  21. G

    Lorentz transform of a scalar in QM

    If you Lorentz transform a scalar: U^{-1}(\Lambda)\phi(x)U(\Lambda)=\phi(\Lambda^{-1}x) If you now perform another Lorentz transform, would it it look like this: U^{-1}(\Lambda')U^{-1}(\Lambda)\phi(x)U(\Lambda)U(\Lambda')=\phi(\Lambda'^{-1}\Lambda^{-1}x) ? But isn't this wrong...
  22. C

    Need to find the Ricci scalar curvature of this metric

    Need to find the Ricci scalar curvature of this metric: ds2 = e2a(z)(dx2 + dy2) + dz2 − e2b(z)dt2I tried to find the solution, but failed to pass the calculation of Riemann curvature tensor: <The Christoffel connection> Here a'(z) denotes the first derivative of a(z) respect to z...
  23. C

    Need to find the Ricci scalar curvature of this metric

    Homework Statement Need to find the Ricci scalar curvature of this metric: ds2 = e2a(z)(dx2 + dy2) + dz2 − e2b(z)dt2 Homework Equations The Attempt at a Solution I tried to find the solution, but failed to pass the calculation of Riemann curvature tensor: <The Christoffel...
  24. P

    Sign of scalar product in electric potential integral?

    the potential difference between b and a is defined as follows: V(b) - V(a) = -∫E \bulletdl the integral is taken from a to b. so the potential of a positive charge, with infinity as reference, is V(r) - V(infinity) = V(r) = -∫E \bulletdl the integral is from infinity to r...
  25. L

    Conformal inv of scalar wave equation

    I'm trying to prove the conformal invariance (under g_{\mu\nu}\to\omega^2 g_{\mu\nu}) of \bar{\Box}{\bar{\phi}}+\frac{1}{4}\frac{n-2}{n-1}\bar{R}\bar{\phi} I've found that this equation is invariant upto a quantity proportional to...
  26. R

    Are Scalar Quantities in Physics really 1-D Affine Spaces in disguise?

    In Physics it seems quantities are either Scalars or Vectors (let's not get into tensors, but if there are other types, please tell me). Vectors I understand reasonably well. Scalars, however, I'm not so sure about: Consider the Scalar quantity of Temperature. There is absolute zero, but...
  27. PerpStudent

    EFE: Why is there a curvature tensor and curvature scalar?

    In the Einstein tensor equation for general relativity, why are there two terms for curvature: specifically the curvature tensor and the curvature scalar multiplied by the metric tensor?
  28. D

    A way to express scalar triple product from inter-vector angles?

    Hi, I'm trying to find a general expression for the scalar triple product for 3 vectors in a simultaneous configuration, that depends only on the inter-vector angles, A1, A2 and A3. I have expressed this quantity in terms of the spherical polar coordinates of the vectors (the length being...
  29. DryRun

    How Do You Solve for Vector x and Scalar λ in Vector Equations?

    Homework Statement Find the vector x and the scalar λ which satisfy the equations x \wedge b = b-λc,\; x.c=-2where b = (-2, 1, -1) and c = (1, -2, 2) Homework Equations Vector algebra.The Attempt at a Solution First, i worked on x.c=-2 Let vector x = (x_1, x_2, x_3) So, i got the first...
  30. R

    Write 3 Scalar Eqns + System of 3 Linear Eqns for r,s,t

    -3i-j+5k = t (root34/102 (11i-13j+4k))-s(-j-k)+r(2i+2j+k) how do i write this eqn as 3 scalar equations and a system of three linear eqns for the three parameters r,s, and t. PLEASE HELP...
  31. E

    Noether current for SO(N) invariant scalar field theory

    Homework Statement I understand the premise of Noether's theorem, and I've read over it in as many online lectures as I can find as well as in An Introduction to Quantum Field Theory; Peskin, Schroeder but I can't seem to figure out how to actually calculate it. I feel like I'm missing a...
  32. A

    Why Work is a Scalar Quantity: Exploring the Reason

    We know that work is the dot product between force and displacement .. so dot product always gives scalar (horizontal projection etc) hence work is a scalar quantity? I want the reason behind it... we always do work in specific direction.. suppose a man in appliying force at the angle...
  33. N

    Dual vector is the covariant derivative of a scalar?

    Homework Statement In Wald's text on General Relativity he makes an assertion that I'm not sure why it is allowed mathematically. Here's the basic setup: Let \omega_{b} be a dual vector, \nabla_{b} and \tilde{\nabla}_{b} be two covariant derivatives and f\in\mathscr{F}. Then we may let...
  34. S

    Can you not separate a scalar into x and y components?

    A scalar like electric potential. Say I have a positive charge, and 4m to the right, and 3m up is a point P. If I wanted to calculate the potential at point P, I'd use V=kQ/r (r=√(4^2 + 3^2)). But I'm confused about why finding the potential at 4m to the right (the x component), and the...
  35. B

    Scalar Equation of Plane Determining the value of k

    Homework Statement Determine the value of k so that the line with parametric equations x = 2 + 3t, y = -2 + 5t, z = k is parallel to the plane with equation 4x + 3y – 3z -12 = 0. Homework Equations The Attempt at a Solution let k=a + bt x=2+3t y=-2+5t z=a+bt direction...
  36. P

    Grad of a generalised scalar function

    Homework Statement r=xi+yj+zk and r =\sqrt{x^2 + y^2 + z^2} Let f(r) be a C2 scalar function Prove that \nablaf = \frac{1}{2}\frac{df}{dr}r Homework Equations Vector identities? The Attempt at a Solution \nablaf = (\frac{df}{dx} , \frac{df}{dy} , \frac{df}{dz})...
  37. J

    Translating scalar torque quantities to their vector analogues (RE: Dipoles)

    My question is at the bottom of this post PREAMBLE: If a dipole is turned by an angle θ (in a uniform electric field) then the torque applied on the dipole by the electric field will be: τ = 2.q.a.E.sin(-θ) = -2.q.a.E.sin(θ) with the negative sign referring to it being a "restoring" torque...
  38. M

    What Is the Correct Theta to Use in Calculating the Scalar Product of Vectors?

    Homework Statement Let vectorB= 5.45 m at 60°. Let C have the same magnitude as A and a direction angle greater than that of A by 25°. Let B·A = 32.4 m2 and B·C = 35.1 m2. Find the magnitude and direction of A . Homework Equations A·B=MagAxMagBcosθ The Attempt at a Solution I just...
  39. A

    Scalar potential of a function F, stuck on curl(F) = 0

    Homework Statement I have a function F defined in a slightly strange way, and I'm not asked to test if curl(F) == 0, but I thought I would do this as part of my working out. Lo and behold, it looks like it doesn't == 0, and this means, as far as I know, that there is no scalar potential but...
  40. A

    How Do You Find the Scalar Equation of a Plane from Two Points and a Vector?

    Homework Statement Find the scalar equation of the plane containing the points A(-3, 1, 1) and B(-4, 0, 3) and the vector u = [1, 2, 3]. Homework Equations I am at a lost, since I can't tell how to figure out the normal vector. I am supposed to find: Ax+By+Cz+D=0, where [A,B,C] is the...
  41. R

    Scalar function satisfying div f=F

    What's the algorithm for finding scalar function satisfying div f=F if I know vector F?
  42. M

    Poisson Equation for a Scalar Field

    We all know that for the gravitational field we can write the Poisson Equation: \nabla^2\phi=-4\pi G\rho But I was wondering if, mathematically, we can write the same equation for a scalar field which scale as r^{-2}. Here is the thing. When you deal with gravity, the Poisson equation is...
  43. I

    What is a scalar (under rotation) 1-chain ?

    What is a "scalar (under rotation) 1-chain"? Hi all, I am trying to make sense of a paper involving differenital geometry and Lie algebras. Here's the part I am confused about: Now things begin with finding the cohomology of a Lie algebra. The galilean algebra is taken as an example, and...
  44. C

    Is \(\nabla \times (\phi \nabla \phi) = 0\) for a Differentiable Scalar Field?

    How to prove that \nabla x (\phi\nabla\phi) = 0? (\phi is a differentiable scalar field) I'm a bit confused by this "differentiable scalar field" thing...
  45. A

    How Does a Line Integral of a Scalar Field Differ from a Regular Integral?

    Okay this might be a nooby question, but it bothers me. What is the difference between the line integral of a scalar field and just a regular integral over the scalar field? For a function of one variable i certainly can't see the difference. But then I thought they might be identical in...
  46. K

    LT of the magnetic vector potential when the scalar potential=0

    Special relativity predicts that electric fields transform into magnetic fields via Lorentz transformations and that the vice versa also occurs. It also has been argued, since experiments verifying the quantum mechanical phenomenon of the Aharonov–Bohm effect, that the vector potentials are more...
  47. F

    If a matrix commutes with all nxn matrices, then A must be scalar.

    Homework Statement Prove: If a matrix A commutes with all matrices B \in M_{nxn}(F), then A must be scalar - i.e., A=diag.(λ,...,λ), for some λ \in F. Homework Equations If two nxn matrices A and B commute, then AB=BA. The Attempt at a Solution I understand that if A is scalar, it...
  48. B

    Understanding the Triple Scalar Product in Vector Calculus

    Homework Statement A x (B dot C) (A x B) dot C They are vectors. Homework Equations A x (B dot C) (A x B) dot C The Attempt at a Solution I know how to do my homework, but I am confused on these formulas. Is the first formula "A x (B dot C)" the same as the second one? I know the...
  49. D

    Product Rule for Scalar Product: Verifying the Functions

    Homework Statement Look up, figure out, or make an intelligent guess at the product rule for the scalar product. That is, a rule of the form d/dt [a(t).b(t)] =?+? Verify your proposed rule on the functions a(t) = ti + sin(t)j + e^(t)k and b(t) = cos(t)i - t^(2)j - e^(t)k: Homework...
  50. H

    Is My Gradient Solution for a Scalar Field Correct?

    Homework Statement Consider the scalar field V = r^n , n ≠ 0 expressed in spherical coordinates. Find it's gradient \nabla V in a.) cartesian coordinates b.) spherical coordinates Homework Equations cartesian version: \nabla V = \frac{\partial V}{\partial x}\hat{x} +...
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