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Explore relationships within this area of study and its connections to other areas.

Showing 1–50 of 811 relationships

Scientific relationships · 50 / 772

The relationship type describes the connection. Its linked statements, sources and conditions define the scope.

  1. Uses definition
    Relationship sources · 1

    Linked statements

    • The slope of the line through two points with different x-coordinates is (y₂ − y₁)/(x₂ − x₁). This ratio is constant along a straight line. A vertical line has no finite slope in this coordinate convention.
  2. Uses definition
    Relationship sources · 1

    Linked statements

    • A plotted point represents values through the graph's stated coordinate system. Read each coordinate against its own axis scale; points on the graph of an equation satisfy that equation.
  3. Uses definition
    Relationship sources · 1

    Linked statements

    • A planar Euclidean vector can be represented by ordered components in a chosen basis. Vector addition adds corresponding components, and scalar multiplication multiplies every component by the scalar.
  4. Uses definition
    Relationship sources · 1

    Linked statements

    • For a real angle θ, the point reached on the unit circle has coordinates (cos θ, sin θ). Therefore cos²θ+sin²θ=1 for every real θ.
  5. Uses definition
    Relationship sources · 1

    Linked statements

    • For a real angle θ, the point reached on the unit circle has coordinates (cos θ, sin θ). Therefore cos²θ+sin²θ=1 for every real θ.
  6. Uses definition
    Relationship sources · 1

    Linked statements

    • A complex number is written z=a+bi with real a and b and i²=−1. Its real part is a and its imaginary part is b. It is represented by (a,b) in the complex plane; addition combines the two components and multiplication uses i²=−1.
  7. Uses definition
    Relationship sources · 1

    Linked statements

    • A real linear map T:ℝⁿ→ℝᵐ satisfies T(αu+βv)=αT(u)+βT(v). After choosing input and output bases, it is represented by an m×n matrix A: the output coordinates are y=Ax, with yᵢ=Σⱼ₌₁ⁿaᵢⱼxⱼ.
  8. Uses definition
    Relationship sources · 1

    Linked statements

    • A real linear map T:ℝⁿ→ℝᵐ satisfies T(αu+βv)=αT(u)+βT(v). After choosing input and output bases, it is represented by an m×n matrix A: the output coordinates are y=Ax, with yᵢ=Σⱼ₌₁ⁿaᵢⱼxⱼ.
  9. Uses definition
    Relationship sources · 1

    Linked statements

    • For a real function defined on an open interval around a, its derivative at a is the limit of [f(a+h)−f(a)]/h as h approaches zero, if that finite limit exists.
  10. Uses definition
    Relationship sources · 1

    Linked statements

    • For a continuous real function on [a,b], the definite integral is the limit of sums f(xᵢ*)Δx over successively finer equal subdivisions. It measures signed accumulation; negative contributions subtract rather than become positive area.
  11. Uses definition
    Relationship sources · 1

    Linked statements

    • If f is continuous on [a,b], the accumulation F(x)=∫ₐˣf(t)dt satisfies F′(x)=f(x) for a<x<b. For any antiderivative G of f, ∫ₐᵇf(x)dx=G(b)−G(a).
  12. Uses definition
    Relationship sources · 1

    Linked statements

    • If f is continuous on [a,b], the accumulation F(x)=∫ₐˣf(t)dt satisfies F′(x)=f(x) for a<x<b. For any antiderivative G of f, ∫ₐᵇf(x)dx=G(b)−G(a).
  13. Uses definition
    Relationship sources · 1

    A conversion changes the numerical representation of the same quantity.

  14. Uses definition
    Relationship sources · 2

    Calibration-chain uncertainty belongs to the traced measurement result.

  15. Uses definition
    Relationship sources · 1

    The first-order formula combines input standard uncertainties and covariances.

  16. Related to
    Relationship sources · 1

    Propagation requires an explicit measurement model containing significant influence quantities.

  17. Derived from
    Relationship sources · 1

    The constant-mass Newtonian particle theorem follows by integrating Newton's second law along the displacement.

  18. Derived from
    Relationship sources · 1

    For a fixed classical particle system, summed Newtonian forces give the total momentum balance.

  19. Related to
    Relationship sources · 1

    In the stated Lagrangian setting, continuous symmetries give conserved quantities; this is not a claim about all visual or discrete symmetries.

  20. Related to
    Relationship sources · 1

    The electric field defines the electric contribution to force on a test charge.

  21. Uses definition
    Relationship sources · 1

    The charge velocity and magnetic field must be evaluated in the same frame.

  22. Uses definition
    Relationship sources · 2

    The linear device relation uses current and potential difference in its stated operating range.

  23. Uses definition
    Relationship sources · 1

    Heat and work cross the specified closed-system boundary under the adopted sign convention.

  24. Uses definition
    Relationship sources · 2

    The total entropy inequality applies to the whole isolated system, not each arbitrary subsystem.

  25. Related to
    Relationship sources · 2

    The gas equation is exact within the ideal model and approximate for a real gas.

  26. Uses definition
    Relationship sources · 2

    The photon relation E = hf uses the Planck constant with energy and frequency in the same frame.

  27. Uses definition
    Relationship sources · 1

    The flat-spacetime interval uses the vacuum speed c; its sign convention is explicit.

  28. Related to
    Relationship sources · 2

    Decimal prefix factors supply exact scale conversions for compatible SI units.

  29. Related to
    Relationship sources · 2

    Equal-and-opposite internal particle-force pairs cancel in the fixed-system Newtonian linear-momentum balance.

  30. Derived from
    Relationship sources · 1

    For a particle about a fixed inertial origin, differentiating r × p and applying Newton's second law gives net torque.

  31. Derived from
    Relationship sources · 1

    Balancing gravity and pressure forces on a static uniform-density fluid element gives the depth pressure difference.

  32. Derived from
    Relationship sources · 1

    The net hydrostatic pressure force equals displaced-fluid weight under the stated static-fluid and boundary assumptions.

  33. Uses definition
    Relationship sources · 2

    E is the electric field in the classical free-space field equations; this is not a material D/H constitutive law.

  34. Uses definition
    Relationship sources · 1

    Current density is charge flow per area and obeys local continuity with charge density in this model.

  35. Uses definition
    Relationship sources · 2

    The one-dimensional scalar-potential time-evolution equation uses ℏ = h/(2π).

  36. Related to
    Relationship sources · 1

    For the stated real-potential evolution and suitable boundaries, the normalized wavefunction's total position probability is preserved.

  37. Uses definition
    Relationship sources · 1

    Linked statements

    • For a linear map T:V→W with finite-dimensional V, dim V = dim ker T + dim im T. Rank is the image dimension and nullity is the kernel dimension.
  38. Uses definition
    Relationship sources · 1

    Linked statements

    • Every real m×n matrix A admits A=UΣVᵀ with orthogonal U,V and rectangular diagonal Σ having nonnegative entries. These singular values are square roots of eigenvalues of AᵀA, with zero padding as needed.
  39. Uses definition
    Relationship sources · 1

    Linked statements

    • For f of class Cⁿ⁺¹ on an interval containing a and x, f(x)=Σₖ₌₀ⁿ f⁽ᵏ⁾(a)(x−a)ᵏ/k! + f⁽ⁿ⁺¹⁾(c)(x−a)ⁿ⁺¹/(n+1)! for some c between a and x.
  40. Uses definition
    Relationship sources · 1

    Linked statements

    • For f:U⊂Rⁿ→Rᵐ with U open, f is differentiable at a if a linear map B satisfies ‖f(a+h)−f(a)−Bh‖/‖h‖→0 as h→0. This B is the total derivative Df(a).
  41. Uses definition
    Relationship sources · 1

    Linked statements

    • On D={(x,y):a≤x≤b, g₁(x)≤y≤g₂(x)}, with continuous g₁≤g₂ and continuous f on D, ∬D f dA=∫ₐᵇ∫g₁(x)ᵍ₂(x) f(x,y)dy dx.
  42. Uses definition
    Relationship sources · 1

    Linked statements

    • If f:Rⁿ→Rⁿ is locally Lipschitz near x₀, the autonomous initial-value problem x′=f(x), x(0)=x₀ has a unique solution on some interval around zero.
  43. Uses definition
    Relationship sources · 1

    Linked statements

    • For a constant real n×n matrix A, x′=Ax with x(0)=x₀ has the unique global solution x(t)=eᵗᴬx₀, where eᵗᴬ=Σₖ₌₀∞ tᵏAᵏ/k! converges for every real t.
  44. Derived from
    Relationship sources · 1

    Linked statements

    • For fixed-mass Newtonian particles with cancelling internal force pairs, the centre of mass R is the mass-weighted position and M R̈ equals total external force.

    Newtonian fixed-particle force sum with cancelling internal pairs.

  45. Uses definition
    Relationship sources · 2

    Linked statements

    • For a rigid body, the inertia tensor about its centre of mass gives rotational angular momentum L = Iω. Its diagonal entries measure squared distances from axes; off-diagonal matrix entries carry minus coordinate products.

    Rigid-body rotational angular momentum about its centre of mass.

  46. Uses definition
    Relationship sources · 1

    Linked statements

    • In principal axes fixed to a rigid body, Euler's equations connect torque components with angular acceleration and products of angular velocities.

    Body-fixed principal moments in the rigid-body torque equation.

  47. Related to
    Relationship sources · 1

    Linked statements

    • For a smooth Lagrangian L(q,q̇,t), stationary action under fixed-endpoint variations gives d(∂L/∂q̇_i)/dt − ∂L/∂q_i = 0.

    Generalized coordinates and rates parameterize classical motion; regular stationary-action assumptions apply.

  48. Derived from
    Relationship sources · 1

    Linked statements

    • For a regular Lagrangian, p_i = ∂L/∂q̇_i and H = Σ_i p_i q̇_i − L lead to q̇_i = ∂H/∂p_i and ṗ_i = −∂H/∂q_i.

    Regular locally invertible momentum map and Legendre transform.

  49. Related to
    Relationship sources · 3

    Linked statements

    • Classical mass conservation without sources gives ∂ρ/∂t + ∇·(ρv) = 0. Constant positive density reduces this to ∇·v = 0; steady constant-density tube flow gives A₁v₁ = A₂v₂.

    A smooth classical no-source mass balance, rather than a claim that every quantity is conserved.

  50. Derived from
    Relationship sources · 2

    Linked statements

    • Along a streamline of steady inviscid incompressible flow under uniform gravity, p + ½ρv² + ρgz is constant.

    Steady inviscid incompressible streamline pressure work and kinetic/gravitational energy balance.