Foundations

Regulated cell states, division and spatial organization produce multicellular form.

Mineral identification combines structural and physical evidence.

Isotopic measurements estimate an age only with a suitable geological model.

Wave travel, refraction and transmission constrain an interior model.

Fluid motion in the outer core maintains much of Earth's magnetic field.

A defined area's water storage changes with total inflow and outflow.

Groundwater flow is often approximately linear in hydraulic-head gradient.

Wind, buoyancy changes and mixing contribute to ocean transport.

Unequal heating and rotation organize broad circulation patterns.

External changes can disturb climate's radiation budget.

Climate responses can amplify or damp a change.

Attribution compares observed changes with plausible drivers and variability.

Stellar structure changes as core fuel and conditions change.

Different evolutionary outcomes leave different compact remnants.

Changing observing position permits geometric distance inference.

Spectral stretching carries expansion information but is not one universal distance ruler.

Measured motions constrain the gravitational mass distribution.

Disk observations and competing growth mechanisms inform planet-formation models.

CMB statistics constrain parameters within a chosen cosmological model.

Motions and lensing constrain mass beyond that accounted for by luminous material.

Gravitational evidence does not yet settle the unseen component's physical identity.

Expansion evidence motivates competing physical descriptions with unresolved origin.

A continuous real function on a closed bounded interval attains a maximum and minimum and takes every value between its endpoint values.

Differentiable real functions satisfy (fg)′=f′g+fg′ and (g∘f)′=(g′∘f)f′.

If f is continuous on [a,b] and differentiable on (a,b), a<b, some c∈(a,b) satisfies f′(c)=(f(b)−f(a))/(b−a).

For f differentiable throughout an open real interval, its derivative takes every value strictly between f′(a) and f′(b) at some c∈(a,b), whenever a<b are points of that interval.

For g of class C¹ on [a,b] and f continuous on an interval containing g([a,b]), ∫ₐᵇ f(g(x))g′(x)dx=∫g(a)ᵍ(b)f(u)du.

For u,v of class C¹ on [a,b], ∫ₐᵇuv′dx=[uv]ₐᵇ−∫ₐᵇu′v dx.

For bounded real f on [a,b], lower and upper Darboux sums use each subinterval's infimum and supremum.

A bounded real function on [a,b] is Riemann integrable exactly when its discontinuity set has Lebesgue measure zero.