(mathematics) A formula giving the expansion of a binomial such as (a+b) raised to any positive integer power, i.e. (a+b)ⁿ. It's possible to expand the power into a sum of terms of the form axᵇyᶜ where the coefficient of each term is a positive integer. For example:

(physics) A theorem in mechanics showing the relationship between the kinetic energy of a system to the virial of Clausius.

(calculus) A generalization of the fundamental theorem of calculus to the two-dimensional plane, which states that given two scalar fields P and Q and a simply connected region R, the area integral of derivatives of the fields equals the line integral of the fields, or

(fluid dynamics) A theorem which states that to achieve propulsion at low Reynolds number in simple (i.e. Newtonian) fluids, a swimmer must deform in a way that is not invariant under time-reversal.

(physics) A theorem, proven by Emmy Noether in 1915 and published in 1918, stating that any differentiable symmetry of the action of a physical system has a corresponding conservation law.

(mathematics) A theorem stating that every Goodstein sequence eventually terminates at zero.

A theorem postulating that all black hole solutions of the Einstein-Maxwell equations in general relativity can be completely characterized by only three externally observable classical parameters: mass, electric charge, and angular momentum.

(logic) A procedure for "discharging" assumptions from an inference, causing them to become antecedents of the conclusion; or vice versa. Symbolically, the conversion of an inference of the form P,A⊢C to an inference of the form P⊢A→C or vice versa, where ⊢ is the turnstile symbol. The validity of the procedure is a metatheorem of the given logical theory.

(logic) The following theorem of Boolean algebra: XY+X'Z+YZ=XY+X'Z where YZ, the algebraically redundant term, is called the "consensus term", or its dual form (X+Y)(X'+Z)(Y+Z)=(X+Y)(X'+Z), in which case Y+Z is the consensus term. (Note: X+Y,X'+Z⊢Y+Z is an example of the resolution inference rule (replacing the + with ∨ and the prime with prefix ¬ might make this more evident).)

(mathematics) a theorem providing conditions under which an operator or matrix can be diagonalized