implicit function theorem
The equation x^2 + y^2 = 1 does not hand you y as a formula in x, yet near most points of the circle you can solve for y as a smooth function of x. The implicit function theorem tells you, without ever finding that formula, when such a local solution exists and that it is as smooth as the equation — turning a tangle of constraints into honest functions.
Precisely: let F: R^n x R^m -> R^m be C^1, and suppose F(a, b) = 0. Split the derivative into the block of partials with respect to the last m variables. If that m-by-m block, the partial Jacobian D_y F(a, b), is invertible, then near a there is a unique C^1 function g with g(a) = b and F(x, g(x)) = 0 for all x near a. Its derivative is Dg(a) = -(D_y F)^{-1} (D_x F), obtained by differentiating the constraint.
The theorem is the workhorse behind level sets being smooth manifolds, behind Lagrange multipliers, and behind solving systems locally. The non-degeneracy hypothesis is essential: where the relevant partial Jacobian is singular — at the points (1, 0) and (-1, 0) of the circle, where the tangent is vertical — you cannot solve for y as a function of x, and the conclusion genuinely fails.
F(x, y) = x^2 + y^2 - 1. At (0, 1), dF/dy = 2y = 2 is nonzero, so y = sqrt(1 - x^2) solves it locally near x = 0. At (1, 0), dF/dy = 0; here y cannot be a function of x (the circle is vertical), but x can be a function of y instead.
Solvable where the partial Jacobian is nonsingular; fails where it vanishes.