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Let a line collection have $M$ joints of a line collection and put $D=\lceil n{M}^{1/n}\rceil$. If $M>LD$, deleting each line with at most $D$ current joints of a line collection, together with those joints of a line collection, leaves a nonempty configuration in which every line has more than $D$ joints of a line collection. A nonzero polynomial of degree at most $D$ vanishes on the retained joints of a line collection by rank-nullity theorem. Choose one of minimum degree. Its restriction vanishes identically on every retained line, so its gradient is orthogonal to a spanning set of directions at every joint of a line collection and hence vanishes there. Each nonzero partial derivative would be a lower-degree polynomial with the same zeros, contradicting minimality. In characteristic zero, all derivatives vanishing forces a constant polynomial, another contradiction. Therefore $M\le LD$, giving the joints theorem. The argument's characteristic-zero assumption is essential to its derivative step.

For a classical solution of $\Delta u=f$ on a half-ball with zero data on its flat face, a tangential Caccioppoli inequality controls every second partial derivative having at least one tangential index. The equation recovers the remaining second normal derivative. The resulting Sobolev norm estimate has a dimension-only constant for the fixed radii one and two. Nonzero data are handled by homogenization of Dirichlet boundary data, adding the $W^{2,2}$ norm of the supplied boundary extension. No condition is imposed on the curved face because the smooth cutoff function vanishes there.

A spatial derivative is a partial derivative with respect to a spatial coordinate.

If an elliptic curve has good reduction over a local field $K$, its integral projective model is smooth. Every point over the residue field lifts by the Hensel lemma, applied in a chart with a unit partial derivative. Thus reduction gives an exact sequence $0\to E_1(K)\to E(K)\to \tilde{E}(k)\to 0$. The kernel of reduction of an elliptic curve is the maximal ideal of the valuation ring, with the formal group of an elliptic curve as its group operation.

A time derivative is the partial derivative of an evolving field with respect to time while holding its spatial coordinates fixed. For a quantity depending only on time, it is an ordinary derivative.