Series and multivariable theorems

Taylor's theorem f(x) = P_n(x) + \frac{f^{(n+1)}(c)}{(n+1)!}(x - a)^{n+1} Taylor's theorem with the Lagrange remainder: how close a Taylor polynomial is to f(x). Statement, error bound, graph and worked examples with e and sin x. Clairaut's theorem f_{xy}(a, b) = f_{yx}(a, b) Clairaut's theorem: if the mixed partials f_xy and f_yx are continuous, they are equal. Statement, a classic counterexample, a graph and worked examples. Green's theorem \oint_C P\,dx + Q\,dy = \iint_D \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) dA Green's theorem turns a line integral around a closed curve into a double integral over the region inside. Statement, hypotheses, area formula, examples. Divergence theorem \iint_S \mathbf{F}\cdot d\mathbf{S} = \iiint_E \nabla\cdot\mathbf{F}\,dV The divergence theorem: the outward flux of a vector field through a closed surface equals the triple integral of its divergence. Statement and examples. Stokes' theorem \oint_C \mathbf{F}\cdot d\mathbf{r} = \iint_S (\nabla\times\mathbf{F})\cdot d\mathbf{S} Stokes' theorem: the circulation of a field around a closed curve equals the flux of its curl through any surface the curve bounds. Statement and examples.