一个向量场可以表示为一个无旋的散度场和一个无散的旋度场的叠加.
在有界区域 $\Omega$ 内, 任一向量场可以由它的散度、旋度和边界条件唯一确定.
In physics and mathematics, the Helmholtz decomposition theorem or the fundamental theorem of vector calculus states that certain differentiable vector fields can be resolved into the sum of an irrotational (curl-free) vector field and a solenoidal (divergence-free) vector field.
For example, the Hodge theorem implies that the cohomology groups with real coefficients of a closed manifold are finite-dimensional. (Admittedly, there are other ways to prove this.) Indeed, the operator $\Delta$ are elliptic, and the kernel of an elliptic operator on a closed manifold is always a finite-dimensional vector space.
A variant of the Hodge theorem is the Hodge decomposition. This says that there is a unique decomposition of any differential form $\omega$ on a closed Riemann manifold as a sum of three parts in the form $$\omega=\md \alpha+\delta\beta+\gamma,$$ in which $\gamma$ is harmonic: $\delta\gamma=0$. In terms of the $L^2$ metric on differential forms, this gives an orthogonal direct sum decomposition $$\Omega^k(M)\cong \mr{Im}\md_{k-1}\oplus \mr{Im}\delta_{k+1}\oplus H_\Delta^k(M).$$ The Hodge decomposition is a generalization of the Helmholtz decomposition for de Rham complex.