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If the cylindrical components of a vector field have dependence $e^{im\varphi }$, differentiation of the rotating basis adds the displayed cross product term. Scalar-component differentiation alone misses this contribution. For $\mathbf{B}=(J/2)s\hat{\mathbf{\varphi }}$, one also has $(\mathbf{v}\cdot \nabla )\mathbf{B}=(J/2)\hat{\mathbf{z}}\times \mathbf{v}$. These identities are useful in cylindrical magnetohydrodynamic waves.

The vector field $\mathbf{G}=(-y,x,0)/(x^2+y^2)$ has zero curl away from its excluded axis, but has circulation $2\pi$ around a positively oriented circle enclosing that axis. Stokes theorem gives zero total circulation for a smooth annular surface avoiding the axis, whose inner and outer boundary circles have opposite orientations. It cannot be applied directly to a disk meeting the axis, because the field is undefined there.

The Lie derivative of a differential form satisfies ${\mathcal{L}}_X=d{\iota }_X+{\iota }_{X}d$. Both sides agree on functions. On a differential 1-form $\eta$, evaluation at a vector field $Y$ gives $X(\eta (Y))-\eta ([X,Y])$ on both sides, by the definition of the exterior derivative and the Lie bracket of vector fields. Both operators are degree-zero derivations of an algebra on the exterior algebra. Agreement on functions and coordinate differential 1-forms therefore proves agreement on every differential form. Since $d^2=0$, the formula also gives ${\mathcal{L}}_{X}d=d{\mathcal{L}}_X$, so the Lie derivative of a differential form commutes with the exterior derivative.

For $a{u}_x+b{u}_y=c$, the projected vector field $(a,b)$ has the characteristic projections as its integral curves. The lifted vector field is $(a,b,c)$ in $(x,y,u)$ coordinates. A vanishing projected vector field can obstruct a classical solution even when the lifted vector field does not vanish.

Let a closed oriented surface $S$ lose a patch $H$, leaving $S^{\prime }$. Assume a smooth vector field on the relevant surfaces and that its total curl flux through $S$ is zero. The Stokes theorem gives the title formula when the edge is oriented as the boundary of the remaining surface $S^{\prime }$. The boundary of the removed patch $H$, with its original normal, has the opposite direction and hence the opposite circulation.

For a cylindrical side of radius $R$, if $(\nabla \times \mathbf{v})\cdot \mathbf{n}=-\alpha R$ everywhere on the removed patch, the circulation around the boundary of the remaining surface is $\alpha R\mathrm{area}(H)$. This uses actual curved area, rather than its planar projection, and does not depend on the patch shape. For multiple boundary components, the formula concerns the sum with their induced orientations.

For a closed polyhedron, let $A_F$ be a face area and ${\mathbf{n}}_F$ its outward unit normal vector. The sum of the face area vectors is zero. To prove it, fix any constant vector $\mathbf{q}$ and apply the divergence theorem to the constant vector field $\mathbf{q}$: its divergence vanishes, while its outward flux is $\mathbf{q}\cdot {\sum }_F{A}_F{\mathbf{n}}_F$. This is zero for every $\mathbf{q}$, so the sum vanishes. For a tetrahedron, the same identity follows directly by expanding the four oriented face cross products; multiplying every face vector by two leaves the identity intact.

A first-order vector field $Z$ is a commutation field for the flat wave equation when $[\square ,Z]$ is zero or a controlled multiple of $\square$. Translations, spatial rotations and Lorentz boost vector fields commute with $\square$; the scaling vector field satisfies $[\square ,S]=2\square$. Repeated commutation preserves the differential order of a semilinear wave equation.

The Lie derivative of a tensor field satisfies $[{\mathcal{L}}_X,{\mathcal{L}}_Y]={\mathcal{L}}_{[X,Y]}$. On smooth functions this is the definition of the Lie bracket of vector fields; on vector fields it follows from the Jacobi identity. The Leibniz rule and contraction compatibility extend the equality to tensor fields. The cyclic double-commutator identity also follows directly from associativity of operator composition.

For a torsion-free connection, the commutator of coordinate covariant derivatives vanishes on a scalar field and acts on a vector field through the displayed Riemann curvature tensor. On a covector field it instead gives $[{\nabla }_{\alpha },{\nabla }_{\beta }]A_{\mu }=-R^{\nu }{}_{\mu \alpha \beta }{A}_{\nu }$. The curvature commutator is a tensor derivation, so each index of a general tensor contributes independently with the corresponding sign. Including the lower derivative index in the second covariant derivative is essential for this identity.

A compact manifold is a smooth manifold whose underlying topological space is compact. Every smooth vector field on a compact manifold without boundary has a flow defined for all real time. On a manifold with boundary, tangency to the boundary or an appropriate inward-flow condition must also be imposed.

A smooth vector field is complete if its maximal integral curves of a vector field exist for every real time. Its local flow is then a global one-parameter group of diffeomorphisms; uniqueness of ordinary differential equations proves the group law and the inverse-time identity. On a compact manifold without boundary, every smooth vector field is complete: compactness gives a uniform positive local existence interval through every point, so a curve can be successively extended past any proposed finite endpoint. On $\mathbb{R}$, $x^2{\partial }_x$ is not complete because its solution $x(t)=x_0/(1-t{x}_0)$ has finite-time blowup for $x_0>0$.

On ${\mathbb{R}}^2$, the vector fields $X=y{\partial }_x$ and $Y=(x^2/2){\partial }_y$ are complete, since their local flows are $(x+ty,y)$ and $(x,y+t{x}^2/2)$. Their Lie bracket of vector fields is $-(x^2/2){\partial }_x+xy{\partial }_y$. Its integral curve of a vector field starting at $(2,0)$ is $(2/(1+t),0)$, which cannot continue through $t=-1$. The sum has an integral curve of a vector fieldthat cannot continue through $t=2\sqrt{3}$. Thus neither operation preserves complete vector fields on a general noncompact smooth manifold. This contrasts with a compact manifold without boundary, where every smooth vector field is complete.

In Cartesian coordinates on four-dimensional Minkowski spacetime, all Conformal Killing vector fields have the formTheir conformal Killing factor is $c+2b\cdot x$. There are four translation generators, six Lorentz transformation generators, one dilation generator, and four special conformal transformation generators. These form a fifteen-dimensional Lie algebra under the Lie bracket of vector fields.

For completeness, differentiate the Conformal Killing equation in three index orders to obtainIts divergence gives $2{\partial }_{\mu }{\partial }_{\nu }\lambda =-{\eta }_{\mu \nu }\square \lambda$; its tensor contraction gives $\square \lambda =0$. Hence $\lambda$ is an affine function. Subtracting the displayed quadratic and dilation terms leaves a Killing vector field whose second partial derivatives vanish, proving completeness of the list. The vector fields are globally smooth, but some special conformal transformations have incomplete flows on affine Minkowski spacetime; the dilation flow $x\mapsto e^{s}x$ is global.

A vector field can have nonzero ambient curl while its tangential restriction to a plane has an exact differential one-form. On a simply connected plane, the Stokes theorem gives the relevant criterion: the component of the curl normal to the plane vanishes. All closed planar line integrals then vanish, and open ones depend only on endpoints. For example, $(y-z,z-x,x-y)$ restricts on $x-z=1$ to $F\cdot dr=d(x-y)$, despite ambient curl $-2(1,1,1)$.

A vector field $X$ is conservative when $X=\nabla \Phi$ for a potential of a conservative vector field $\Phi$. On a simply connected open subset of ${\mathbb{R}}^3$, every continuously differentiable curl-free vector field is conservative. Locally this follows from the Poincaré lemma applied to the differential one-form $X_{i}d{x}^i$; globally, simple connectivity makes its line integral independent of the path. The fundamental theorem for line integrals expresses that path independence as the difference of the potential at the endpoints.

A vector field on an open subset of Euclidean space whose component functions have continuous first partial derivatives. Its Jacobian matrix is continuous; this is a standard regularity assumption for the divergence theorem and Stokes theorem.

If a locally integrable function has weak derivatives represented by a continuous vector field $g$, it has a representative $U\in C^1$ with $\nabla U=g$. The assertion is local, so work on a ball compactly contained in the domain and regularize using a mollifier. The regularizations satisfy the segment identity $u_{\epsilon }(x)-u_{\epsilon }(y)={\int }_0^1\nabla u_{\epsilon }(y+t(x-y))\cdot (x-y)dt$. Averaging over a smaller ball in $y$ fixes the additive constant through local $L^1$ convergence. The gradients converge uniformly on compact sets, so the averaged identity gives local uniform convergence to $U$. Passing to the limit on each segment then gives $U(x+h)-U(x)-g(x)\cdot h=o(|h|)$. Continuity of $g$ proves that $U$ is continuously differentiable.

Under a change of coordinates, the second-derivative terms arising from the chain rule cancel because their coordinate indices are symmetric while the coefficient $X^i{Y}^j-Y^i{X}^j$ is antisymmetric. Consequently $[X,Y]$ transforms as a vector field.

For every contravariant slot of a tensor field, the tensor Lie derivative contributes a negative derivative of its generating vector field; every covariant slot contributes a positive one. The remaining term differentiates the tensor's components along the generator. This follows from the flow definition of the Lie derivative of a tensor field and its product and contraction rules, without choosing a affine connection.

The Christoffel trace identity converts the covariant derivative of a vector field into a coordinate divergence weighted by the metric volume density. This is valid for the Levi-Civita connection. Applied to the raised gradient of a scalar field, it gives $\square \varphi =|g{|}^{-1/2}{\partial }_{\mu }(\sqrt{|g|}{g}^{\mu \nu }{\partial }_{\nu }\varphi )$ in Lorentzian signature, the curved d'Alembert operator.