$$\iint_\mathcal{A} \boldsymbol{\nabla} \cdot \mathbf{u} \, \mathrm{d} A=\oint_\mathcal{L} \mathbf{u} \cdot \mathrm{d}\mathbf{n} $$
and Kelvin-Stokes theorem
$$\iint_\mathcal{A} \mathbf{k}\cdot\boldsymbol{\nabla} \times \mathbf{u} \, \mathrm{d} A=\oint_\mathcal{L} \mathbf{u} \cdot \mathrm{d}\mathbf{x} $$
are fundamental results relating the area integral of a vector $\mathbf{u}$ to its values along the boundary.
Note that the quantities being integrated on both the left- and right-hand sides are scalars.
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##The Gradient Tensor Theorem
A related result, we which term the *gradient tensor theorem*, is
$$\boxed{\iint_\mathcal{A} \boldsymbol{\nabla} \otimes \mathbf{u} \, \mathrm{d} A=\oint_\mathcal{L} \mathrm{d}\mathbf{n} \otimes \mathbf{u}} $$
with $\otimes$ being the tensor product operator, defined subsequently.
The gradient tensor theorem has a two-by-two *tensor* on each side.
It includes in its components the divergence and Kelvin-Stokes theorems, as well as two less familiar identities.
Note the similarity with the divergence theorem:
$$\iint_\mathcal{A} \boldsymbol{\nabla} \cdot \mathbf{u} \, \mathrm{d} A=\oint_\mathcal{L} \mathrm{d}\mathbf{n} \cdot \mathbf{u} $$
This is an important result that should be more widely known!
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## This Has Been Known for $\sim$150 Years
From J. W. Gibbs (1881), *Elements of Vector Analysis*, p. 65
$$\delta \equiv \frac{\partial u}{\partial x} +\frac{\partial v}{\partial y},\quad\quad
\zeta \equiv \frac{\partial v}{\partial x} -\frac{\partial u}{\partial y},\quad\quad
\nu\equiv \frac{\partial u}{\partial x} -\frac{\partial v}{\partial y},\quad\quad
\sigma \equiv\frac{\partial v}{\partial x}+ \frac{\partial u}{\partial y}$$
These are the four components of the velocity gradient tensor.
There are four corresponding scalar-valued integral theorems
$$
\begin{array}{ll}\displaystyle \iint_\mathcal{A} \displaystyle\delta\, \mathrm{d} A = \oint_\mathcal{L}\left(-v \mathrm{d} x+u \mathrm{d} y \right) &\quad\quad \displaystyle
\iint_\mathcal{A} \zeta\, \mathrm{d} A = \oint_\mathcal{L}\left(u \mathrm{d} x + v \mathrm{d} y\right) \\
\boxed{\displaystyle \iint_\mathcal{A} \nu\, \mathrm{d} A =\oint_\mathcal{L} \left(v \mathrm{d} x+u \mathrm{d} y \right)}&\quad\quad\boxed{\displaystyle\iint_\mathcal{A} \sigma\,\mathrm{d} A =\oint_\mathcal{L}\left(-u \mathrm{d} x + v \mathrm{d} y\right)}
\end{array}$$
all of which are readily derived via Green's theorem
$$\iint_\mathcal{A} \left(\frac{\partial M}{\partial x}-\frac{\partial L}{\partial y}\right)\mathrm{d} x\mathrm{d} y = \oint_\mathcal{L} \left(L \,\mathrm{d} x +M \,\mathrm{d} y\right).$$
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##An Application to an Oceanic Vortex