What are correct ways to represent differential operations related to vector fields?

"ISO 80000-2 Quantities and Units - Mathematics" specifies that differential operations should be represented as described below:

Del (Nabla) Operator - The nabla operator is a vector operator and should be identified by the Nabla symbol ∇ (U+2207, ∇ and ∇). It can be defined as a vector:

<math><mo>∇</mo> <mo>≔</mo> <mo>(</mo> 
  <mfrac> <mo>∂</mo> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <mo>,</mo>
  <mfrac> <mo>∂</mo> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <mo>,</mo>
  <mfrac> <mo>∂</mo> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <mo>)</mo>
  </math>
    $\nabla ≔(\frac{\partial }{\partial x},\frac{\partial }{\partial y},\frac{\partial }{\partial y})$

Gradient Operation - The gradient operation measures the rate and direction of change of a scalar field at a given point. It should be identified as the grad(f) function or the multiplication of the Nabla operator ∇:

<math> <mi>grad</mi> <mi>f</mi> <mo>=</mo>
  <mo>&#x2207;</mo> <mi>f</mi> <mo>=</mo> <mo>(</mo> 
  <mfrac> <mrow> <mo>&#x2202;</mo> <mi>f</mi> </mrow> <mrow> <mo>&#x2202;</mo> <mi>x</mi> </mrow> </mfrac> <mo>,</mo>
  <mfrac> <mrow> <mo>&#x2202;</mo> <mi>f</mi> </mrow> <mrow> <mo>&#x2202;</mo> <mi>y</mi> </mrow> </mfrac> <mo>,</mo>
  <mfrac> <mrow> <mo>&#x2202;</mo> <mi>f</mi> </mrow> <mrow> <mo>&#x2202;</mo> <mi>y</mi> </mrow> </mfrac> <mo>)</mo>
  </math>
    $\mathrm{grad}f=\nabla f=(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial f}{\partial y})$

Divergence Operation - The divergence operation measures the rate at which a vector field flows out of a give point. It should be identified as the div(F) function or the dot product of the Nabla operator ∇:

<math> <mi>div</mi> <mi mathvariant="bold-italic">F</mi> <mo>=</mo>
  <mo>&#x2207;</mo> <mo>&#x22C5;</mo> <mi mathvariant="bold-italic">F</mi> <mo>=</mo>
  <mfrac> <mrow> <mo>&#x2202;</mo> <msub> <mi>F</mi> <mi>x</mi> </msub> </mrow> 
    <mrow> <mo>&#x2202;</mo> <mi>x</mi> </mrow></mfrac> <mo>+</mo>
  <mfrac> <mrow> <mo>&#x2202;</mo> <msub> <mi>F</mi> <mi>y</mi> </msub> </mrow> 
    <mrow> <mo>&#x2202;</mo> <mi>y</mi> </mrow></mfrac> <mo>+</mo>
  <mfrac> <mrow> <mo>&#x2202;</mo> <msub> <mi>F</mi> <mi>z</mi> </msub> </mrow> 
    <mrow> <mo>&#x2202;</mo> <mi>z</mi> </mrow></mfrac>&/math>
    $\mathrm{div}\mathit{F}=\nabla \cdot \mathit{F}=\frac{\partial F_x}{\partial x}+\frac{\partial F_y}{\partial y}+\frac{\partial F_z}{\partial z}$

Rotation Operation - The rotation (or curl) operation measures the tendency of rotation of a vector field at a give point. It should be identified as the rot(F) function or the cross product of the Nabla operator ∇:

<math> <mi>rot</mi> <mi mathvariant="bold-italic">F</mi> <mo>=</mo>
  <mo>&#x2207;</mo> <mo>&#x00D7;</mo> <mi mathvariant="bold-italic">F</mi> <mo>=</mo>
  <mo>(</mo>
    <mfrac> <mrow> <mo>&#x2202;</mo> <msub> <mi>F</mi> <mi>z</mi> </msub> </mrow>
      <mrow> <mo>&#x2202;</mo> <mi>y</mi> </mrow> </mfrac> <mo>-</mo>
    <mfrac> <mrow> <mo>&#x2202;</mo> <msub> <mi>F</mi> <mi>y</mi> </msub> </mrow>
      <mrow> <mo>&#x2202;</mo> <mi>z</mi> </mrow> </mfrac> <mo>,</mo>
    <mfrac> <mrow> <mo>&#x2202;</mo> <msub> <mi>F</mi> <mi>x</mi> </msub> </mrow>
      <mrow> <mo>&#x2202;</mo> <mi>z</mi> </mrow> </mfrac> <mo>-</mo>
    <mfrac> <mrow> <mo>&#x2202;</mo> <msub> <mi>F</mi> <mi>z</mi> </msub> </mrow>
      <mrow> <mo>&#x2202;</mo> <mi>x</mi> </mrow> </mfrac> <mo>,</mo>
    <mfrac> <mrow> <mo>&#x2202;</mo> <msub> <mi>F</mi> <mi>y</mi> </msub> </mrow>
      <mrow> <mo>&#x2202;</mo> <mi>x</mi> </mrow> </mfrac> <mo>-</mo>
    <mfrac> <mrow> <mo>&#x2202;</mo> <msub> <mi>F</mi> <mi>x</mi> </msub> </mrow>
      <mrow> <mo>&#x2202;</mo> <mi>y</mi> </mrow> </mfrac>
  <mo>)</mo></math>
    $\mathrm{rot}\mathit{F}=\nabla \times \mathit{F}=(\frac{\partial F_z}{\partial y}-\frac{\partial F_y}{\partial z},\frac{\partial F_x}{\partial z}-\frac{\partial F_z}{\partial x},\frac{\partial F_y}{\partial x}-\frac{\partial F_x}{\partial y})$
 

Laplacian Operation - The Laplacian operation measures the difference between the value of the scalar field with its average on infinitesimal balls at a given point. It should be identified as ∇²:

<math> 
  <msup><mo>&#x2207;</mo><mn>2</mn></msup><mi>f</mi><mo>=</mo>
  <mo>&#x2207;</mo><mo>&#x22C5;</mo> <mo>&#x2207;</mo> <mi>f</mi> <mo>=</mo>
  <mfrac> <mrow> <msup><mo>&#x2202;</mo><mn>2</mn></msup> <mi>f</mi> </mrow> 
    <mrow> <mo>&#x2202;</mo> <msup><mi>x</mi><mn>2</mn></msup> </mrow></mfrac> <mo>+</mo>
  <mfrac> <mrow> <msup><mo>&#x2202;</mo><mn>2</mn></msup> <mi>f</mi> </mrow> 
    <mrow> <mo>&#x2202;</mo> <msup><mi>y</mi><mn>2</mn></msup> </mrow></mfrac> <mo>+</mo>
  <mfrac> <mrow> <msup><mo>&#x2202;</mo><mn>2</mn></msup> <mi>f</mi> </mrow> 
    <mrow> <mo>&#x2202;</mo> <msup><mi>z</mi><mn>2</mn></msup> </mrow></mfrac></math>
    ${\nabla }^{2}f=\nabla \cdot \nabla f=\frac{{\partial }^{2}f}{\partial x^2}+\frac{{\partial }^{2}f}{\partial y^2}+\frac{{\partial }^{2}f}{\partial z^2}$

⇒ Integrals over Curve, Surface and Volume

⇐ Vectors and Vector Operators"

⇑ ISO 80000-2 Quantities and Units - Mathematics

⇑⇑ MathML (Mathematical Markup Language)