Del

 ( Vector Calculus ) 

Del, or nabla, is an operator used in mathematics (particularly in vector calculus) as a vector differential operator, usually represented by ∇ (the nabla symbol). When applied to a function defined on a one-dimensional domain, it denotes the standard derivative of the function as defined in calculus. When applied to a field (a function defined on a multi-dimensional domain), it may denote any one of three operations depending on the way it is applied: the gradient or (locally) steepest slope of a scalar field (or sometimes of a vector field, as in the Navier–Stokes equations); the divergence of a vector field; or the curl (rotation) of a vector field.

Del is a very convenient mathematical notation for those three operations (gradient, divergence, and curl) that makes many equations easier to write and remember. The del symbol (or nabla) can be formally defined as a vector operator whose components are the corresponding partial derivative operators. As a vector operator, it can act on scalar and vector fields in three different ways, giving rise to three different differential operations: first, it can act on scalar fields by a formal scalar multiplication—to give a vector field called the gradient; second, it can act on vector fields by a formal dot product—to give a scalar field called the divergence; and lastly, it can act on vector fields by a formal cross product—to give a vector field called the curl. These formal products do not necessarily commute with other operators or products. These three uses are summarized as:

• Gradient: $\backslash operatorname\{grad\}f\; =\; \backslash nabla\; f$
• Divergence: $\backslash operatorname\{div\}\backslash mathbf\; v\; =\; \backslash nabla\; \backslash cdot\; \backslash mathbf\; v$
• Curl: $\backslash operatorname\{curl\}\backslash mathbf\; v\; =\; \backslash nabla\; \backslash times\; \backslash mathbf\; v$

---

Definition In the Cartesian coordinate system $\backslash mathbb\{R\}^n$ with coordinates $(x\_1,\; \backslash dots,\; x\_n)$ and standard basis $\backslash \{\backslash mathbf\; e\_1,\; \backslash dots,\; \backslash mathbf\; e\_n\; \backslash \}$, del is a vector operator whose $x\_1,\; \backslash dots,\; x\_n$ components are the partial derivative operators $\{\backslash partial\; \backslash over\; \backslash partial\; x\_1\},\; \backslash dots,\; \{\backslash partial\; \backslash over\; \backslash partial\; x\_n\}$; that is,

$\backslash nabla\; =\; \backslash sum\_\{i=1\}^n\; \backslash mathbf\; e\_i\; \{\backslash partial\; \backslash over\; \backslash partial\; x\_i\}\; =\; \backslash left(\{\backslash partial\; \backslash over\; \backslash partial\; x\_1\},\; \backslash ldots,\; \{\backslash partial\; \backslash over\; \backslash partial\; x\_n\}\; \backslash right)$

where the expression in parentheses is a row vector. In three-dimensional Cartesian coordinate system $\backslash mathbb\{R\}^3$ with coordinates $(x,\; y,\; z)$ and standard basis or unit vectors of axes $\backslash \{\backslash mathbf\; e\_x,\; \backslash mathbf\; e\_y,\; \backslash mathbf\; e\_z\; \backslash \}$, del is written as:

$\backslash nabla\; =\; \backslash mathbf\{e\}\_x\; \{\backslash partial\; \backslash over\; \backslash partial\; x\}\; +\; \backslash mathbf\{e\}\_y\; \{\backslash partial\; \backslash over\; \backslash partial\; y\}\; +\; \backslash mathbf\{e\}\_z\; \{\backslash partial\; \backslash over\; \backslash partial\; z\}=\; \backslash left(\{\backslash partial\; \backslash over\; \backslash partial\; x\},\; \{\backslash partial\; \backslash over\; \backslash partial\; y\},\; \{\backslash partial\; \backslash over\; \backslash partial\; z\}\; \backslash right)$

As a vector operator, del naturally acts on scalar fields via scalar multiplication, and naturally acts on vector fields via dot products and cross products.

More specifically, for any scalar field $f$ and any vector field $\backslash mathbf\{F\}=(F\_x,\; F\_y,\; F\_z)$, if one defines

$\backslash left(\backslash mathbf\{e\}\_i\; \{\backslash partial\; \backslash over\; \backslash partial\; x\_i\}\backslash right)\; f\; :=\; \{\backslash partial\; \backslash over\; \backslash partial\; x\_i\}(\backslash mathbf\{e\}\_i\; f)\; =\; \{\backslash partial\; f\; \backslash over\; \backslash partial\; x\_i\}\backslash mathbf\{e\}\_i$

$\backslash left(\backslash mathbf\{e\}\_i\; \{\backslash partial\; \backslash over\; \backslash partial\; x\_i\}\backslash right)\; \backslash cdot\; \backslash mathbf\{F\}\; :=\; \{\backslash partial\; \backslash over\; \backslash partial\; x\_i\}(\backslash mathbf\{e\}\_i\backslash cdot\; \backslash mathbf\{F\})\; =\; \{\backslash partial\; F\_i\; \backslash over\; \backslash partial\; x\_i\}$

$\backslash left(\backslash mathbf\{e\}\_x\; \{\backslash partial\; \backslash over\; \backslash partial\; x\}\backslash right)\; \backslash times\; \backslash mathbf\{F\}\; :=\; \{\backslash partial\; \backslash over\; \backslash partial\; x\}(\backslash mathbf\{e\}\_x\backslash times\; \backslash mathbf\{F\})\; =\; \{\backslash partial\; \backslash over\; \backslash partial\; x\}(0,\; -F\_z,\; F\_y)$

$\backslash left(\backslash mathbf\{e\}\_y\; \{\backslash partial\; \backslash over\; \backslash partial\; y\}\backslash right)\; \backslash times\; \backslash mathbf\{F\}\; :=\; \{\backslash partial\; \backslash over\; \backslash partial\; y\}(\backslash mathbf\{e\}\_y\backslash times\; \backslash mathbf\{F\})\; =\; \{\backslash partial\; \backslash over\; \backslash partial\; y\}(F\_z,0,-F\_x)$

$\backslash left(\backslash mathbf\{e\}\_z\; \{\backslash partial\; \backslash over\; \backslash partial\; z\}\backslash right)\; \backslash times\; \backslash mathbf\{F\}\; :=\; \{\backslash partial\; \backslash over\; \backslash partial\; z\}(\backslash mathbf\{e\}\_z\backslash times\; \backslash mathbf\{F\})\; =\; \{\backslash partial\; \backslash over\; \backslash partial\; z\}(-F\_y,F\_x,0),$

then using the above definition of $\backslash nabla$, one may write

$$

\nabla f =\left(\mathbf{e}_x {\partial \over \partial x}\right)f + \left(\mathbf{e}_y {\partial \over \partial y}\right)f + \left(\mathbf{e}_z {\partial \over \partial z}\right)f = {\partial f \over \partial x}\mathbf{e}_x + {\partial f \over \partial y}\mathbf{e}_y + {\partial f \over \partial z}\mathbf{e}_z and

$$

\nabla \cdot \mathbf{F} = \left(\mathbf{e}_x {\partial \over \partial x}\cdot \mathbf{F}\right) + \left(\mathbf{e}_y {\partial \over \partial y}\cdot \mathbf{F}\right) + \left(\mathbf{e}_z {\partial \over \partial z}\cdot \mathbf{F}\right)= {\partial F_x \over \partial x} + {\partial F_y \over \partial y} + {\partial F_z \over \partial z} and

$\backslash begin\{align\}$

\nabla \times \mathbf{F} &= \left(\mathbf{e}_x {\partial \over \partial x}\times \mathbf{F}\right) + \left(\mathbf{e}_y {\partial \over \partial y}\times \mathbf{F}\right) + \left(\mathbf{e}_z {\partial \over \partial z}\times \mathbf{F}\right)\\ &= {\partial \over \partial x}(0, -F_z, F_y) + {\partial \over \partial y}(F_z,0,-F_x) + {\partial \over \partial z}(-F_y,F_x,0)\\ &= \left({\partial F_z \over \partial y}-{\partial F_y \over \partial z}\right)\mathbf{e}_x + \left({\partial F_x \over \partial z}-{\partial F_z \over \partial x}\right)\mathbf{e}_y + \left({\partial F_y \over \partial x}-{\partial F_x \over \partial y}\right)\mathbf{e}_z \end{align}

Example:

$f(x,\; y,\; z)\; =\; x\; +\; y\; +\; z$

$\backslash nabla\; f\; =\; \backslash mathbf\{e\}\_x\; \{\backslash partial\; f\; \backslash over\; \backslash partial\; x\}\; +\; \backslash mathbf\{e\}\_y\; \{\backslash partial\; f\; \backslash over\; \backslash partial\; y\}\; +\; \backslash mathbf\{e\}\_z\; \{\backslash partial\; f\; \backslash over\; \backslash partial\; z\}\; =\; \backslash left(1,\; 1,\; 1\; \backslash right)$

Del can also be expressed in other coordinate systems, see for example del in cylindrical and spherical coordinates.

---

Notational uses Del is used as a shorthand form to simplify many long mathematical expressions. It is most commonly used to simplify expressions for the gradient, divergence, curl, directional derivative, and Laplacian.

---

Gradient The vector derivative of a scalar field $f$ is called the gradient, and it can be represented as:

$\backslash operatorname\{grad\}f\; =\; \{\backslash partial\; f\; \backslash over\; \backslash partial\; x\}\; \backslash hat\backslash mathbf\; x\; +\; \{\backslash partial\; f\; \backslash over\; \backslash partial\; y\}\; \backslash hat\backslash mathbf\; y\; +\; \{\backslash partial\; f\; \backslash over\; \backslash partial\; z\}\; \backslash hat\backslash mathbf\; z=\backslash nabla\; f$

It always points in the direction of greatest increase of $f$, and it has a magnitude equal to the maximum rate of increase at the point—just like a standard derivative. In particular, if a hill is defined as a height function over a plane $h(x,y)$, the gradient at a given location will be a vector in the xy-plane (visualizable as an arrow on a map) pointing along the steepest direction. The magnitude of the gradient is the value of this steepest slope.

In particular, this notation is powerful because the gradient product rule looks very similar to the 1d-derivative case:

$\backslash nabla(f\; g)\; =\; f\; \backslash nabla\; g\; +\; g\; \backslash nabla\; f$

However, the rules for do not turn out to be simple, as illustrated by:

$\backslash nabla\; (\backslash mathbf\; u\; \backslash cdot\; \backslash mathbf\; v)\; =\; (\backslash mathbf\; u\; \backslash cdot\; \backslash nabla)\; \backslash mathbf\; v\; +\; (\backslash mathbf\; v\; \backslash cdot\; \backslash nabla)\; \backslash mathbf\; u\; +\; \backslash mathbf\; u\; \backslash times\; (\backslash nabla\; \backslash times\; \backslash mathbf\; v)\; +\; \backslash mathbf\; v\; \backslash times\; (\backslash nabla\; \backslash times\; \backslash mathbf\; u)$

---

Divergence The divergence of a vector field $\backslash mathbf\; v(x,\; y,\; z)\; =\; v\_x\; \backslash hat\backslash mathbf\; x\; +\; v\_y\; \backslash hat\backslash mathbf\; y\; +\; v\_z\; \backslash hat\backslash mathbf\; z$ is a scalar field that can be represented as:

$\backslash operatorname\{div\}\backslash mathbf\; v\; =\; \{\backslash partial\; v\_x\; \backslash over\; \backslash partial\; x\}\; +\; \{\backslash partial\; v\_y\; \backslash over\; \backslash partial\; y\}\; +\; \{\backslash partial\; v\_z\; \backslash over\; \backslash partial\; z\}\; =\; \backslash nabla\; \backslash cdot\; \backslash mathbf\; v$

The divergence is roughly a measure of a vector field's increase in the direction it points; but more accurately, it is a measure of that field's tendency to converge toward or diverge from a point.

The power of the del notation is shown by the following product rule:

$\backslash nabla\; \backslash cdot\; (f\; \backslash mathbf\; v)\; =\; (\backslash nabla\; f)\; \backslash cdot\; \backslash mathbf\; v\; +\; f\; (\backslash nabla\; \backslash cdot\; \backslash mathbf\; v)$

The formula for the vector product is slightly less intuitive, because this product is not commutative:

$\backslash nabla\; \backslash cdot\; (\backslash mathbf\; u\; \backslash times\; \backslash mathbf\; v)\; =\; (\backslash nabla\; \backslash times\; \backslash mathbf\; u)\; \backslash cdot\; \backslash mathbf\; v\; -\; \backslash mathbf\; u\; \backslash cdot\; (\backslash nabla\; \backslash times\; \backslash mathbf\; v)$

---

Curl The curl of a vector field $\backslash mathbf\; v(x,\; y,\; z)\; =\; v\_x\backslash hat\backslash mathbf\; x\; +\; v\_y\backslash hat\backslash mathbf\; y\; +\; v\_z\backslash hat\backslash mathbf\; z$ is a vector field function that can be represented as:

$\backslash operatorname\{curl\}\backslash mathbf\; v\; =\; \backslash left(\{\backslash partial\; v\_z\; \backslash over\; \backslash partial\; y\}\; -\; \{\backslash partial\; v\_y\; \backslash over\; \backslash partial\; z\}\; \backslash right)\; \backslash hat\backslash mathbf\; x\; +\; \backslash left(\{\backslash partial\; v\_x\; \backslash over\; \backslash partial\; z\}\; -\; \{\backslash partial\; v\_z\; \backslash over\; \backslash partial\; x\}\; \backslash right)\; \backslash hat\backslash mathbf\; y\; +\; \backslash left(\{\backslash partial\; v\_y\; \backslash over\; \backslash partial\; x\}\; -\; \{\backslash partial\; v\_x\; \backslash over\; \backslash partial\; y\}\; \backslash right)\; \backslash hat\backslash mathbf\; z\; =\; \backslash nabla\; \backslash times\; \backslash mathbf\; v$

The curl at a point is proportional to the on-axis torque that a tiny pinwheel would be subjected to if it were centered at that point.

The vector product operation can be visualized as a pseudo-determinant:

Again the power of the notation is shown by the product rule:

$\backslash nabla\; \backslash times\; (f\; \backslash mathbf\; v)\; =\; (\backslash nabla\; f)\; \backslash times\; \backslash mathbf\; v\; +\; f\; (\backslash nabla\; \backslash times\; \backslash mathbf\; v)$

The rule for the vector product does not turn out to be simple:

$\backslash nabla\; \backslash times\; (\backslash mathbf\; u\; \backslash times\; \backslash mathbf\; v)\; =\; \backslash mathbf\; u\; \backslash ,\; (\backslash nabla\; \backslash cdot\; \backslash mathbf\; v)\; -\; \backslash mathbf\; v\; \backslash ,\; (\backslash nabla\; \backslash cdot\; \backslash mathbf\; u)\; +\; (\backslash mathbf\; v\; \backslash cdot\; \backslash nabla)\; \backslash ,\; \backslash mathbf\; u\; -\; (\backslash mathbf\; u\; \backslash cdot\; \backslash nabla)\; \backslash ,\; \backslash mathbf\; v$

---

Directional derivative The directional derivative of a scalar field $f(x,y,z)$ in the direction $\backslash mathbf\; a(x,y,z)\; =\; a\_x\; \backslash hat\backslash mathbf\; x\; +\; a\_y\; \backslash hat\backslash mathbf\; y\; +\; a\_z\; \backslash hat\backslash mathbf\; z$ is defined as:

$(\backslash mathbf\; a\backslash cdot\backslash nabla)f=\backslash lim\_\{h\; \backslash to\; 0\}\{\backslash frac\{f(x+a\_xh,y+a\_yh,z+a\_zh)\; -\; f(x,y,z)\}\{h\}\}.$

Which is equal to the following when the gradient exists

$\backslash mathbf\; a\backslash cdot\backslash operatorname\{grad\}f\; =\; a\_x\; \{\backslash partial\; f\; \backslash over\; \backslash partial\; x\}\; +\; a\_y\; \{\backslash partial\; f\; \backslash over\; \backslash partial\; y\}\; +\; a\_z\; \{\backslash partial\; f\; \backslash over\; \backslash partial\; z\}\; =\; \backslash mathbf\; a\; \backslash cdot\; (\backslash nabla\; f)$

This gives the rate of change of a field $f$ in the direction of $\backslash mathbf\; a$, scaled by the magnitude of $\backslash mathbf\; a$. In operator notation, the element in parentheses can be considered a single coherent unit; fluid dynamics uses this convention extensively, terming it the convective derivative—the "moving" derivative of the fluid.

Note that $(\backslash mathbf\; a\; \backslash cdot\; \backslash nabla)$ is an operator that maps scalars to scalars. It can be extended to act on a vector field by applying the operator component-wise to each component of the vector.

---

Laplacian The Laplace operator is a scalar operator that can be applied to either vector or scalar fields; for cartesian coordinate systems it is defined as:

$\backslash Delta\; =\; \{\backslash partial^2\; \backslash over\; \backslash partial\; x^2\}\; +\; \{\backslash partial^2\; \backslash over\; \backslash partial\; y^2\}\; +\; \{\backslash partial^2\; \backslash over\; \backslash partial\; z^2\}\; =\; \backslash nabla\; \backslash cdot\; \backslash nabla\; =\; \backslash nabla^2$

and the definition for more general coordinate systems is given in vector Laplacian.

The Laplacian is ubiquitous throughout modern mathematical physics, appearing for example in Laplace's equation, Poisson's equation, the heat equation, the wave equation, and the Schrödinger equation.

---

Hessian matrix While $\backslash nabla^2$ usually represents the Laplacian, sometimes $\backslash nabla^2$ also represents the Hessian matrix. The former refers to the inner product of $\backslash nabla$, while the latter refers to the dyadic product of $\backslash nabla$:

$\backslash nabla^2\; =\; \backslash nabla\; \backslash cdot\; \backslash nabla^T$.

So whether $\backslash nabla^2$ refers to a Laplacian or a Hessian matrix depends on the context.

---

Tensor derivative Del can also be applied to a vector field with the result being a tensor. The tensor derivative of a vector field $\backslash mathbf\{v\}$ (in three dimensions) is a 9-term second-rank tensor – that is, a 3×3 matrix – but can be denoted simply as $\backslash nabla\; \backslash otimes\; \backslash mathbf\{v\}$, where $\backslash otimes$ represents the dyadic product. This quantity is equivalent to the transpose of the Jacobian matrix of the vector field with respect to space. The divergence of the vector field can then be expressed as the trace of this matrix.

For a small displacement $\backslash delta\; \backslash mathbf\{r\}$, the change in the vector field is given by:

$\backslash delta\; \backslash mathbf\{v\}\; =\; (\backslash nabla\; \backslash otimes\; \backslash mathbf\{v\})^T\; \backslash sdot\; \backslash delta\; \backslash mathbf\{r\}$

---

Product rules For vector calculus:

$\backslash begin\{align\}$

                          \nabla (fg) &= f\nabla g + g\nabla f \\
          \nabla(\mathbf u \cdot \mathbf v) &= \mathbf u \times (\nabla \times \mathbf v) + \mathbf v \times (\nabla \times \mathbf u) + (\mathbf u \cdot \nabla) \mathbf v + (\mathbf v \cdot \nabla)\mathbf u \\
              \nabla \cdot (f \mathbf v) &= f (\nabla \cdot \mathbf v) + \mathbf v \cdot (\nabla f) \\
  \nabla \cdot (\mathbf u \times \mathbf v) &= \mathbf v \cdot (\nabla \times \mathbf u) - \mathbf u \cdot (\nabla \times \mathbf v) \\
             \nabla \times (f \mathbf v) &= (\nabla f) \times \mathbf v + f (\nabla \times \mathbf v) \\
 \nabla \times (\mathbf u \times \mathbf v) &= \mathbf u \, (\nabla \cdot \mathbf v) - \mathbf v \, (\nabla \cdot \mathbf u) + (\mathbf v \cdot \nabla) \, \mathbf u - (\mathbf u \cdot \nabla) \, \mathbf v
     

\end{align}

For matrix calculus (for which $\backslash mathbf\; u\; \backslash cdot\; \backslash mathbf\; v$ can be written $\backslash mathbf\; u^\backslash text\{T\}\; \backslash mathbf\; v$):

$\backslash begin\{align\}$

 \left(\mathbf{A}\nabla\right)^\text{T} \mathbf u &= \nabla^\text{T} \left(\mathbf{A}^\text{T}\mathbf u\right) - \left(\nabla^\text{T} \mathbf{A}^\text{T}\right) \mathbf u
     

\end{align}

Another relation of interest (see e.g. Euler equations) is the following, where $\backslash mathbf\; u\; \backslash otimes\; \backslash mathbf\; v$ is the outer product tensor:

$\backslash begin\{align\}$

   \nabla \cdot (\mathbf u \otimes \mathbf v) = (\nabla \cdot \mathbf u) \mathbf v + (\mathbf u \cdot \nabla) \mathbf v
     

\end{align}

---

Second derivatives [[File:DCG chart.svg|thumb|DCG chart:

A simple chart depicting all rules pertaining to second derivatives. D, C, G, L and CC stand for divergence, curl, gradient, Laplacian and curl of curl, respectively.

Arrows indicate existence of second derivatives. Blue circle in the middle represents curl of curl, whereas the other two red circles (dashed) mean that DD and GG do not exist. ]] When del operates on a scalar or vector, either a scalar or vector is returned. Because of the diversity of vector products (scalar, dot, cross) one application of del already gives rise to three major derivatives: the gradient (scalar product), divergence (dot product), and curl (cross product). Applying these three sorts of derivatives again to each other gives five possible second derivatives, for a scalar field f or a vector field v; the use of the scalar Laplacian and vector Laplacian gives two more:

$\backslash begin\{align\}$

       \operatorname{div}(\operatorname{grad}f) &= \nabla \cdot (\nabla f) = \nabla^2 f \\
      \operatorname{curl}(\operatorname{grad}f) &= \nabla \times (\nabla f) \\
  \operatorname{grad}(\operatorname{div}\mathbf v) &= \nabla (\nabla \cdot \mathbf v) \\
  \operatorname{div}(\operatorname{curl}\mathbf v) &= \nabla \cdot (\nabla \times \mathbf v) \\
 \operatorname{curl}(\operatorname{curl}\mathbf v) &= \nabla \times (\nabla \times \mathbf v) \\
                                       \Delta f &= \nabla^2 f \\
                                  \Delta \mathbf v &= \nabla^2 \mathbf v
     

\end{align}

These are of interest principally because they are not always unique or independent of each other. As long as the functions are well-behaved ($C^\backslash infty$ in most cases), two of them are always zero:

$\backslash begin\{align\}$

     \operatorname{curl}(\operatorname{grad}f) &= \nabla \times (\nabla f) = 0 \\
 \operatorname{div}(\operatorname{curl}\mathbf v) &= \nabla \cdot (\nabla \times \mathbf v) = 0
     

\end{align}

Two of them are always equal:

$\backslash operatorname\{div\}(\backslash operatorname\{grad\}f)\; =\; \backslash nabla\; \backslash cdot\; (\backslash nabla\; f)\; =\; \backslash nabla^2\; f\; =\; \backslash Delta\; f$

The 3 remaining vector derivatives are related by the equation:

$\backslash nabla\; \backslash times\; \backslash left(\backslash nabla\; \backslash times\; \backslash mathbf\; v\backslash right)\; =\; \backslash nabla\; (\backslash nabla\; \backslash cdot\; \backslash mathbf\; v)\; -\; \backslash nabla^2\; \backslash mathbf\{v\}$

And one of them can even be expressed with the tensor product, if the functions are well-behaved:

$\backslash nabla\; (\backslash nabla\; \backslash cdot\; \backslash mathbf\; v)\; =\; \backslash nabla\; \backslash cdot\; (\backslash mathbf\; v\; \backslash otimes\; \backslash nabla\; )$

---

Precautions Most of the above vector properties (except for those that rely explicitly on del's differential properties—for example, the product rule) rely only on symbol rearrangement, and must necessarily hold if the del symbol is replaced by any other vector. This is part of the value to be gained in notationally representing this operator as a vector.

Though one can often replace del with a vector and obtain a vector identity, making those identities mnemonic, the reverse is not necessarily reliable, because del does not commute in general.

A counterexample that demonstrates the divergence ($\backslash nabla\; \backslash cdot\; \backslash mathbf\; v$) and the advection operator ($\backslash mathbf\; v\; \backslash cdot\; \backslash nabla$) are not commutative:

$\backslash begin\{align\}$

             (\mathbf u \cdot \mathbf v) f &\equiv (\mathbf v \cdot \mathbf u) f \\
             (\nabla \cdot \mathbf v) f &= \left (\frac{\partial v_x}{\partial x} + \frac{\partial v_y}{\partial y} + \frac{\partial v_z}{\partial z} \right)f
                                       = \frac{\partial v_x}{\partial x}f + \frac{\partial v_y}{\partial y}f + \frac{\partial v_z}{\partial z}f \\
             (\mathbf v \cdot \nabla) f &= \left (v_x \frac{\partial}{\partial x} + v_y \frac{\partial}{\partial y} + v_z \frac{\partial}{\partial z} \right)f
                                       = v_x \frac{\partial f}{\partial x} + v_y \frac{\partial f}{\partial y} + v_z \frac{\partial f}{\partial z} \\
 \Rightarrow (\nabla \cdot \mathbf v) f &\ne (\mathbf v \cdot \nabla) f \\
     

\end{align}

A counterexample that relies on del's differential properties:

$\backslash begin\{align\}$

 (\nabla x) \times (\nabla y) &= \left (\mathbf e_x \frac{\partial x}{\partial x}+\mathbf e_y \frac{\partial x}{\partial y}+\mathbf e_z \frac{\partial x}{\partial z} \right) \times \left (\mathbf e_x \frac{\partial y}{\partial x}+\mathbf e_y \frac{\partial y}{\partial y}+\mathbf e_z \frac{\partial y}{\partial z} \right) \\
                              &= (\mathbf e_x \cdot 1 +\mathbf e_y \cdot 0+\mathbf e_z \cdot 0) \times (\mathbf e_x \cdot 0+\mathbf e_y \cdot 1+\mathbf e_z \cdot 0) \\
                              &= \mathbf e_x  \times \mathbf e_y \\
                              &= \mathbf e_z \\
 (\mathbf u x)\times (\mathbf u y) &= x y (\mathbf u \times \mathbf u) \\
                              &= x y \mathbf 0 \\
                              &= \mathbf 0
     

\end{align}

Central to these distinctions is the fact that del is not simply a vector; it is a vector operator. Whereas a vector is an object with both a magnitude and direction, del has neither a magnitude nor a direction until it operates on a function.

For that reason, identities involving del must be derived with care, using both vector identities and differentiation identities such as the product rule.

---

See also

• Nabla symbol
• Dirac operator
• Del in cylindrical and spherical coordinates
• Notation for differentiation
• Vector calculus identities
• Maxwell's equations
• Navier–Stokes equations
• Table of mathematical symbols
• Quabla operator

• Willard Gibbs & Edwin Bidwell Wilson (1901) Vector Analysis, Yale University Press, 1960: Dover Publications.
• H. M. Schey (1997). 9780393969979, Norton. ISBN 9780393969979

---

External links

Post Comment