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The weirdest thing ever.

I think my parents abandon me ,a different set.

I remember another man barely. He left me with my parents, they took another kid home and didn't want to take me back. I don't know. This is a really early memory then they moved. There was some child switching going on. I remember my sister hannah born but I never remember this other sister that disappeared till school. I can't remember the guys face or another mother. I got a weird message that president trump was my father at 16 but I have no recalection of this till garage near the school. He abandoned me because didn't like I didn't like tools. I told him to start a construction company. I ended up with better parents. It didn't look like trump. Maybe his dad or a lookalike of his dad. I had other memories colonial farms and  don't remember running into the fire hydrant at 3 but does look like I had surgery on my mouth. I tried waiting for that dad but david put me in the car. This was the day after the other one left. I remember maybe a cover up by Karen.I didn't know what she was talking about. I don't remember my brother but it seemed there was two brothers one day. I was confused. My mom wondered where her kid was. I think he teleported away at school. People were teleporting into the room. One was an adult the secretary of state today. 5 people left teleported out, 5 teleported in and 5-7 entered the room from the outside of the elementary. It was very confusing time. I then had a vision then dream about trump. I also remember being abducted by a ufo. at my house one night they said they took me at 5. My mom met Trump in jail in San Angelo in the 80s. She wouldnt say much, David said he gave her a black eye. He was running for office. My mom also mentioned time traveling relatives. I had a weird time traveling experience or two, there was some of that in dreams and a group at school. Trump is one of Chuck Norris's relatives but my dad used his sperm, I'm not sure who my mother is trump said my mother. Dad said it was a research project they met him orginally. So the story doesn't make sense. I though the time traveling on their part at the school was covering up crime. Some of it did legit things that were supernatural that I couldn't explain. Like one of my classmates traveled through time but he was a criminal. I and others in vision traveled through time as time cops from the year 25000. Some type alien time traveling ring that immigrated people through time and alien immigration from other worlds.

Trump got a contract through the school because of me. My original family may have been time travelers.

My life is confusing as hell. My mom today said I retiti. She was half alien. I thought I became Zeta-Retiti part angel man. I battled with time traveler from another world who want abduct the world and send it to the future, the guy couldn't be trusted. I battled with him and his intendants the top secret core that got corrupted by the time traveler. 

Miracle Balm known to sometimes save a leg.

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Del Math originally from wikipedia

 

https://en.wikipedia.org/wiki/Del

Del

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From Wikipedia, the free encyclopedia

Not to be confused with Dell. For other uses, see Del (disambiguation).

This article includes a list of references, related reading, or external links, but its sources remain unclear because it lacks inline citations. Please help improve this article by introducing more precise citations. (March 2010) (Learn how and when to remove this message)

Del operator,
represented by
the nabla symbol

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: ${\displaystyle \mathrm{grad}f=\mathrm{\nabla }f}$
• Divergence: ${\displaystyle \mathrm{div}\mathbf{v}=\mathrm{\nabla }\cdot \mathbf{v}}$
• Curl: ${\displaystyle \mathrm{curl}\mathbf{v}=\mathrm{\nabla }\times \mathbf{v}}$

Definition

In the Cartesian coordinate system ${\displaystyle {\mathbb{R}}^n}$ with coordinates ${\displaystyle (x_1,\dots ,x_n)}$ and standard basis ${\displaystyle \{{\mathbf{e}}_1,\dots ,{\mathbf{e}}_n\}}$, del is a vector operator whose ${\displaystyle x_1,\dots ,x_n}$ components are the partial derivative operators ${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }{x}_1},\dots ,\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_n}}$; that is,

${\displaystyle \mathrm{\nabla }=\sum _{i=1}^n{\mathbf{e}}_i\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_i}=\left(\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_1},\dots ,\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_n}\right)}$

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

${\displaystyle \mathrm{\nabla }={\mathbf{e}}_x\frac{\mathrm{\partial }}{\mathrm{\partial }x}+{\mathbf{e}}_y\frac{\mathrm{\partial }}{\mathrm{\partial }y}+{\mathbf{e}}_z\frac{\mathrm{\partial }}{\mathrm{\partial }z}=\left(\frac{\mathrm{\partial }}{\mathrm{\partial }x},\frac{\mathrm{\partial }}{\mathrm{\partial }y},\frac{\mathrm{\partial }}{\mathrm{\partial }z}\right)}$

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, in three dimensions, for any scalar field ${\displaystyle f}$ and any vector field ${\displaystyle \mathbf{F}=(F_x,F_y,F_z)}$, if one defines

${\displaystyle \left({\mathbf{e}}_i\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_i}\right)f:=\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_i}({\mathbf{e}}_{i}f)=\frac{\mathrm{\partial }f}{\mathrm{\partial }{x}_i}{\mathbf{e}}_i}$

${\displaystyle \left({\mathbf{e}}_i\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_i}\right)\cdot \mathbf{F}:=\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_i}({\mathbf{e}}_i\cdot \mathbf{F})=\frac{\mathrm{\partial }{F}_i}{\mathrm{\partial }{x}_i}}$

${\displaystyle \left({\mathbf{e}}_x\frac{\mathrm{\partial }}{\mathrm{\partial }x}\right)\times \mathbf{F}:=\frac{\mathrm{\partial }}{\mathrm{\partial }x}({\mathbf{e}}_x\times \mathbf{F})=\frac{\mathrm{\partial }}{\mathrm{\partial }x}(0,-F_z,F_y)}$

${\displaystyle \left({\mathbf{e}}_y\frac{\mathrm{\partial }}{\mathrm{\partial }y}\right)\times \mathbf{F}:=\frac{\mathrm{\partial }}{\mathrm{\partial }y}({\mathbf{e}}_y\times \mathbf{F})=\frac{\mathrm{\partial }}{\mathrm{\partial }y}(F_z,0,-F_x)}$

${\displaystyle \left({\mathbf{e}}_z\frac{\mathrm{\partial }}{\mathrm{\partial }z}\right)\times \mathbf{F}:=\frac{\mathrm{\partial }}{\mathrm{\partial }z}({\mathbf{e}}_z\times \mathbf{F})=\frac{\mathrm{\partial }}{\mathrm{\partial }z}(-F_y,F_x,0),}$

then using the above definition of ${\displaystyle \mathrm{\nabla }}$, one may write

${\displaystyle \mathrm{\nabla }f=\left({\mathbf{e}}_x\frac{\mathrm{\partial }}{\mathrm{\partial }x}\right)f+\left({\mathbf{e}}_y\frac{\mathrm{\partial }}{\mathrm{\partial }y}\right)f+\left({\mathbf{e}}_z\frac{\mathrm{\partial }}{\mathrm{\partial }z}\right)f=\frac{\mathrm{\partial }f}{\mathrm{\partial }x}{\mathbf{e}}_x+\frac{\mathrm{\partial }f}{\mathrm{\partial }y}{\mathbf{e}}_y+\frac{\mathrm{\partial }f}{\mathrm{\partial }z}{\mathbf{e}}_z}$

and

${\displaystyle \mathrm{\nabla }\cdot \mathbf{F}=\left({\mathbf{e}}_x\frac{\mathrm{\partial }}{\mathrm{\partial }x}\cdot \mathbf{F}\right)+\left({\mathbf{e}}_y\frac{\mathrm{\partial }}{\mathrm{\partial }y}\cdot \mathbf{F}\right)+\left({\mathbf{e}}_z\frac{\mathrm{\partial }}{\mathrm{\partial }z}\cdot \mathbf{F}\right)=\frac{\mathrm{\partial }{F}_x}{\mathrm{\partial }x}+\frac{\mathrm{\partial }{F}_y}{\mathrm{\partial }y}+\frac{\mathrm{\partial }{F}_z}{\mathrm{\partial }z}}$

and

Example:

${\displaystyle f(x,y,z)=x+y+z}$

${\displaystyle \mathrm{\nabla }f={\mathbf{e}}_x\frac{\mathrm{\partial }f}{\mathrm{\partial }x}+{\mathbf{e}}_y\frac{\mathrm{\partial }f}{\mathrm{\partial }y}+{\mathbf{e}}_z\frac{\mathrm{\partial }f}{\mathrm{\partial }z}=\left(1,1,1\right)}$

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 ${\displaystyle f}$ is called the gradient, and it can be represented as:

${\displaystyle \mathrm{\nabla }f=\frac{\mathrm{\partial }f}{\mathrm{\partial }x}\hat{\mathbf{x}}+\frac{\mathrm{\partial }f}{\mathrm{\partial }y}\hat{\mathbf{y}}+\frac{\mathrm{\partial }f}{\mathrm{\partial }z}\hat{\mathbf{z}}=\mathrm{grad}f}$

It always points in the direction of greatest increase of ${\displaystyle f}$, 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 ${\displaystyle h(x,y)}$, 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:

${\displaystyle \mathrm{\nabla }(fg)=f\mathrm{\nabla }g+g\mathrm{\nabla }f}$

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

${\displaystyle \mathrm{\nabla }(\mathbf{u}\cdot \mathbf{v})=(\mathbf{u}\cdot \mathrm{\nabla })\mathbf{v}+(\mathbf{v}\cdot \mathrm{\nabla })\mathbf{u}+\mathbf{u}\times (\mathrm{\nabla }\times \mathbf{v})+\mathbf{v}\times (\mathrm{\nabla }\times \mathbf{u})}$

Divergence

The divergence of a vector field ${\displaystyle \mathbf{v}(x,y,z)=v_x\hat{\mathbf{x}}+v_y\hat{\mathbf{y}}+v_z\hat{\mathbf{z}}}$ is a scalar field that can be represented as:

${\displaystyle \mathrm{div}\mathbf{v}=\frac{\mathrm{\partial }{v}_x}{\mathrm{\partial }x}+\frac{\mathrm{\partial }{v}_y}{\mathrm{\partial }y}+\frac{\mathrm{\partial }{v}_z}{\mathrm{\partial }z}=\mathrm{\nabla }\cdot \mathbf{v}}$

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:

${\displaystyle \mathrm{\nabla }\cdot (f\mathbf{v})=(\mathrm{\nabla }f)\cdot \mathbf{v}+f(\mathrm{\nabla }\cdot \mathbf{v})}$

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

${\displaystyle \mathrm{\nabla }\cdot (\mathbf{u}\times \mathbf{v})=(\mathrm{\nabla }\times \mathbf{u})\cdot \mathbf{v}-\mathbf{u}\cdot (\mathrm{\nabla }\times \mathbf{v})}$

Curl

The curl of a vector field ${\displaystyle \mathbf{v}(x,y,z)=v_x\hat{\mathbf{x}}+v_y\hat{\mathbf{y}}+v_z\hat{\mathbf{z}}}$ is a vector function that can be represented as:

${\displaystyle \mathrm{curl}\mathbf{v}=\left(\frac{\mathrm{\partial }{v}_z}{\mathrm{\partial }y}-\frac{\mathrm{\partial }{v}_y}{\mathrm{\partial }z}\right)\hat{\mathbf{x}}+\left(\frac{\mathrm{\partial }{v}_x}{\mathrm{\partial }z}-\frac{\mathrm{\partial }{v}_z}{\mathrm{\partial }x}\right)\hat{\mathbf{y}}+\left(\frac{\mathrm{\partial }{v}_y}{\mathrm{\partial }x}-\frac{\mathrm{\partial }{v}_x}{\mathrm{\partial }y}\right)\hat{\mathbf{z}}=\mathrm{\nabla }\times \mathbf{v}}$

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:

${\displaystyle \mathrm{\nabla }\times (f\mathbf{v})=(\mathrm{\nabla }f)\times \mathbf{v}+f(\mathrm{\nabla }\times \mathbf{v})}$

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

${\displaystyle \mathrm{\nabla }\times (\mathbf{u}\times \mathbf{v})=\mathbf{u}(\mathrm{\nabla }\cdot \mathbf{v})-\mathbf{v}(\mathrm{\nabla }\cdot \mathbf{u})+(\mathbf{v}\cdot \mathrm{\nabla })\mathbf{u}-(\mathbf{u}\cdot \mathrm{\nabla })\mathbf{v}}$

Directional derivative

The directional derivative of a scalar field ${\displaystyle f(x,y,z)}$ in the direction ${\displaystyle \mathbf{a}(x,y,z)=a_x\hat{\mathbf{x}}+a_y\hat{\mathbf{y}}+a_z\hat{\mathbf{z}}}$ is defined as:

${\displaystyle (\mathbf{a}\cdot \mathrm{\nabla })f=\underset{h\to 0}{lim}\frac{f(x+a_{x}h,y+a_{y}h,z+a_{z}h)-f(x,y,z)}{h}.}$

Which is equal to the following when the gradient exists

${\displaystyle \mathbf{a}\cdot \mathrm{grad}f=a_x\frac{\mathrm{\partial }f}{\mathrm{\partial }x}+a_y\frac{\mathrm{\partial }f}{\mathrm{\partial }y}+a_z\frac{\mathrm{\partial }f}{\mathrm{\partial }z}=\mathbf{a}\cdot (\mathrm{\nabla }f)}$

This gives the rate of change of a field ${\displaystyle f}$ in the direction of ${\displaystyle \mathbf{a}}$, scaled by the magnitude of ${\displaystyle \mathbf{a}}$. 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 ${\displaystyle (\mathbf{a}\cdot \mathrm{\nabla })}$ 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:

${\displaystyle \mathrm{\Delta }=\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{y}^2}+\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{z}^2}=\mathrm{\nabla }\cdot \mathrm{\nabla }={\mathrm{\nabla }}^2}$

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

${\displaystyle {\mathrm{\nabla }}^2=\mathrm{\nabla }\cdot {\mathrm{\nabla }}^T}$.

So whether ${\displaystyle {\mathrm{\nabla }}^2}$ 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 ${\displaystyle \mathbf{v}}$ (in three dimensions) is a 9-term second-rank tensor – that is, a 3×3 matrix – but can be denoted simply as ${\displaystyle \mathrm{\nabla }\otimes \mathbf{v}}$, where ${\displaystyle \otimes }$ 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 ${\displaystyle \delta \mathbf{r}}$, the change in the vector field is given by:

${\displaystyle \delta \mathbf{v}=(\mathrm{\nabla }\otimes \mathbf{v}{)}^T\cdot \delta \mathbf{r}}$

Product rules

For vector calculus:

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

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

${\displaystyle \begin{array}{r}\mathrm{\nabla }\cdot (\mathbf{u}\otimes \mathbf{v})=(\mathrm{\nabla }\cdot \mathbf{u})\mathbf{v}+(\mathbf{u}\cdot \mathrm{\nabla })\mathbf{v}\end{array}}$

Second derivatives

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:

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

Two of them are always equal:

${\displaystyle \mathrm{div}(\mathrm{grad}f)=\mathrm{\nabla }\cdot (\mathrm{\nabla }f)={\mathrm{\nabla }}^{2}f=\mathrm{\Delta }f}$

The 3 remaining vector derivatives are related by the equation:

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

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

${\displaystyle \mathrm{\nabla }(\mathrm{\nabla }\cdot \mathbf{v})=\mathrm{\nabla }\cdot (\mathbf{v}\otimes \mathrm{\nabla })}$

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 (${\displaystyle \mathrm{\nabla }\cdot \mathbf{v}}$) and the advection operator (${\displaystyle \mathbf{v}\cdot \mathrm{\nabla }}$) are not commutative:

A counterexample that relies on del's differential properties:

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

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

References

• Willard Gibbs & Edwin Bidwell Wilson (1901) Vector Analysis, Yale University Press, 1960: Dover Publications.
• Schey, H. M. (1997). Div, Grad, Curl, and All That: An Informal Text on Vector Calculus. New York: Norton. ISBN 0-393-96997-5.
• Miller, Jeff. "Earliest Uses of Symbols of Calculus".
• Arnold Neumaier (January 26, 1998). Cleve Moler (ed.). "History of Nabla". NA Digest, Volume 98, Issue 03. netlib.org.

External links

• Tai, Chen-To (1994). A survey of the improper use of ∇ in vector analysis (Report). Radiation Laboratory, University of Michigan. hdl:2027.42/7869.

Categories: 

• Vector calculus
• Mathematical notation
• Differential operators

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From Wikipedia, the free encyclopedia

Not to be confused with Dell. For other uses, see Del (disambiguation).

This article includes a list of references, related reading, or external links, but its sources remain unclear because it lacks inline citations. Please help improve this article by introducing more precise citations. (March 2010) (Learn how and when to remove this message)

Del operator,
represented by
the nabla symbol

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: ${\displaystyle \mathrm{grad}f=\mathrm{\nabla }f}$
• Divergence: ${\displaystyle \mathrm{div}\mathbf{v}=\mathrm{\nabla }\cdot \mathbf{v}}$
• Curl: ${\displaystyle \mathrm{curl}\mathbf{v}=\mathrm{\nabla }\times \mathbf{v}}$

Definition

In the Cartesian coordinate system ${\displaystyle {\mathbb{R}}^n}$ with coordinates ${\displaystyle (x_1,\dots ,x_n)}$ and standard basis ${\displaystyle \{{\mathbf{e}}_1,\dots ,{\mathbf{e}}_n\}}$, del is a vector operator whose ${\displaystyle x_1,\dots ,x_n}$ components are the partial derivative operators ${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }{x}_1},\dots ,\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_n}}$; that is,

${\displaystyle \mathrm{\nabla }=\sum _{i=1}^n{\mathbf{e}}_i\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_i}=\left(\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_1},\dots ,\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_n}\right)}$

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

${\displaystyle \mathrm{\nabla }={\mathbf{e}}_x\frac{\mathrm{\partial }}{\mathrm{\partial }x}+{\mathbf{e}}_y\frac{\mathrm{\partial }}{\mathrm{\partial }y}+{\mathbf{e}}_z\frac{\mathrm{\partial }}{\mathrm{\partial }z}=\left(\frac{\mathrm{\partial }}{\mathrm{\partial }x},\frac{\mathrm{\partial }}{\mathrm{\partial }y},\frac{\mathrm{\partial }}{\mathrm{\partial }z}\right)}$

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, in three dimensions, for any scalar field ${\displaystyle f}$ and any vector field ${\displaystyle \mathbf{F}=(F_x,F_y,F_z)}$, if one defines

${\displaystyle \left({\mathbf{e}}_i\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_i}\right)f:=\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_i}({\mathbf{e}}_{i}f)=\frac{\mathrm{\partial }f}{\mathrm{\partial }{x}_i}{\mathbf{e}}_i}$

${\displaystyle \left({\mathbf{e}}_i\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_i}\right)\cdot \mathbf{F}:=\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_i}({\mathbf{e}}_i\cdot \mathbf{F})=\frac{\mathrm{\partial }{F}_i}{\mathrm{\partial }{x}_i}}$

${\displaystyle \left({\mathbf{e}}_x\frac{\mathrm{\partial }}{\mathrm{\partial }x}\right)\times \mathbf{F}:=\frac{\mathrm{\partial }}{\mathrm{\partial }x}({\mathbf{e}}_x\times \mathbf{F})=\frac{\mathrm{\partial }}{\mathrm{\partial }x}(0,-F_z,F_y)}$

${\displaystyle \left({\mathbf{e}}_y\frac{\mathrm{\partial }}{\mathrm{\partial }y}\right)\times \mathbf{F}:=\frac{\mathrm{\partial }}{\mathrm{\partial }y}({\mathbf{e}}_y\times \mathbf{F})=\frac{\mathrm{\partial }}{\mathrm{\partial }y}(F_z,0,-F_x)}$

${\displaystyle \left({\mathbf{e}}_z\frac{\mathrm{\partial }}{\mathrm{\partial }z}\right)\times \mathbf{F}:=\frac{\mathrm{\partial }}{\mathrm{\partial }z}({\mathbf{e}}_z\times \mathbf{F})=\frac{\mathrm{\partial }}{\mathrm{\partial }z}(-F_y,F_x,0),}$

then using the above definition of ${\displaystyle \mathrm{\nabla }}$, one may write

${\displaystyle \mathrm{\nabla }f=\left({\mathbf{e}}_x\frac{\mathrm{\partial }}{\mathrm{\partial }x}\right)f+\left({\mathbf{e}}_y\frac{\mathrm{\partial }}{\mathrm{\partial }y}\right)f+\left({\mathbf{e}}_z\frac{\mathrm{\partial }}{\mathrm{\partial }z}\right)f=\frac{\mathrm{\partial }f}{\mathrm{\partial }x}{\mathbf{e}}_x+\frac{\mathrm{\partial }f}{\mathrm{\partial }y}{\mathbf{e}}_y+\frac{\mathrm{\partial }f}{\mathrm{\partial }z}{\mathbf{e}}_z}$

and

${\displaystyle \mathrm{\nabla }\cdot \mathbf{F}=\left({\mathbf{e}}_x\frac{\mathrm{\partial }}{\mathrm{\partial }x}\cdot \mathbf{F}\right)+\left({\mathbf{e}}_y\frac{\mathrm{\partial }}{\mathrm{\partial }y}\cdot \mathbf{F}\right)+\left({\mathbf{e}}_z\frac{\mathrm{\partial }}{\mathrm{\partial }z}\cdot \mathbf{F}\right)=\frac{\mathrm{\partial }{F}_x}{\mathrm{\partial }x}+\frac{\mathrm{\partial }{F}_y}{\mathrm{\partial }y}+\frac{\mathrm{\partial }{F}_z}{\mathrm{\partial }z}}$

and

Example:

${\displaystyle f(x,y,z)=x+y+z}$

${\displaystyle \mathrm{\nabla }f={\mathbf{e}}_x\frac{\mathrm{\partial }f}{\mathrm{\partial }x}+{\mathbf{e}}_y\frac{\mathrm{\partial }f}{\mathrm{\partial }y}+{\mathbf{e}}_z\frac{\mathrm{\partial }f}{\mathrm{\partial }z}=\left(1,1,1\right)}$

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 ${\displaystyle f}$ is called the gradient, and it can be represented as:

${\displaystyle \mathrm{\nabla }f=\frac{\mathrm{\partial }f}{\mathrm{\partial }x}\hat{\mathbf{x}}+\frac{\mathrm{\partial }f}{\mathrm{\partial }y}\hat{\mathbf{y}}+\frac{\mathrm{\partial }f}{\mathrm{\partial }z}\hat{\mathbf{z}}=\mathrm{grad}f}$

It always points in the direction of greatest increase of ${\displaystyle f}$, 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 ${\displaystyle h(x,y)}$, 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:

${\displaystyle \mathrm{\nabla }(fg)=f\mathrm{\nabla }g+g\mathrm{\nabla }f}$

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

${\displaystyle \mathrm{\nabla }(\mathbf{u}\cdot \mathbf{v})=(\mathbf{u}\cdot \mathrm{\nabla })\mathbf{v}+(\mathbf{v}\cdot \mathrm{\nabla })\mathbf{u}+\mathbf{u}\times (\mathrm{\nabla }\times \mathbf{v})+\mathbf{v}\times (\mathrm{\nabla }\times \mathbf{u})}$

Divergence

The divergence of a vector field ${\displaystyle \mathbf{v}(x,y,z)=v_x\hat{\mathbf{x}}+v_y\hat{\mathbf{y}}+v_z\hat{\mathbf{z}}}$ is a scalar field that can be represented as:

${\displaystyle \mathrm{div}\mathbf{v}=\frac{\mathrm{\partial }{v}_x}{\mathrm{\partial }x}+\frac{\mathrm{\partial }{v}_y}{\mathrm{\partial }y}+\frac{\mathrm{\partial }{v}_z}{\mathrm{\partial }z}=\mathrm{\nabla }\cdot \mathbf{v}}$

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:

${\displaystyle \mathrm{\nabla }\cdot (f\mathbf{v})=(\mathrm{\nabla }f)\cdot \mathbf{v}+f(\mathrm{\nabla }\cdot \mathbf{v})}$

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

${\displaystyle \mathrm{\nabla }\cdot (\mathbf{u}\times \mathbf{v})=(\mathrm{\nabla }\times \mathbf{u})\cdot \mathbf{v}-\mathbf{u}\cdot (\mathrm{\nabla }\times \mathbf{v})}$

Curl

The curl of a vector field ${\displaystyle \mathbf{v}(x,y,z)=v_x\hat{\mathbf{x}}+v_y\hat{\mathbf{y}}+v_z\hat{\mathbf{z}}}$ is a vector function that can be represented as:

${\displaystyle \mathrm{curl}\mathbf{v}=\left(\frac{\mathrm{\partial }{v}_z}{\mathrm{\partial }y}-\frac{\mathrm{\partial }{v}_y}{\mathrm{\partial }z}\right)\hat{\mathbf{x}}+\left(\frac{\mathrm{\partial }{v}_x}{\mathrm{\partial }z}-\frac{\mathrm{\partial }{v}_z}{\mathrm{\partial }x}\right)\hat{\mathbf{y}}+\left(\frac{\mathrm{\partial }{v}_y}{\mathrm{\partial }x}-\frac{\mathrm{\partial }{v}_x}{\mathrm{\partial }y}\right)\hat{\mathbf{z}}=\mathrm{\nabla }\times \mathbf{v}}$

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:

${\displaystyle \mathrm{\nabla }\times (f\mathbf{v})=(\mathrm{\nabla }f)\times \mathbf{v}+f(\mathrm{\nabla }\times \mathbf{v})}$

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

${\displaystyle \mathrm{\nabla }\times (\mathbf{u}\times \mathbf{v})=\mathbf{u}(\mathrm{\nabla }\cdot \mathbf{v})-\mathbf{v}(\mathrm{\nabla }\cdot \mathbf{u})+(\mathbf{v}\cdot \mathrm{\nabla })\mathbf{u}-(\mathbf{u}\cdot \mathrm{\nabla })\mathbf{v}}$

Directional derivative

The directional derivative of a scalar field ${\displaystyle f(x,y,z)}$ in the direction ${\displaystyle \mathbf{a}(x,y,z)=a_x\hat{\mathbf{x}}+a_y\hat{\mathbf{y}}+a_z\hat{\mathbf{z}}}$ is defined as:

${\displaystyle (\mathbf{a}\cdot \mathrm{\nabla })f=\underset{h\to 0}{lim}\frac{f(x+a_{x}h,y+a_{y}h,z+a_{z}h)-f(x,y,z)}{h}.}$

Which is equal to the following when the gradient exists

${\displaystyle \mathbf{a}\cdot \mathrm{grad}f=a_x\frac{\mathrm{\partial }f}{\mathrm{\partial }x}+a_y\frac{\mathrm{\partial }f}{\mathrm{\partial }y}+a_z\frac{\mathrm{\partial }f}{\mathrm{\partial }z}=\mathbf{a}\cdot (\mathrm{\nabla }f)}$

This gives the rate of change of a field ${\displaystyle f}$ in the direction of ${\displaystyle \mathbf{a}}$, scaled by the magnitude of ${\displaystyle \mathbf{a}}$. 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 ${\displaystyle (\mathbf{a}\cdot \mathrm{\nabla })}$ 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:

${\displaystyle \mathrm{\Delta }=\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{y}^2}+\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{z}^2}=\mathrm{\nabla }\cdot \mathrm{\nabla }={\mathrm{\nabla }}^2}$

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

${\displaystyle {\mathrm{\nabla }}^2=\mathrm{\nabla }\cdot {\mathrm{\nabla }}^T}$.

So whether ${\displaystyle {\mathrm{\nabla }}^2}$ 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 ${\displaystyle \mathbf{v}}$ (in three dimensions) is a 9-term second-rank tensor – that is, a 3×3 matrix – but can be denoted simply as ${\displaystyle \mathrm{\nabla }\otimes \mathbf{v}}$, where ${\displaystyle \otimes }$ 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 ${\displaystyle \delta \mathbf{r}}$, the change in the vector field is given by:

${\displaystyle \delta \mathbf{v}=(\mathrm{\nabla }\otimes \mathbf{v}{)}^T\cdot \delta \mathbf{r}}$

Product rules

For vector calculus:

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

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

${\displaystyle \begin{array}{r}\mathrm{\nabla }\cdot (\mathbf{u}\otimes \mathbf{v})=(\mathrm{\nabla }\cdot \mathbf{u})\mathbf{v}+(\mathbf{u}\cdot \mathrm{\nabla })\mathbf{v}\end{array}}$

Second derivatives

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:

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

Two of them are always equal:

${\displaystyle \mathrm{div}(\mathrm{grad}f)=\mathrm{\nabla }\cdot (\mathrm{\nabla }f)={\mathrm{\nabla }}^{2}f=\mathrm{\Delta }f}$

The 3 remaining vector derivatives are related by the equation:

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

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

${\displaystyle \mathrm{\nabla }(\mathrm{\nabla }\cdot \mathbf{v})=\mathrm{\nabla }\cdot (\mathbf{v}\otimes \mathrm{\nabla })}$

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 (${\displaystyle \mathrm{\nabla }\cdot \mathbf{v}}$) and the advection operator (${\displaystyle \mathbf{v}\cdot \mathrm{\nabla }}$) are not commutative:

A counterexample that relies on del's differential properties:

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

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

References

• Willard Gibbs & Edwin Bidwell Wilson (1901) Vector Analysis, Yale University Press, 1960: Dover Publications.
• Schey, H. M. (1997). Div, Grad, Curl, and All That: An Informal Text on Vector Calculus. New York: Norton. ISBN 0-393-96997-5.
• Miller, Jeff. "Earliest Uses of Symbols of Calculus".
• Arnold Neumaier (January 26, 1998). Cleve Moler (ed.). "History of Nabla". NA Digest, Volume 98, Issue 03. netlib.org.

External links

• Tai, Chen-To (1994). A survey of the improper use of ∇ in vector analysis (Report). Radiation Laboratory, University of Michigan. hdl:2027.42/7869.

Categories: 

• Vector calculus
• Mathematical notation
• Differential operators