# Nabla operator

The Nabla operator is a symbol that is used in vector and tensor analysis to note one of the three differential operators gradient , divergence or rotation , depending on the context . The symbol of the operator is the Nabla symbol (also or , to emphasize the formal similarity to common vector quantities). ${\ displaystyle \ nabla}$${\ displaystyle {\ vec {\ nabla}}}$${\ displaystyle {\ underline {\ nabla}}}$

The name "Nabla" is derived from a harp-like Phoenician stringed instrument that roughly had the shape of this symbol. The spelling was introduced by the mathematician Peter Guthrie Tait (1831–1901). In English the operator is called "del".

## definition

Formally, the Nabla operator is a vector whose components are the partial derivative operators : ${\ displaystyle \ textstyle {\ frac {\ partial} {\ partial x_ {i}}}}$

${\ displaystyle {\ vec {\ nabla}} = \ left ({\ frac {\ partial} {\ partial x_ {1}}}, \ ldots, {\ frac {\ partial} {\ partial x_ {n}} } \ right)}$

It can appear both as a column vector (for example grad) and as a row vector (for example div). In the three-dimensional Cartesian coordinate system one also writes:

${\ displaystyle {\ vec {\ nabla}} = \ left ({\ frac {\ partial} {\ partial x}}, {\ frac {\ partial} {\ partial y}}, {\ frac {\ partial} {\ partial z}} \ right) = {\ vec {e}} _ {x} {\ frac {\ partial} {\ partial x}} + {\ vec {e}} _ {y} {\ frac { \ partial} {\ partial y}} + {\ vec {e}} _ {z} {\ frac {\ partial} {\ partial z}}}$

Where , and are the unit vectors of the coordinate system. In generally curvilinear coordinates , the unit vectors are to be replaced by the contravariant basis vectors: ${\ displaystyle {\ vec {e}} _ {x}}$${\ displaystyle {\ vec {e}} _ {y}}$${\ displaystyle {\ vec {e}} _ {z}}$ ${\ displaystyle \ Theta _ {i}}$

${\ displaystyle {\ vec {\ nabla}} = \ sum _ {i = 1} ^ {n} {\ vec {g}} ^ {i} {\ frac {\ partial} {\ partial \ Theta _ {i }}} \ quad {\ text {with}} \ quad {\ vec {g}} ^ {i}: = \ operatorname {grad} \ Theta _ {i} \ ,.}$

Here grad is the gradient operator. When applying this Nabla operator to a vector field, it should be noted that the base vectors in curvilinear coordinate systems generally depend on the coordinates and must also be differentiated. ${\ displaystyle \ Theta _ {i}}$

The Nabla operator is used for calculations like a vector, whereby the “product” of, for example , a function to the right of it is interpreted as a partial derivative . ${\ displaystyle \ textstyle {\ frac {\ partial} {\ partial x_ {i}}}}$${\ displaystyle f}$${\ displaystyle \ textstyle {\ frac {\ partial f} {\ partial x_ {i}}}}$

## Representation of other differential operators

### In n-dimensional space

Let be an open subset , a differentiable function and a differentiable vector field . The superscript denotes transposition . ${\ displaystyle D \ subset \ mathbb {R} ^ {n}}$${\ displaystyle f \ colon D \ to \ mathbb {R}}$${\ displaystyle {\ vec {V}} = (V_ {1}, \ dots, V_ {n}) ^ {\ top} \ colon D \ to \ mathbb {R} ^ {n}}$

The (formal) product of with the function gives its gradient: ${\ displaystyle {\ vec {\ nabla}}}$${\ displaystyle f}$

${\ displaystyle {\ vec {\ nabla}} f = \ operatorname {grad} f = \ left ({\ frac {\ partial f} {\ partial x_ {1}}}, \ ldots, {\ frac {\ partial f} {\ partial x_ {n}}} \ right) ^ {\ top} \ ,.}$

The transposed (formal) dyadic product " " of with the vector field gives its gradient or Jacobi matrix : ${\ displaystyle \ otimes}$${\ displaystyle {\ vec {\ nabla}}}$${\ displaystyle {\ vec {V}}}$

${\ displaystyle ({\ vec {\ nabla}} \ otimes {\ vec {V}}) ^ {\ top} = \ operatorname {grad} {\ vec {V}} = J _ {\ vec {V}} = {\ begin {pmatrix} {\ frac {\ partial V_ {1}} {\ partial x_ {1}}} & \ ldots & {\ frac {\ partial V_ {1}} {\ partial x_ {n}}} \\\ vdots & \ ddots & \ vdots \\ {\ frac {\ partial V_ {n}} {\ partial x_ {1}}} & \ ldots & {\ frac {\ partial V_ {n}} {\ partial x_ {n}}} \ end {pmatrix}} \ ,.}$

The (formal) scalar product with the vector field results in its divergence: ${\ displaystyle {\ vec {V}}}$

${\ displaystyle {\ vec {\ nabla}} \ cdot {\ vec {V}} = \ operatorname {div} {\ vec {V}} = \ sum _ {i = 1} ^ {n} {\ frac { \ partial V_ {i}} {\ partial x_ {i}}} \ ,.}$

She is the trace of the gradient.

The (formal) scalar product of with itself results in the Laplace operator , because it holds ${\ displaystyle {\ vec {\ nabla}} ^ {2}}$${\ displaystyle {\ vec {\ nabla}}}$ ${\ displaystyle \ Delta}$

${\ displaystyle {\ vec {\ nabla}} ^ {2} = {\ vec {\ nabla}} \ cdot {\ vec {\ nabla}} = \ sum _ {i = 1} ^ {n} {\ frac {\ partial ^ {2}} {\ partial x_ {i} ^ {2}}} = \ Delta \ ,.}$

Given a vector , the operator ${\ displaystyle {\ vec {H}}}$

${\ displaystyle \ operatorname {D} _ {\ vec {H}}: = {\ vec {H}} \ cdot {\ vec {\ nabla}} = \ sum _ {i = 1} ^ {n} H_ { i} {\ frac {\ partial} {\ partial x_ {i}}} \ ,,}$

the directional derivative of differentiable functions in the direction of the vector can be calculated: ${\ displaystyle f}$${\ displaystyle {\ vec {H}}}$

${\ displaystyle \ operatorname {D} _ {\ vec {H}} f = ({\ vec {H}} \ cdot {\ vec {\ nabla}}) f = {\ vec {H}} \ cdot {\ vec {\ nabla}} f = {\ vec {H}} \ cdot \ operatorname {grad} (f) = \ operatorname {grad} (f) \ cdot {\ vec {H}} \ ,,}$

see the relationship between gradient and directional derivative . If the function is a vector field , then the product is calculated from the Jacobi matrix of the field and the vector: ${\ displaystyle {\ vec {V}}}$

{\ displaystyle {\ begin {aligned} \ operatorname {D} _ {\ vec {H}} {\ vec {V}} & = \ underbrace {({\ vec {H}} \ cdot {\ vec {\ nabla }}) {\ vec {V}}} = ({\ vec {H}} ^ {\ top} ({\ vec {\ nabla}} \ otimes {\ vec {V}})) ^ {\ top} = ({\ vec {\ nabla}} \ otimes {\ vec {V}}) ^ {\ top} {\ vec {H}} & = & \ underbrace {J _ {\ vec {V}}} {\ vec {H}} \\ & = {\ begin {pmatrix} H_ {1} {\ frac {\ partial} {\ partial x_ {1}}} V_ {1} + & \ ldots & + H_ {n} {\ frac {\ partial} {\ partial x_ {n}}} V_ {1} \\\ vdots & \ ddots & \ vdots \\ H_ {1} {\ frac {\ partial} {\ partial x_ {1}}} V_ {n} + & \ ldots & + H_ {n} {\ frac {\ partial} {\ partial x_ {n}}} V_ {n} \ end {pmatrix}} & = & {\ begin {pmatrix} { \ frac {\ partial V_ {1}} {\ partial x_ {1}}} & \ ldots & {\ frac {\ partial V_ {1}} {\ partial x_ {n}}} \\\ vdots & \ ddots & \ vdots \\ {\ frac {\ partial V_ {n}} {\ partial x_ {1}}} & \ ldots & {\ frac {\ partial V_ {n}} {\ partial x_ {n}}} \ end {pmatrix}} {\ begin {pmatrix} H_ {1} \\\ vdots \\ H_ {n} \ end {pmatrix}} \ ,, \ end {aligned}}}

see vector gradient and its application in continuum mechanics below.

### In three-dimensional space

Now let be an open subset, a differentiable function and a differentiable vector field. The indices ... x, y, z denote the vector components and not derivatives. In the three-dimensional space with the Cartesian coordinates , , to provide the above formulas are as follows: ${\ displaystyle D \ subset \ mathbb {R} ^ {3}}$${\ displaystyle f \ colon D \ to \ mathbb {R}}$${\ displaystyle {\ vec {V}} = (V_ {x}, V_ {y}, V_ {z}) ^ {\ top} \ colon D \ to \ mathbb {R} ^ {3}}$${\ displaystyle \ mathbb {R} ^ {3}}$${\ displaystyle x}$${\ displaystyle y}$${\ displaystyle z}$

The Nabla operator applied to the scalar field gives the gradient of the scalar field ${\ displaystyle f}$

${\ displaystyle \ operatorname {grad} f = {\ vec {\ nabla}} f = \ left ({\ frac {\ partial f} {\ partial x}}, {\ frac {\ partial f} {\ partial y }}, {\ frac {\ partial f} {\ partial z}} \ right) ^ {\ top} = {\ frac {\ partial f} {\ partial x}} {\ vec {e}} _ {x } + {\ frac {\ partial f} {\ partial y}} {\ vec {e}} _ {y} + {\ frac {\ partial f} {\ partial z}} {\ vec {e}} _ {z} \ ,.}$

The result is a vector field. Here are the unit vectors of . ${\ displaystyle {\ vec {e}} _ {x}, \, {\ vec {e}} _ {y}, \, {\ vec {e}} _ {z}}$${\ displaystyle \ mathbb {R} ^ {3}}$

The Nabla operator applied to the vector field results in the divergence of the vector field as a formal scalar product with the vector field ${\ displaystyle {\ vec {V}}}$

${\ displaystyle \ operatorname {div} {\ vec {V}} = {\ vec {\ nabla}} \ cdot {\ vec {V}} = {\ frac {\ partial V_ {x}} {\ partial x} } + {\ frac {\ partial V_ {y}} {\ partial y}} + {\ frac {\ partial V_ {z}} {\ partial z}},}$

so a scalar field.

A special feature of three-dimensional space is the rotation of a vector field. It results from the (right-hand) link via the formal cross product as

${\ displaystyle \ operatorname {rot} {\ vec {V}} = {\ vec {\ nabla}} \ times {\ vec {V}} = {\ begin {pmatrix} {\ frac {\ partial V_ {z} } {\ partial y}} - {\ frac {\ partial V_ {y}} {\ partial z}} \\ {\ frac {\ partial V_ {x}} {\ partial z}} - {\ frac {\ partial V_ {z}} {\ partial x}} \\ {\ frac {\ partial V_ {y}} {\ partial x}} - {\ frac {\ partial V_ {x}} {\ partial y}} \ end {pmatrix}} \ ,,}$

so again a vector field.

Cylindrical coordinates (ρ, φ, z) and spherical coordinates (r, θ, φ) are examples of curvilinear coordinates. The formulas for the gradient in cylindrical and spherical coordinates result from the Nabla operators

{\ displaystyle {\ begin {aligned} {\ text {in cylinder coordinates:}} \ quad & {\ vec {\ nabla}} = {\ vec {e}} _ {\ rho} {\ frac {\ partial} { \ partial \ rho}} + {\ frac {1} {\ rho}} {\ vec {e}} _ {\ varphi} {\ frac {\ partial} {\ partial \ varphi}} + {\ vec {e }} _ {z} {\ frac {\ partial} {\ partial z}} \\ {\ text {or. Spherical coordinates:}} \ quad & {\ vec {\ nabla}} = {\ vec {e}} _ {r} {\ frac {\ partial} {\ partial r}} + {\ frac {1} {r} } {\ vec {e}} _ {\ theta} {\ frac {\ partial} {\ partial \ theta}} + {\ frac {1} {r \ sin \ theta}} {\ vec {e}} _ {\ varphi} {\ frac {\ partial} {\ partial \ varphi}} \,. \ end {aligned}}}

When applying to a vector field, as mentioned above, it should be noted that the base vectors in curvilinear coordinate systems generally depend on the coordinates, as is the case here, and must also be differentiated. For example, the divergence of a vector field in cylindrical coordinates results, where the basis vectors and depend on the angle φ and the following applies: ${\ displaystyle {\ vec {e}} _ {\ rho}}$${\ displaystyle {\ vec {e}} _ {\ varphi}}$${\ displaystyle {\ tfrac {\ partial} {\ partial \ varphi}} {\ vec {e}} _ {\ rho} = {\ vec {e}} _ {\ varphi}, \, {\ tfrac {\ partial} {\ partial \ varphi}} {\ vec {e}} _ {\ varphi} = - {\ vec {e}} _ {\ rho}}$

{\ displaystyle {\ begin {aligned} \ operatorname {div} {\ vec {V}} = & {\ vec {\ nabla}} \ cdot {\ vec {V}} = \ left ({\ vec {e} } _ {\ rho} {\ frac {\ partial} {\ partial \ rho}} + {\ frac {1} {\ rho}} {\ vec {e}} _ {\ varphi} {\ frac {\ partial } {\ partial \ varphi}} + {\ vec {e}} _ {z} {\ frac {\ partial} {\ partial z}} \ right) \ cdot (V _ {\ rho} {\ vec {e} } _ {\ rho} + V _ {\ varphi} {\ vec {e}} _ {\ varphi} + V_ {z} {\ vec {e}} _ {z}) \\ = & {\ frac {\ partial} {\ partial \ rho}} V _ {\ rho} + {\ frac {1} {\ rho}} {\ vec {e}} _ {\ varphi} \ cdot {\ frac {\ partial} {\ partial \ varphi}} (V _ {\ rho} {\ vec {e}} _ {\ rho} + V _ {\ varphi} {\ vec {e}} _ {\ varphi} + V_ {z} {\ vec {e }} _ {z}) + {\ frac {\ partial} {\ partial z}} V_ {z} \\ = & {\ frac {\ partial} {\ partial \ rho}} V _ {\ rho} + { \ frac {1} {\ rho}} {\ vec {e}} _ {\ varphi} \ cdot \ left ({\ frac {\ partial} {\ partial \ varphi}} V _ {\ rho} {\ vec { e}} _ {\ rho} + V _ {\ rho} {\ vec {e}} _ {\ varphi} + {\ frac {\ partial} {\ partial \ varphi}} V _ {\ varphi} {\ vec { e}} _ {\ varphi} -V _ {\ varphi} {\ vec {e}} _ {\ rho} + {\ frac {\ partial} {\ partial \ varphi}} V_ {z} {\ vec {e }} _ {z} \ right) + {\ frac { \ partial} {\ partial z}} V_ {z} \\ = & {\ frac {\ partial} {\ partial \ rho}} V _ {\ rho} + {\ frac {1} {\ rho}} V_ { \ rho} + {\ frac {1} {\ rho}} {\ frac {\ partial} {\ partial \ varphi}} V _ {\ varphi} + {\ frac {\ partial} {\ partial z}} V_ { z} \,. \ end {aligned}}}

## Notation with subscription

If the Nabla operator only affects certain components of a function with a multi-dimensional argument, this is indicated by a subscript . For example, for a function with${\ displaystyle f ({\ vec {r}}, t)}$${\ displaystyle {\ vec {r}} = (x_ {1}, x_ {2}, \ dotsc, x_ {n})}$

${\ displaystyle {\ vec {\ nabla}} _ {\ vec {r}} f = \ left ({\ frac {\ partial f} {\ partial x_ {1}}}, {\ frac {\ partial f} {\ partial x_ {2}}}, \ dots, {\ frac {\ partial f} {\ partial x_ {n}}} \ right) ^ {\ top}}$

in contrast to

${\ displaystyle {\ vec {\ nabla}} f = \ left ({\ frac {\ partial f} {\ partial x_ {1}}}, {\ frac {\ partial f} {\ partial x_ {2}} }, \ dots, {\ frac {\ partial f} {\ partial x_ {n}}}, {\ frac {\ partial {f}} {\ partial t}} \ right) ^ {\ top} \ ,. }$

This term is common when the Nabla symbol denotes the simple differential (i.e. the one-row Jacobian matrix ) or part of it.

Occasionally, an alternative for the spelling with the Nabla symbol spelling on. ${\ displaystyle {\ vec {\ nabla}} _ {\ vec {r}}}$${\ displaystyle {\ frac {\ partial} {\ partial {\ vec {r}}}}}$

## Representation as a quaternion

Sir William Rowan Hamilton defined the Nabla operator as a pure quaternion

${\ displaystyle \ nabla: = \ mathrm {i} \, {\ frac {\ partial} {\ partial x}} + \ mathrm {j} \, {\ frac {\ partial} {\ partial y}} + \ mathrm {k} \, {\ frac {\ partial} {\ partial z}}}$

with the complex imaginary units , and , which are not commutatively linked by the Hamilton rules . For example . ${\ displaystyle \ mathrm {i}}$${\ displaystyle \ mathrm {j}}$${\ displaystyle \ mathrm {k}}$ ${\ displaystyle \ mathrm {i ^ {2} = j ^ {2} = k ^ {2} = i \, j \, k = -1}}$${\ displaystyle \ mathrm {j \, k = -k \, j = i}}$

Application to a real-valued function (formal multiplication) provides the quaternionic equivalent for its gradient and Laplace derivative: ${\ displaystyle f}$

{\ displaystyle {\ begin {aligned} \ nabla f = & \ mathrm {i} \, {\ frac {\ partial f} {\ partial x}} + \ mathrm {j} \, {\ frac {\ partial f } {\ partial y}} + \ mathrm {k} \, {\ frac {\ partial f} {\ partial z}} = \ operatorname {grad} f \\\ nabla \ nabla 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}}} = - \ Delta f \ end {aligned}}}

Application to a pure quaternion (formal multiplication) yields: ${\ displaystyle q = \ mathrm {i} \, u + \ mathrm {j} \, v + \ mathrm {k} \, w}$

{\ displaystyle {\ begin {aligned} \ nabla q = & - {\ frac {\ partial u} {\ partial x}} - {\ frac {\ partial v} {\ partial y}} - {\ frac {\ partial w} {\ partial z}} + \ mathrm {i} \, \ left ({\ frac {\ partial w} {\ partial y}} - {\ frac {\ partial v} {\ partial z}} \ right) + \ mathrm {j} \, \ left ({\ frac {\ partial u} {\ partial z}} - {\ frac {\ partial w} {\ partial x}} \ right) + \ mathrm {k } \, \ left ({\ frac {\ partial v} {\ partial x}} - {\ frac {\ partial u} {\ partial y}} \ right) \\ = & - \ nabla \ cdot q + \ nabla \ times q = - \ operatorname {div} q + \ operatorname {red} q \ end {aligned}}}

The definitions of the scalar product and cross product of quaternions used here can be looked up in the main article.

## Calculation rules

Calculation rules for the Nabla operator can be formally derived from the calculation rules for scalar and cross product together with the derivation rules. The product rule must be used when the Nabla operator is to the left of a product.

If and are differentiable scalar fields (functions) and as well as differentiable vector fields, then: ${\ displaystyle \ psi}$${\ displaystyle \ varphi}$${\ displaystyle {\ vec {A}}}$${\ displaystyle {\ vec {B}}}$

${\ displaystyle {\ vec {\ nabla}} \ varphi (\ psi) = {\ frac {\ mathrm {d} \ varphi} {\ mathrm {d} \ psi}} {\ vec {\ nabla}} \ psi }$ (Chain rule for gradient)
${\ displaystyle {\ vec {\ nabla}} (\ psi \ varphi) = \ psi {\ vec {\ nabla}} \ varphi + \ varphi {\ vec {\ nabla}} \ psi}$ (Product rule for gradient)
${\ displaystyle {\ vec {\ nabla}} ({\ vec {A}} \ cdot {\ vec {B}}) = ({\ vec {A}} \ cdot {\ vec {\ nabla}}) { \ vec {B}} + ({\ vec {B}} \ cdot {\ vec {\ nabla}}) {\ vec {A}} + {\ vec {A}} \ times ({\ vec {\ nabla }} \ times {\ vec {B}}) + {\ vec {B}} \ times ({\ vec {\ nabla}} \ times {\ vec {A}})}$
${\ displaystyle {\ vec {\ nabla}} \ cdot (\ varphi {\ vec {A}}) = \ varphi {\ vec {\ nabla}} \ cdot {\ vec {A}} + {\ vec {A }} \ cdot {\ vec {\ nabla}} \ varphi}$
${\ displaystyle {\ vec {\ nabla}} \ cdot ({\ vec {A}} \ times {\ vec {B}}) = {\ vec {B}} \ cdot ({\ vec {\ nabla}} \ times {\ vec {A}}) - {\ vec {A}} \ cdot ({\ vec {\ nabla}} \ times {\ vec {B}})}$
${\ displaystyle {\ vec {\ nabla}} \ cdot ({\ vec {\ nabla}} \ varphi) = \ operatorname {div (grad} \ varphi) = \ Delta \ varphi}$(see also Laplace operator )
${\ displaystyle {\ vec {\ nabla}} \ cdot ({\ vec {\ nabla}} \ times {\ vec {A}}) = \ operatorname {div (red} {\ vec {A}}) = 0 }$
${\ displaystyle {\ vec {\ nabla}} \ times ({\ vec {\ nabla}} \ varphi) = \ operatorname {red (grad} \ varphi) = 0}$
${\ displaystyle {\ vec {\ nabla}} \ times \ varphi {\ vec {A}} = \ varphi {\ vec {\ nabla}} \ times {\ vec {A}} - {\ vec {A}} \ times {\ vec {\ nabla}} \ varphi}$
${\ displaystyle {\ vec {\ nabla}} \ times ({\ vec {A}} \ times {\ vec {B}}) = ({\ vec {B}} \ cdot {\ vec {\ nabla}} ) {\ vec {A}} - {\ vec {B}} ({\ vec {\ nabla}} \ cdot {\ vec {A}}) + {\ vec {A}} ({\ vec {\ nabla }} \ cdot {\ vec {B}}) - ({\ vec {A}} \ cdot {\ vec {\ nabla}}) {\ vec {B}}}$
${\ displaystyle {\ vec {\ nabla}} \ times ({\ vec {\ nabla}} \ times {\ vec {A}}) = \ operatorname {red (red} {\ vec {A}}) = \ operatorname {grad (div} {\ vec {A}}) - \ Delta {\ vec {A}}}$(see also vectorial Laplace operator )

For further calculation rules see under gradient , divergence and rotation .

## Application in continuum mechanics

In continuum mechanics , the Nabla operator is used to define the gradient of a vector field and the divergence and rotation of a tensor field in addition to the operators mentioned above. Here the Nabla operator can occasionally also act to the left.

Because of the importance of rotation for continuum mechanics, the representation is made in three dimensions. So let us be an open subset , a differentiable vector field with components V x, y, z , which are numbered as usual according to the scheme x → 1, y → 2 and z → 3, and a differentiable tensor field of the second level with components with respect to a Cartesian coordinate system . ${\ displaystyle D \ subset \ mathbb {R} ^ {3}}$${\ displaystyle {\ vec {V}} = (V_ {x}, V_ {y}, V_ {z}) ^ {\ top} \ colon D \ to \ mathbb {R} ^ {3}}$${\ displaystyle \ mathbf {T} \ colon D \ to \ mathbb {R} ^ {3 \ times 3}}$${\ displaystyle T_ {ij} \ ,, \; i, j = x, y, z}$

The transposed dyadic product of the Nabla operator with a vector field gives - as explained above - its gradient ${\ displaystyle {\ vec {V}}}$

${\ displaystyle ({\ vec {\ nabla}} \ otimes {\ vec {V}}) ^ {\ top} = \ operatorname {grad} ({\ vec {V}}): = \ sum _ {j = 1} ^ {3} {\ frac {\ partial {\ vec {V}}} {\ partial x_ {j}}} \ otimes {\ vec {e}} _ {j} = \ sum _ {i, j = 1} ^ {3} {\ frac {\ partial V_ {i}} {\ partial x_ {j}}} {\ vec {e}} _ {i} \ otimes {\ vec {e}} _ {j } = {\ begin {pmatrix} {\ frac {\ partial V_ {x}} {\ partial x}} & {\ frac {\ partial V_ {x}} {\ partial y}} & {\ frac {\ partial V_ {x}} {\ partial z}} \\ {\ frac {\ partial V_ {y}} {\ partial x}} & {\ frac {\ partial V_ {y}} {\ partial y}} & { \ frac {\ partial V_ {y}} {\ partial z}} \\ {\ frac {\ partial V_ {z}} {\ partial x}} & {\ frac {\ partial V_ {z}} {\ partial y}} & {\ frac {\ partial V_ {z}} {\ partial z}} \ end {pmatrix}}}$

thus a second order tensor field. The gradient defined in this way agrees with the Fréchet derivation :

${\ displaystyle \ operatorname {grad} ({\ vec {V}}) \ colon \ quad \ operatorname {grad} ({\ vec {V}}) \ cdot {\ vec {h}} = \ left. {\ frac {\ mathrm {d}} {\ mathrm {d} s}} {\ vec {V}} ({\ vec {x}} + s {\ vec {h}}) \ right | _ {s = 0 } = \ lim _ {s \ rightarrow 0} {\ frac {{\ vec {V}} ({\ vec {x}} + s {\ vec {h}}) - {\ vec {V}} ({ \ vec {x}})} {s}} \ quad {\ text {for all}} \; {\ vec {x}}, {\ vec {h}} \ in D \ ,.}$

The left scalar product of the Nabla operator with a second order tensor field gives formally the divergence of the tensor field:

${\ displaystyle {\ vec {\ nabla}} \ cdot \ mathbf {T} = \ operatorname {div} \ mathbf {T}: = \ sum _ {k = 1} ^ {3} {\ vec {e}} _ {k} \ cdot {\ frac {\ partial \ mathbf {T}} {\ partial x_ {k}}} = \ sum _ {i, j, k} ^ {3} {\ vec {e}} _ {k} \ cdot {\ frac {\ partial T_ {ij}} {\ partial x_ {k}}} ({\ vec {e}} _ {i} \ otimes {\ vec {e}} _ {j} ) = \ sum _ {i, j} ^ {3} {\ frac {\ partial T_ {ij}} {\ partial x_ {i}}} {\ vec {e}} _ {j} = {\ begin { pmatrix} {\ frac {\ partial T_ {xx}} {\ partial x}} + {\ frac {\ partial T_ {yx}} {\ partial y}} + {\ frac {\ partial T_ {zx}} { \ partial z}} \\ {\ frac {\ partial T_ {xy}} {\ partial x}} + {\ frac {\ partial T_ {yy}} {\ partial y}} + {\ frac {\ partial T_ {zy}} {\ partial z}} \\ {\ frac {\ partial T_ {xz}} {\ partial x}} + {\ frac {\ partial T_ {yz}} {\ partial y}} + {\ frac {\ partial T_ {zz}} {\ partial z}} \ end {pmatrix}}}$

so a vector field.

The cross product of the Nabla operator with a second order tensor yields its rotation:

${\ displaystyle {\ vec {\ nabla}} \ times \ mathbf {T} = \ operatorname {red} \ mathbf {T}: = \ sum _ {i = 1} ^ {3} {\ vec {e}} _ {i} \ times {\ frac {\ partial \ mathbf {T}} {\ partial x_ {i}}} = \ sum _ {i, j, l = 1} ^ {3} {\ vec {e} } _ {i} \ times {\ frac {\ partial T_ {jl}} {\ partial x_ {i}}} ({\ vec {e}} _ {j} \ otimes {\ vec {e}} _ { l}) = \ sum _ {i, j, k, l = 1} ^ {3} \ epsilon _ {ijk} {\ frac {\ partial T_ {jl}} {\ partial x_ {i}}} ({ \ vec {e}} _ {k} \ otimes {\ vec {e}} _ {l})}$

thus a second order tensor field. Inside is the permutation symbol . The Tensoranalysis formula collection contains further formulas and definitions . ${\ displaystyle \ epsilon _ {ijk} = ({\ vec {e}} _ {i} \ times {\ vec {e}} _ {j}) \ cdot {\ vec {e}} _ {k}}$

These operators are not uniformly defined in the literature. Occasionally there are transposed versions:

gradient ${\ displaystyle {\ tilde {\ operatorname {grad}}} {\ vec {V}}: = {\ vec {\ nabla}} \ otimes {\ vec {V}} = \ operatorname {grad} ({\ vec {V}}) ^ {\ top}}$ ${\ displaystyle {\ tilde {\ operatorname {div}}} \ mathbf {T}: = {\ vec {\ nabla}} \ cdot (\ mathbf {T} ^ {\ top}) = \ operatorname {div} ( \ mathbf {T} ^ {\ top})}$ ${\ displaystyle {\ tilde {\ operatorname {red}}} \ mathbf {T}: = {\ vec {\ nabla}} \ times (\ mathbf {T} ^ {\ top}) = \ operatorname {red} ( \ mathbf {T} ^ {\ top})}$

The transposed tensors must then be used in the formulas of Wikipedia in order to compare them with the formulas in the literature.

## literature

• Bronstein, Semendjajew, Musiol, Mühlig: Taschenbuch der Mathematik . 5th edition. Harri Deutsch, 2001, ISBN 3-8171-2005-2 (Contains all properties mentioned here, but without proof.).
• Jänich: Vector analysis . Springer, 1992, ISBN 3-540-55530-7 (Contains only the basic definition.).
• Großmann: Mathematical introductory course for physics . Teubner, Stuttgart 1991 (see in particular Section 3.6).
• H. Altenbach: Continuum Mechanics . Springer, 2012, ISBN 978-3-642-24118-5 (see Section 2.3 Tensoranalysis).

Wikibooks: Physics formula collection: Nabla operator  - learning and teaching materials

## References and footnotes

1. KE Georges: Detailed Latin-German concise dictionary . 8th edition. tape 4 (MQ). Hofenberg, 1913, ISBN 978-3-8430-4923-8 (complete new edition by K.-M. Guth 2014).
2. Eric Weisstein: Del . In: MathWorld (English).
3. Row and column vectors are also referred to as covariant and contravariant in differential geometry and in the mathematical formalism of the theory of relativity . The derivative operator according to the covariant coordinates forms a contravariant vector and vice versa.
4. Jürgen Schnakenberg: Electrodynamics. John Wiley & Sons, 2003, ISBN 3-527-40369-8 , pp. 31 ff., Google Books .
5. H.-D. Ebbinghaus, H. Hermes, F. Hirzebruch, M. Koecher, K. Mainzer, A. Prestel, R. Remmert: Numbers . tape 1 basic knowledge and mathematics . Springer-Verlag, Berlin a. a. 1983, ISBN 978-3-540-12666-9 , doi : 10.1007 / 978-3-642-96783-2 ( limited preview in Google book search).
6. P. Haupt: Continuum Mechanics and Theory of Materials . 2nd Edition. Springer, 2002, ISBN 978-3-540-43111-4 .