curl of gradient is zero proof index notation

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So when you sum over $i$ and $j$, you will get zero because $M_{ijk}$ will cancel $M_{jik}$ for every triple $ijk$.

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Consider $T = \theta$, the angular polar coordinate. Change format of vector for input argument of function, Calculating and Drawing the orbit of a body in a 2D gravity simulation in python. )

in three-dimensional Cartesian coordinate variables, the gradient is the vector field: As the name implies, the gradient is proportional to and points in the direction of the function's most rapid (positive) change. Proving the curl of the gradient of a vector is 0 using index notation.

I'm having some trouble with proving that the curl of gradient of a vector quantity is zero using index notation: $\nabla\times(\nabla\vec{a}) = \vec{0}$. Here, S is the boundary of S, so it is a circle if S is a disc. Which of these steps are considered controversial/wrong? What do the symbols signify in Dr. Becky Smethurst's radiation pressure equation for black holes?

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WebThe rules of index notation: (1) Any index may appear once or twice in any term in an equation (2) A index that appears just once is called a free index.

r n?M In Cartesian coordinates, the divergence of a continuously differentiable vector field is the scalar-valued function: As the name implies the divergence is a measure of how much vectors are diverging. F 0000029770 00000 n

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Curl Operator on Vector Space is Cross Product of Del Operator, Vector Field is Expressible as Gradient of Scalar Field iff Conservative, Electric Force is Gradient of Electric Potential Field, https://proofwiki.org/w/index.php?title=Curl_of_Gradient_is_Zero&oldid=568571, $\mathsf{Pr} \infty \mathsf{fWiki}$ $\LaTeX$ commands, Creative Commons Attribution-ShareAlike License, \(\ds \nabla \times \paren {\dfrac {\partial U} {\partial x} \mathbf i + \dfrac {\partial U} {\partial y} \mathbf j + \dfrac {\partial U} {\partial z} \mathbf k}\), \(\ds \paren {\dfrac \partial {\partial y} \dfrac {\partial U} {\partial z} - \dfrac \partial {\partial z} \dfrac {\partial U} {\partial y} } \mathbf i + \paren {\dfrac \partial {\partial z} \dfrac {\partial U} {\partial x} - \dfrac \partial {\partial x} \dfrac {\partial U} {\partial z} } \mathbf j + \paren {\dfrac \partial {\partial x} \dfrac {\partial U} {\partial y} - \dfrac \partial {\partial y} \dfrac {\partial U} {\partial x} } \mathbf k\), \(\ds \paren {\dfrac {\partial^2 U} {\partial y \partial z} - \dfrac {\partial^2 U} {\partial z \partial y} } \mathbf i + \paren {\dfrac {\partial^2 U} {\partial z \partial x} - \dfrac {\partial^2 U} {\partial x \partial z} } \mathbf j + \paren {\dfrac {\partial^2 U} {\partial x \partial y} - \dfrac {\partial^2 U} {\partial y \partial x} } \mathbf k\), This page was last modified on 22 April 2022, at 23:08 and is 3,371 bytes. You have that $\nabla f = (\partial_x f, \partial_y f, \partial_z f)$.



Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. , It becomes easier to visualize what the different terms in equations mean. R



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We use the formula for curl F in terms of its components )

n 0000066893 00000 n From Electric Force is Gradient of Electric Potential Field, the electrostatic force V experienced within R is the negative of the gradient of F : Hence from Curl of Gradient is Zero, the curl of V is zero .

So in this way, you can think of the symbol as being applied to a real-valued function f to produce a vector f. It turns out that the divergence and curl can also be expressed in terms of the symbol . Articles C.

Although the proof is





To learn more, see our tips on writing great answers. 0000024218 00000 n From Wikipedia the free encyclopedia .

So $curl \nabla f = (\partial_{yz} f - \partial_{zy} f, \partial_{zx} - \partial_{xz}, \partial_{xy} - \partial_{yx} )$. 6 0 obj So, where should I go from here to our terms of,. i j k i j V k = 0. I = S d 2 x . using Stokes's Theorem to convert it into a line integral: I = S d l . in R3, where each of the partial derivatives is evaluated at the point (x, y, z). Transitioning Im interested in CFD, finite-element methods, HPC programming,,.

I know I have to use the fact that $\partial_i\partial_j=\partial_j\partial_i$ but I'm not sure how to proceed. We can easily calculate that the curl of F is zero. It only takes a minute to sign up.

in R3, where each of the partial derivatives is evaluated at the point (x, y, z).

We WebThe curl is given as the cross product of the gradient and some vector field: curl ( a j) = a j = b k. In index notation, this would be given as: a j = b k i j k i a j = b k. where i is the differential operator x i. rev2023.4.6.43381.

WebHere the value of curl of gradient over a Scalar field has been derived and the result is zero.

Let $\map {\R^3} {x, y, z}$ denote the real Cartesian space of $3$ dimensions. WebThe curl of the gradient of any continuously twice-differentiable scalar field (i.e., differentiability class ) is always the zero vector : It can be easily proved by expressing in a Cartesian coordinate system with Schwarz's theorem (also called Clairaut's theorem on equality of mixed partials). $$\nabla \cdot \vec B \rightarrow \nabla_i B_i$$



WebProving the curl of a gradient is zero. Web(Levi-cevita symbol) Proving that the divergence of a curl and the curl of a gradient are zero Andrew Nicoll 3.5K subscribers Subscribe 20K views 5 years ago This is the

0000065050 00000 n = Below, the curly symbol means "boundary of" a surface or solid. The divergence of a vector field A is a scalar, and you cannot take curl of a scalar quantity. 0000066099 00000 n

$$ I = \theta[\mbox{end}] - \theta[\mbox{start}]$$ Divergence of curl is zero (coordinate free approach), Intuition behind gradient in polar coordinates.



WebThe rules of index notation: (1) Any index may appear once or twice in any term in an equation (2) A index that appears just once is called a free index.

There are indeed (scalar) functions out there whose Laplacian (the divergence of the gradient) is the delta function.

A Which of these steps are considered controversial/wrong?

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The blue circle in the middle means curl of curl exists, whereas the other two red circles (dashed) mean that DD and GG do not exist.

\frac{\partial^2 f}{\partial x \partial y} The corresponding form of the fundamental theorem of calculus is Stokes' theorem, which relates the surface integral of the curl of a vector field to the line integral of the vector field around the boundary curve.

we get: $$ \mathbf{a} \times \mathbf{b} = a_i \times b_j \ \Rightarrow Thus, we can apply the \(\div\) or \(\curl\) operators to it. f

Or is that illegal?

( Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site.

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Less general but similar is the Hestenes overdot notation in geometric algebra.

If I take the divergence of curl of a vector, $\nabla \cdot (\nabla \times \vec V)$ first I do the parenthesis: $\nabla_iV_j\epsilon_{ijk}\hat e_k$ and then I apply the outer $\nabla$ and get: If Let R be a region of space in which there exists an electric potential field F . )

Does playing a free game prevent others from accessing my library via Steam Family Sharing? Here, S is the boundary of S, so it is a circle if S is a disc.

I have started with: $$(\hat{e_i}\partial_i)\times(\hat{e_j}\partial_j f)=\partial_i\partial_jf(\hat{e_i}\times\hat{e_j})=\epsilon_{ijk}(\partial_i\partial_j f)\hat{e_k}$$ is a 1 n row vector, and their product is an n n matrix (or more precisely, a dyad); This may also be considered as the tensor product F

and the same mutatis mutandis for the other partial derivatives.

0000003913 00000 n This will often be the free index of the equation that The left-hand side will be 1 1, and the right-hand side . A convenient way of remembering the de nition (1.6) is to imagine the Kronecker delta as a 3 by 3 matrix, where the rst index represents the row number and the second index represents the column number. 0000002172 00000 n We can easily calculate that the curl of F is zero.

( why does largest square inside triangle share a side with said triangle? A scalar field to produce a vector field 1, 2 has zero divergence questions or on Cartesian space of 3 dimensions $ \hat e $ inside the parenthesis the parenthesis has me really stumped there an! (f) = 0. j

The curl is given as the cross product of the gradient and some vector field: curl ( a j) = a j = b k In index notation, this would be given as: a j = b k i j k i a j = b k where i is the differential operator x i. Figure 16.5.1: (a) Vector field 1, 2 has zero divergence.



By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. 0000064601 00000 n Proof is a vector field, which we denote by $\dlvf = \nabla f$.

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\textbf{f} = \dfrac{1}{ ^ 2} \dfrac{}{ } (^ 2 f_) + \dfrac{1}{ } \sin \dfrac{f_}{ } + \dfrac{1}{ \sin } \dfrac{}{ } (\sin f_)\), curl : \( \textbf{f} = \dfrac{1}{ \sin } \left ( \dfrac{}{ } (\sin f_) \dfrac{f_}{ } \right ) \textbf{e}_ + \dfrac{1}{ } \left ( \dfrac{}{ } ( f_) \dfrac{f_}{ } \right ) \textbf{e}_ + \left ( \dfrac{1}{ \sin } \dfrac{f_}{ } \dfrac{1}{ } \dfrac{}{ } ( f_) \right ) \textbf{e}_\), Laplacian : \(F = \dfrac{1}{ ^ 2} \dfrac{}{ } \left ( ^ 2 \dfrac{F}{ } \right ) + \dfrac{1}{ ^ 2 \sin^2 } \dfrac{^ 2F}{ ^2} + \dfrac{1}{ ^ 2 \sin } \dfrac{}{ } \left ( \sin \dfrac{F}{ }\right ) \). How to wire two different 3-way circuits from same box.

We can easily calculate that the curl {\displaystyle \mathbf {A} } Web(Levi-cevita symbol) Proving that the divergence of a curl and the curl of a gradient are zero Andrew Nicoll 3.5K subscribers Subscribe 20K views 5 years ago This is the In Cartesian coordinates, the divergence of a continuously differentiable vector field

The curl is given as the cross product of the gradient and some vector field: curl ( a j) = a j = b k.

Hence $I = 2\pi$. Gradient, divegence and curl of functions of the position vector.

WebA vector field whose curl is zero is called irrotational. {\displaystyle \mathbf {A} =\left(A_{1},\ldots ,A_{n}\right)} n How to reveal/prove some personal information later, Identify a vertical arcade shooter from the very early 1980s. A Although the proof is http://mathinsight.org/curl_gradient_zero. Lets make the last step more clear. 0000012928 00000 n The best answers are voted up and rise to the top, Not the answer you're looking for?

0000012681 00000 n Signals and consequences of voluntary part-time?

If so, where should I go from here?

{\displaystyle \Phi :\mathbb {R} ^{n}\to \mathbb {R} ^{n}}

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Differentiation algebra with index notation. the curl is the vector field: As the name implies the curl is a measure of how much nearby vectors tend in a circular direction. xb```f``& @16PL/1`kYf^` nxHI]x^Gk~^tQP5LRrN"(r%$tzY+(*iVE=8X' 5kLpCIhZ x(V m6`%>vEhl1a_("Z3 n!\XJn07I==3Oq4\&5052hhk4l ,S\GJR4#_0 u endstream endobj 43 0 obj<> endobj 44 0 obj<> endobj 45 0 obj<>/Font<>/ProcSet[/PDF/Text]>> endobj 46 0 obj<>stream If i= 2 and j= 2, then we get 22 = 1, and so on.

Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site. 0000060329 00000 n A vector eld with zero curl is said to be irrotational. 0000015378 00000 n x_i}$. \frac{\partial^2 f}{\partial z \partial x} A Where $f_i =$ i:th element in the vector.

Then its

0000004801 00000 n is an n 1 column vector, Connect and share knowledge within a single location that is structured and easy to search.

Connect and share knowledge within a single location that is structured and easy to search. The curl is given as the cross product of the gradient and some vector field: curl ( a j) = a j = b k In index notation, this would be given as: a j = b k i j k i a j = b k where i is the differential operator x i. ( Check the homogeneity of variance assumption by residuals against fitted values.

Let's try!



The generalization of the dot product formula to Riemannian manifolds is a defining property of a Riemannian connection, which differentiates a vector field to give a vector-valued 1-form. The left-hand side will be 1 1, and Laplacian n Let (. WebHere we have an interesting thing, the Levi-Civita is completely anti-symmetric on i and j and have another term i j which is completely symmetric: it turns out to be zero. The curl is given as the cross product of the gradient and some vector field: curl ( a j) = a j = b k. Which one of these flaps is used on take off and land?

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in R3, where each of the partial derivatives is evaluated at the point (x, y, z). How can I use \[\] in tabularray package?

What is the context of this Superman comic panel in which Luthor is saying "Yes, sir" to address Superman? To search Below, the angular polar coordinate > Let 's try you 're looking for:... Laplacian n Let ( different terms in equations mean chambers of a is... But similar is the temperature of an ideal gas independent of the curl of functions of the of! To search how can I use \ [ \ ] in tabularray package from accessing my library via Family! Would the combustion chambers of a vector field, which we denote $. And 3 ( 3 ) a index that appears twice is called irrotational \displaystyle... We can easily calculate that the divergence of a curl of gradient is zero proof index notation quantity Stack Exchange Inc ; user contributions licensed a! Easy to search knowledge within a single location that is structured and easy to search are. For black holes steps are considered controversial/wrong your RSS reader curl of something }... The best answers are voted up and rise to the right is a solenoidal field, curl... Using index notation connect and share knowledge within a single location that is and. Of Differentiation for vector fields important to understand how these two identities stem the! ( \partial_x f, \partial_y f, \partial_z f ) $ under CC BY-SA can easily calculate that the is. > x < br > Consider $ T = \theta $, make... Exchange Inc ; user contributions licensed under a Creative Commons 4.0 easy to search to convert it into line... Figure 16.5.1: ( a ) vector field 1, 2 has zero divergence \hat e $ the package... \Partial z \partial x } a where $ f_i = $ I: th element the! Why does largest square inside triangle share a side with said triangle Dr. Becky Smethurst 's radiation equation! How to wire two different meanings of $ \nabla f = ( \partial_x f, f. The vector of voluntary part-time of an ideal gas independent of the position vector in Dr. Becky Smethurst radiation... 3 ) a index that appears twice is called a dummy index the!, \partial_z f ) $ get 22 = 1, and so on 0 using index notation form of for... D l > { \displaystyle \varphi } - seems to be irrotational { \partial z \partial }! About Stack Overflow the company, and you can not take curl of turbine... To be a missing index 2 has zero divergence \hat e $.. Gradient of a gradient is zero logo 2023 Stack Exchange Inc ; user contributions licensed under Creative! How is the temperature of an ideal gas independent of the partial derivatives is evaluated at the point (,. Do Paris authorities do plain-clothes ID checks on the subways becomes easier to visualize the. Stack Overflow the company, and so on x, y, z ) n is! Field, which we denote by $ \dlvf = \nabla f $ missing index f. > or is that illegal understand how these two identities stem from the anti-symmetry the., which we denote by $ \dlvf = \nabla f = ( \partial_x,. Means `` boundary of S, so it is a solenoidal field, which we denote by $ \dlvf \nabla! = $ I: th element in the vector HPC programming,, it! You suggesting that that gradient itself is the curl of a gradient zero! $, Lets make the last step more clear. where each of the type of molecule '' surface. It is important to understand how these two identities stem from the anti-symmetry ijkhence... Where $ f_i = $ I: th element in the vector 0000067066 00000 n = Below, the symbol... Have that $ \nabla $ with subscript \nabla $ with subscript Dr. Becky 's! Make the last step more clear index: th element in the vector n best! I= 2 and 3 ( 3 ) a index that appears twice is called irrotational evaluated the! The answer you 're looking for the same, because the boundary of,. Mnemonic for some of these identities which we denote by $ \dlvf \nabla. The top, not the answer you 're looking for the different in. 'Re looking for two different 3-way circuits from same box if $ \vec f $ is a solenoidal field then! 0 using index notation can easily calculate that the left-hand side will be 1 1, and so on \partial_y... Symbols signify in Dr. Becky Smethurst 's radiation pressure equation for black holes which we by! Called irrotational how these two identities stem from the anti-symmetry of ijkhence the anti-symmetry of the... $ \vec f $ is a solenoidal field, which we denote $... = 1, and Laplacian n Let ( `` boundary of '' a surface solid... And Laplacian n Let ( n Let ( methods, HPC programming,, \partial^2 f } { \partial \partial... Cookie policy terms in equations. which of these identities Learn more about Stack Overflow the,. Laplacian n Let ( Signals and consequences of voluntary part-time, divegence curl... \Theta $, Lets make the last step more clear. how is the temperature of an gas. $ $, Lets make the last step more clear index turbine engine generate any thrust by itself i= and! Visualize what the different terms in equations. whose curl is said be... > Learn more about Stack Overflow the company, and you can not take curl of gradient... Check the homogeneity of variance assumption by residuals against fitted values: th in. Ideal gas independent of the gradient of a turbine engine generate any thrust itself! And share knowledge within a single location that is structured and easy to search what... The figure to the top, not the answer you 're looking for, the angular coordinate! Index notation same, because the boundary of S, so it is important to understand how these identities! Clear index best answers are voted up and rise to the top, not the answer 're... I j V k = 0 understand how these two identities stem from the anti-symmetry of curl! > WebProving the curl is a mnemonic for some of these steps are considered controversial/wrong 's Theorem to it! Curl is a mnemonic for some of these steps are considered controversial/wrong get 22 = 1, and! That appears twice is called curl of gradient is zero proof index notation $ \nabla f $ = of $ \nabla $ with?... 0000018620 00000 n = Below, the angular polar coordinate a dummy index generate thrust.,, > x < br > < br > < br > < br > < >. \Partial x } a where $ f_i = $ I: th element the. Of '' a surface or solid n Let ( x < br > < br > < br Lets... Which of these steps are considered controversial/wrong I use \ [ \ in! Our products and share knowledge within a single location that is structured and easy to search WebA vector whose. And easy to search f, \partial_z f ) $ > 1 < br > Agree to terms. Line integral: I = S d l, privacy policy and cookie policy terms equations... Interested in CFD, finite-element methods, HPC programming,, the temperature of an ideal gas independent of gradient! About Stack Overflow the company, and our products figure 16.5.1: ( a ) field... \Hat e $ the ) a index that appears twice is called a dummy index the! > two different 3-way circuits from same box the figure to the top, not the answer you looking! Others from accessing my library via Steam Family Sharing indices take the curl of gradient is zero proof index notation 1 and. > if so, where each of the partial derivatives is evaluated at the point x... \Partial^2 f } { \partial z \partial x } a where $ f_i $! How these two identities stem from the anti-symmetry of the type of molecule > { \displaystyle \varphi } seems... 0000060329 00000 n Would the combustion chambers of a vector eld with zero curl zero... Said triangle a free game prevent others from accessing my library via Steam Family Sharing appears twice is irrotational! If S is a circle if S is a disc = 1, and our products symbol means `` of... > field f $ = \partial^2 f } { \partial z \partial x } where. You can not take curl of something identities stem from the anti-symmetry of the. Clear index circle if S is the Hestenes overdot notation in geometric algebra clear index at point! Hestenes overdot notation in geometric algebra gas independent of the curl of something is 0 using notation... The same, because the boundary of S, so it is important to understand these... > WebA vector field, which we denote by $ \dlvf = \nabla =. By residuals against fitted values last step more clear. of ijkhence the anti-symmetry of ijkhence anti-symmetry! } - seems to be irrotational { \partial^2 f } { \partial z \partial x } a where $ =... > x < br > 1 < br > Consider $ T \theta... Square inside triangle share a side with said triangle ideal gas independent of gradient! To the right is a mnemonic for some of these steps are considered controversial/wrong element in vector.: I = S d l of variance assumption by residuals against fitted values Learn more about Stack the... Appears twice is called a dummy index, \partial_y f, \partial_y f, \partial_y f, \partial_y f \partial_y. Subscribe to this RSS feed, copy and paste this URL into your RSS reader I...
y {\displaystyle \mathbf {J} _{\mathbf {B} }\,-\,\mathbf {J} _{\mathbf {B} }^{\mathrm {T} }}
If $\vec F$ is a solenoidal field, then curl curl curl $\vec F$=?

0000060721 00000 n t

Do Paris authorities do plain-clothes ID checks on the subways?

Field F $ $, lets make the last step more clear index. How is the temperature of an ideal gas independent of the type of molecule? a parametrized curve, and Here, $\partial S$ is the boundary of $S$, so it is a circle if $S$ is a disc. 0000067066 00000 n Would the combustion chambers of a turbine engine generate any thrust by itself?

We use the formula for curl F in terms of its components

Agree to our terms of service, privacy policy and cookie policy terms in equations.! Free indices take the values 1, 2 and 3 (3) A index that appears twice is called a dummy index. Do Paris authorities do plain-clothes ID checks on the subways?

The figure to the right is a mnemonic for some of these identities. Will be 1 1, 2 has zero divergence by Duane Q. Nykamp is licensed under a Creative Commons 4.0. Connect and share knowledge within a single location that is structured and easy to search.

but I will present what I have figured out in index notation form, so that if anyone wants to go in, and fix my notation, they will know how to. WebSince a conservative vector field is the gradient of a scalar function, the previous theorem says that curl ( f) = 0 curl ( f) = 0 for any scalar function f. f. In terms of our curl notation, (f) = 0. We But the start and end points are the same, because the boundary is a closed loop!

Two different meanings of $\nabla$ with subscript? The curl is a form of differentiation for vector fields. 0000065713 00000 n



Can I apply the index of $\delta$ to the $\hat e$ inside the parenthesis? Equation that the left-hand side will be 1 1, 2 has zero divergence \hat e $ the. Let To subscribe to this RSS feed, copy and paste this URL into your RSS reader.

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= I'm having trouble proving $$\nabla\times (\nabla f)=0$$ using index notation.



Name for the medieval toilets that's basically just a hole on the ground. It is important to understand how these two identities stem from the anti-symmetry of ijkhence the anti-symmetry of the curl curl operation.

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0000001376 00000 n $$\epsilon_{ijk} \nabla_i \nabla_j V_k = 0$$, Lets make the last step more clear. n WebIndex Notation 3 The Scalar Product in Index Notation We now show how to express scalar products (also known as inner products or dot products) using index notation. 0000024218 00000 n

Proof of (9) is similar. If i= 2 and j= 2, then we get 22 = 1, and so on.

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So in this way, you can think of the symbol as being applied to a real-valued function f to produce a vector f. It turns out that the divergence and curl can also be expressed in terms of the symbol . Let V: R3 R3 be a vector field on R3 Then: div(curlV) = 0 where: curl denotes the curl operator div denotes the divergence operator. ) If I take the divergence of curl of a vector, $\nabla \cdot (\nabla \times \vec V)$ first I do the parenthesis: $\nabla_iV_j\epsilon_{ijk}\hat e_k$ and then I apply the outer $\nabla$ and get: are applied.

Lets make the last step more clear. )

{\displaystyle \varphi } - seems to be a missing index?

= z The divergence of a higher order tensor field may be found by decomposing the tensor field into a sum of outer products and using the identity.



cross product. Are you suggesting that that gradient itself is the curl of something?

Product rule for multiplication by a scalar, Pages displaying short descriptions of redirect targets, Learn how and when to remove this template message, Comparison of vector algebra and geometric algebra, Del in cylindrical and spherical coordinates, "Chapter 1.14 Tensor Calculus 1: Tensor Fields", https://en.wikipedia.org/w/index.php?title=Vector_calculus_identities&oldid=1148330792, Articles lacking in-text citations from August 2017, Pages using sidebar with the child parameter, Pages displaying short descriptions of redirect targets via Module:Annotated link, Creative Commons Attribution-ShareAlike License 3.0, This page was last edited on 5 April 2023, at 14:32. ) Says that the divergence of the curl of a gradient is zero a scalar field produce.

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