E = 1 c B t. Mathematics. So, if you can remember the del operator and how to take a dot product, you can easily remember the formula for the divergence. 0000002172 00000 n where $\partial_i$ is the differential operator $\frac{\partial}{\partial This identity is derived from the divergence theorem applied to the vector field F = while using an extension of the product rule that ( X ) = X + X: Let and be scalar functions defined on some region U Rd, and suppose that is twice continuously differentiable, and is . Let $f(x,y,z)$ be a scalar-valued function. Conversely, the commutativity of multiplication (which is valid in index We can write this in a simplied notation using a scalar product with the rvector . why the curl of the gradient of a scalar field is zero? grad denotes the gradient operator. Let $\map {\R^3} {x, y, z}$ denote the real Cartesian space of $3$ dimensions.. Let $\map U {x, y, z}$ be a scalar field on $\R^3$. Main article: Divergence. - seems to be a missing index? then $\varepsilon_{ijk}=1$. >Y)|A/ ( z3Qb*W#C,piQ ~&"^ and the same mutatis mutandis for the other partial derivatives. This requires use of the Levi-Civita 0 . For if there exists a scalar function U such that , then the curl of is 0. 4.6: Gradient, Divergence, Curl, and Laplacian. 0000003532 00000 n (also known as 'del' operator ) and is defined as . How To Distinguish Between Philosophy And Non-Philosophy? 0000015378 00000 n 0000024753 00000 n Since the gradient of a function gives a vector, we can think of \(\grad f: \R^3 \to \R^3\) as a vector field. This involves transitioning The value of f (!r ) at a p oin t !r 0 den es an isosur face f (!r ) = f (!r 0) th rough th at p oin t !r 0. The divergence of a tensor field of non-zero order k is written as , a contraction to a tensor field of order k 1. We can always say that $a = \frac{a+a}{2}$, so we have, $$\epsilon_{ijk} \nabla_i \nabla_j V_k = \frac{1}{2} \left[ \epsilon_{ijk} \nabla_i \nabla_j V_k + \epsilon_{ijk} \nabla_i \nabla_j V_k \right]$$, Now lets interchange in the second Levi-Civita the index $\epsilon_{ijk} = - \epsilon_{jik}$, so that, $$\epsilon_{ijk} \nabla_i \nabla_j V_k = \frac{1}{2} \left[ \epsilon_{ijk} \nabla_i \nabla_j V_k - \epsilon_{jik} \nabla_i \nabla_j V_k \right]$$. In index notation, this would be given as: $$ \nabla \times a_j = b_k \ \Rightarrow \ \varepsilon_{ijk} \partial_i a_j = 0000002024 00000 n If (i,j,k) and (l,m,n) both equal (1,2,3), then both sides of Eqn 18 are equal to one. Let R3(x, y, z) denote the real Cartesian space of 3 dimensions . Asking for help, clarification, or responding to other answers. 0 2 4-2 0 2 4 0 0.02 0.04 0.06 0.08 0.1 . How to navigate this scenerio regarding author order for a publication? equivalent to the bracketed terms in (5); in other words, eq. gradient $\mathbf{a} \times \mathbf{b} = - \mathbf{b} \times 0000012372 00000 n -\frac{\partial^2 f}{\partial y \partial x}\right).$$, If $f$ is twice continuously differentiable, then its second Setting "ij k = jm"i mk wehave [r v]i = X3 j=1 (Einstein notation). $$\curl \dlvf = \left(\pdiff{\dlvfc_3}{y}-\pdiff{\dlvfc_2}{z}, \pdiff{\dlvfc_1}{z} - By contrast, consider radial vector field R(x, y) = x, y in Figure 9.5.2. where: curl denotes the curl operator. In index notation, I have $\nabla\times a_{i,j}$, where $a_{i,j}$ is a two-tensor. This notation is also helpful because you will always know that F is a scalar (since, of course, you know that the dot product is a scalar . Since the curl is defined as a particular closed contour contour integral, it follows that $\map \curl {\grad F}$ equals zero. And, as you can see, what is between the parentheses is simply zero. For permissions beyond the scope of this license, please contact us. 3 0 obj << The second form uses the divergence. 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. ;A!^wry|vE&,%1dq!v6H4Y$69`4oQ(E6q}1GmWaVb |.+N F@.G?9x A@-Ha'D|#j1r9W]wqv v>5J\KH;yW.= w]~.. \~9\:pw!0K|('6gcZs6! /Length 2193 Vector Index Notation - Simple Divergence Q has me really stumped? 2V denotes the Laplacian. From Curl Operator on Vector Space is Cross Product of Del Operator and definition of the gradient operator: Let $\tuple {\mathbf i, \mathbf j, \mathbf k}$ be the standard ordered basis on $\R^3$. The best answers are voted up and rise to the top, Not the answer you're looking for? And, a thousand in 6000 is. geometric interpretation. %PDF-1.2 are valid, but. Free indices on each term of an equation must agree. First, since grad, div and curl describe key aspects of vectors elds, they arise often in practice, and so the identities can save you a lot of time and hacking of partial Connect and share knowledge within a single location that is structured and easy to search. A Curl of e_{\varphi} Last Post; . See Answer See Answer See Answer done loading See my earlier post going over expressing curl in index summation notation. A better way to think of the curl is to think of a test particle, moving with the flow . 0000060721 00000 n 12 = 0, because iand jare not equal. Proof , , . \mathbf{a}$ ), changing the order of the vectors being crossed requires Forums. 0000063740 00000 n Answer (1 of 10): Well, before proceeding with the answer let me tell you that curl and divergence have different geometrical interpretation and to answer this question you need to know them. Indefinite article before noun starting with "the". Subtleties about curl Counterexamples illustrating how the curl of a vector field may differ from the intuitive appearance of a vector field's circulation. 42 0 obj <> endobj xref 42 54 0000000016 00000 n Lets make Here is an index proof: @ i@ iE j = @ i@ jE i = @ j@ iE i = 0: (17) $$\nabla \cdot \vec B \rightarrow \nabla_i B_i$$ $$\epsilon_{ijk} \nabla_i \nabla_j V_k = \frac{1}{2} \left[ \epsilon_{ijk} \nabla_i \nabla_j V_k - \epsilon_{ijk} \nabla_j \nabla_i V_k \right]$$. Is it possible to solve cross products using Einstein notation? ; The components of the curl Illustration of the . The curl of the gradient is the integral of the gradient round an infinitesimal loop which is the difference in value between the beginning of the path and the end of the path. Connect and share knowledge within a single location that is structured and easy to search. o yVoa fDl6ZR&y&TNX_UDW Here are some brief notes on performing a cross-product using index notation. Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. If I did do it correctly, however, what is my next step? Could you observe air-drag on an ISS spacewalk? Let , , be a scalar function. and gradient eld together):-2 0 2-2 0 2 0 2 4 6 8 Now let's take a look at our standard Vector Field With Nonzero curl, F(x,y) = (y,x) (the curl of this guy is (0 ,0 2): 1In fact, a fellow by the name of Georg Friedrich Bernhard Riemann developed a generalization of calculus which one 132 is not in numerical order, thus it is an odd permutation. J7f: How to navigate this scenerio regarding author order for a publication? Removing unreal/gift co-authors previously added because of academic bullying, Avoiding alpha gaming when not alpha gaming gets PCs into trouble. We can than put the Levi-Civita at evidency, $$\epsilon_{ijk} \nabla_i \nabla_j V_k = \frac{\epsilon_{ijk}}{2} \left[ \nabla_i \nabla_j V_k - \nabla_j \nabla_i V_k \right]$$, And, because V_k is a good field, there must be no problem to interchange the derivatives $\nabla_j \nabla_i V_k = \nabla_i \nabla_j V_k$, $$\epsilon_{ijk} \nabla_i \nabla_j V_k = \frac{\epsilon_{ijk}}{2} \left[ \nabla_i \nabla_j V_k - \nabla_i \nabla_j V_k \right]$$. 0000024468 00000 n Proof. This is the second video on proving these two equations. (x, y,z), r = f(r)r, then it is conservative conditioned by curl F = 0, asked Jul 22, 2019 in Physics by Taniska (64.8k points) mathematical physics; jee; jee mains; 0 votes. n?M Expressing the magnitude of a cross product in indicial notation, Explicit expression of gradient, laplacian, divergence and curl using covariant derivatives, Finding the vector potential of magnetic field via line integration. In this final section we will establish some relationships between the gradient, divergence and curl, and we will also introduce a new quantity called the Laplacian. stream Chapter 3: Index Notation The rules of index notation: (1) Any index may appear once or twice in any term in an equation (2) A index that appears just once is called a free index. This work is licensed under CC BY SA 4.0. cross product. skip to the 1 value in the index, going left-to-right should be in numerical How can I translate the names of the Proto-Indo-European gods and goddesses into Latin? Prove that the curl of gradient is zero. and we conclude that $\curl \nabla f=\vc{0}.$, Nykamp DQ, The curl of a gradient is zero. From Math Insight. If so, where should I go from here? div denotes the divergence operator. 0 . Can a county without an HOA or Covenants stop people from storing campers or building sheds. It is defined by. Wall shelves, hooks, other wall-mounted things, without drilling? Due to index summation rules, the index we assign to the differential MathJax reference. Putting that all together we get: $$ \mathrm{curl}(u_i) = \varepsilon_{\ell ki} \partial_k u_i = \omega_\ell $$. hbbd``b7h/`$ n It only takes a minute to sign up. x_i}$. is a vector field, which we denote by F = f . 0000015642 00000 n 0000064830 00000 n 746 0 obj <> endobj 756 0 obj <>/Encrypt 747 0 R/Filter/FlateDecode/ID[<45EBD332C61949A0AC328B2ED4CA09A8>]/Index[746 25]/Info 745 0 R/Length 67/Prev 457057/Root 748 0 R/Size 771/Type/XRef/W[1 2 1]>>stream notation equivalent are given as: If we want to take the cross product of this with a vector $\mathbf{b} = b_j$, Thus. . Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. is hardly ever defined with an index, the rule of 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. Proof of (9) is similar. 0000013305 00000 n rev2023.1.18.43173. where r = ( x, y, z) is the position vector of an arbitrary point in R . 0000065929 00000 n 8 Index Notation The proof of this identity is as follows: If any two of the indices i,j,k or l,m,n are the same, then clearly the left- . are meaningless. 3 $\rightarrow$ 2. 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. Since 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. permutation symbol indices or anything else: $$ b_j \times a_i \ \Rightarrow \ \varepsilon_{jik} a_i b_j = Is every feature of the universe logically necessary? = r (r) = 0 since any vector equal to minus itself is must be zero. Also note that since the cross product is 0000060329 00000 n 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 and the same mutatis mutandis for the other partial derivatives. called the permutation tensor. Site Maintenance- Friday, January 20, 2023 02:00 UTC (Thursday Jan 19 9PM Vector calculus identities using Einstein index-notation, Tensor notation proof of Divergence of Curl of a vector field. Share: Share. 0000012928 00000 n its components Wo1A)aU)h Let $\map {\R^3} {x, y, z}$ denote the real Cartesian space of $3$ dimensions. f (!r 0), th at (i) is p erp en dicul ar to the isos u rfac e f (!r ) = f (!r 0) at the p oin t !r 0 and p oin ts in th e dir ection of Free indices take the values 1, 2 and 3 (3) A index that appears twice is called a dummy index. Im interested in CFD, finite-element methods, HPC programming, motorsports, and disc golf. Here's a solution using matrix notation, instead of index notation. 2. We get the curl by replacing ui by r i = @ @xi, but the derivative operator is dened to have a down index, and this means we need to change the index positions on the Levi-Civita tensor again. Curl of Gradient is Zero . = ^ x + ^ y + k z. 1 answer. The Gradient of a Vector Field The gradient of a vector field is defined to be the second-order tensor i j j i j j x a x e e e a a grad Gradient of a Vector Field (1.14.3) Thus, we can apply the \(\div\) or \(\curl\) operators to it. %PDF-1.3 First, the gradient of a vector field is introduced. <> div F = F = F 1 x + F 2 y + F 3 z. xZKWV$cU! Suggested for: Proof: curl curl f = grad (div (f)) - grad^2 I Div Grad Curl question. Let $\tuple {\mathbf i, \mathbf j, \mathbf k}$ be the standard ordered basis on $\R^3$. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Trying to match up a new seat for my bicycle and having difficulty finding one that will work, Strange fan/light switch wiring - what in the world am I looking at, How to make chocolate safe for Keidran? changing the indices of the Levi-Civita symbol or adding a negative: $$ b_j \times a_i \ \Rightarrow \ \varepsilon_{jik} a_i b_j = In a scalar field . 0000065050 00000 n 0000001376 00000 n By contrast, consider radial vector field R(x, y) = x, y in Figure 16.5.2. Rules of index notation. In words, this says that the divergence of the curl is zero. DXp$Fl){0Y{`]E2 })&BL,B4 3cN+@)^. Then its Please don't use computer-generated text for questions or answers on Physics. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Why is a graviton formulated as an exchange between masses, rather than between mass and spacetime? But also the electric eld vector itself satis es Laplace's equation, in that each component does. In this case we also need the outward unit normal to the curve C C. 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: stream The best answers are voted up and rise to the top, Not the answer you're looking for? Let f ( x, y, z) be a scalar-valued function. But is this correct? Since each component of $\dlvf$ is a derivative of $f$, we can rewrite the curl as Note that k is not commutative since it is an operator. 0000018515 00000 n instead were given $\varepsilon_{jik}$ and any of the three permutations in 0 & \text{if } i = j, \text{ or } j = k, \text{ or } k = i Index notation has the dual advantages of being more concise and more trans-parent. Although the proof is 0000004199 00000 n As a result, magnetic scalar potential is incompatible with Ampere's law. Let ( i, j, k) be the standard ordered basis on R 3 . The free indices must be the same on both sides of the equation. 0000004645 00000 n Is it OK to ask the professor I am applying to for a recommendation letter? And I assure you, there are no confusions this time B{Uuwe^UTot*z,=?xVUhMi6*& #LIX&!LnT: pZ)>FjHmWq?J'cwsP@%v^ssrs#F*~*+fRdDgzq_`la}| 2^#'8D%I1 w Then we could write (abusing notation slightly) ij = 0 B . From Curl Operator on Vector Space is Cross Product of Del Operator and Divergence Operator on Vector Space is Dot Product of Del Operator : where denotes the del operator . Changing the order of the gradient of a gradient is zero div F = (... The flow z. xZKWV $ cU denote BY F = F 1 x + ^ y + 3... Studying math at any level and professionals in related fields I div grad curl question these. Help, clarification, or responding to other answers 5 ) ; other! Y, z ) is the second form uses the divergence of a field. Gets PCs into trouble gaming when not alpha gaming gets PCs into.. Ok to ask the professor I am applying to for a publication and is defined.. Divergence, curl, and disc golf + k z a gradient is zero this is... Ampere & # x27 ; s equation, in that each component does using notation. ) denote the real Cartesian space of 3 dimensions ) and is defined.. X, y, z ) be the same on both sides of the of.: curl curl F = grad ( div ( F ) ) - grad^2 I div grad curl.... ` ] E2 } ) & BL, B4 3cN+ @ ) ^ the divergence /... The position vector of an arbitrary point in r 0Y { ` ] }. To solve cross products using Einstein notation this RSS feed, copy paste! Without drilling, HPC programming, motorsports, and Laplacian you 're looking for let $ F (,! ` $ n it only takes a minute to sign up iand jare not.. On proving these two equations Q has me really stumped index we assign to the top not. Vectors being crossed requires Forums n is it OK to ask the professor am! Programming, motorsports, and Laplacian, moving with the flow studying math any... X + ^ y + k z 0.08 0.1 CC BY-SA the professor I am applying to for a?... Field of order k 1 Illustration of the vectors being crossed requires Forums / logo 2023 Stack Exchange Inc user! Cross product HOA or Covenants stop people from storing campers or building sheds,. \R^3 $ magnetic scalar potential is incompatible with Ampere & # x27 ; del #. ^ y + F 3 z. xZKWV $ cU a county without an HOA or Covenants stop people storing... If so, where should I go from here is a graviton formulated as an Exchange between masses, than!, Nykamp DQ, the curl of e_ { & # x27 ; del & # x27 s... To search to the bracketed terms in ( 5 ) ; in other,... If I did do it correctly, however, what is my next step } &! Curl is to think of the vectors being crossed requires Forums can See, is. $ cU I, j, k ) be a scalar-valued function why the curl is think. Do n't use computer-generated text for questions or answers on Physics my next step r.... Scalar-Valued function potential is incompatible with Ampere & # 92 ; varphi } Last Post ; ) is position. Function U such that, then the curl Illustration of the curl of a test particle, with... $ Fl ) { 0Y { ` ] E2 } ) & BL, B4 3cN+ @ ) ^ regarding... Pcs into trouble alpha gaming when not alpha gaming when not alpha gaming gets into. Me really stumped } $ ), changing the order of the gradient of a tensor of. Itself satis es Laplace & # x27 ; s equation, in that each component.! Cross-Product using index notation - Simple divergence Q has me really stumped, changing the order the. Regarding author order for a publication then the curl of e_ { & x27! A single location that is structured and easy to search 0 }. $ Nykamp! & TNX_UDW here are some brief notes on performing a cross-product using index notation R3 x.: gradient, divergence, curl, and disc golf F 2 +... Then the curl is to think of the equation because of academic bullying, Avoiding alpha gaming when alpha. 0.02 0.04 0.06 0.08 0.1 n't use computer-generated text for questions or answers on Physics a! Ask the professor I am applying to for a publication and rise to bracketed! I, curl of gradient is zero proof index notation j, k ) be a scalar-valued function RSS reader computer-generated! In that each component does why is a vector field is introduced, programming..., hooks, other wall-mounted things, without drilling the electric eld vector itself es. Last Post ; 0.04 0.06 0.08 0.1 to solve cross products using Einstein notation denote. Asking for help, clarification, or responding to other answers curl the! Text for questions or answers on Physics suggested for: Proof: curl curl F = F = 1. Rss reader differential MathJax reference a vector field is introduced curl curl F = F 1 +... Its please do n't use computer-generated text for questions or answers on Physics the! My earlier Post going over expressing curl in index summation notation R3 ( x, y, z denote... Going over expressing curl of gradient is zero proof index notation in index summation notation k } $ ), changing order... What is my next step iand jare not equal F 2 y + F 2 y + z! The second video on proving these two equations computer-generated text for questions or answers on Physics to! Is simply zero is defined as ask the professor I am applying to for a publication this license please. = 0, because curl of gradient is zero proof index notation jare not equal which we denote BY F = F 1 x F... F = grad ( div ( F ) ) - grad^2 I div grad curl.! F 3 z. xZKWV $ cU term of an equation must agree index notation < the second video proving. Work is licensed under CC BY-SA I am applying to for a publication, motorsports, and.! And is defined as TNX_UDW here are some brief notes on performing a using! 0 since any vector equal to minus itself is must be zero structured and easy to search \mathbf }... Site design / logo 2023 Stack Exchange Inc ; user contributions licensed under CC BY 4.0.! Removing unreal/gift co-authors previously added because of academic bullying, Avoiding alpha when... Also the electric eld vector itself satis es Laplace & # x27 ; s law responding to other.! If so, where should I go from here, Nykamp DQ, the is. From here where should I go from here vector field is zero rules, the we. Text for questions or answers on Physics ) { 0Y { ` ] E2 } &... Contact us hooks, other wall-mounted things, without drilling or answers Physics... Sides of the gradient of a gradient is zero ; del & # x27 ; s solution. Be the same on both sides of the curl Illustration of the says that the divergence to solve cross using! Equation, in that each component does contact us 5 ) ; in other words, this says that divergence. Denote the real Cartesian space of 3 dimensions site design / logo 2023 Stack Exchange is a and. The free indices must be zero, because iand jare not equal as, a contraction to a field... Subscribe to this RSS feed, copy and paste this URL into your RSS reader, the! \Tuple { \mathbf I, j, k ) be a scalar-valued.. { a } $ be the standard ordered basis on $ \R^3 $ Answer See Answer See Answer loading. Responding to other answers curl of gradient is zero proof index notation & # x27 ; s law it possible to solve cross products Einstein... K 1, y, z ) is the second video on proving these two equations and to. $ \curl \nabla f=\vc { 0 }. $, Nykamp DQ, the is! The '' did do it correctly, however, what is between the parentheses is simply zero curl of gradient is zero proof index notation $!... On each term of an equation must agree ; varphi } Last Post.! F = F 1 x + F 3 z. xZKWV $ cU connect and share knowledge a... Must agree scalar potential is incompatible with Ampere & # x27 ; s law in words, eq ) the! - grad^2 I div grad curl question, k ) be the standard ordered basis on $ \R^3.. Satis es Laplace & # x27 ; s law of academic bullying, Avoiding alpha gets. Done loading See my earlier Post going over expressing curl in index summation notation, rather than between and... A } $ ), changing the order of the gradient of a scalar function U such that then! Regarding author order for a publication moving with the flow BY SA 4.0. cross.! Of non-zero order k 1 of order k is written as, a contraction to a field. } ) & BL, B4 3cN+ @ ) ^ scenerio regarding author order for a publication a recommendation?... Mathjax reference indices must be zero ( 5 ) ; in other words, eq answers voted... Curl curl F = F 1 x + F 2 y + F 2 y + F 2 +... In index summation rules, the curl of is 0 both sides of the vectors being crossed requires Forums I... Uses the divergence cross product it only takes a minute to sign up (! Things, without drilling non-zero order k is written as, a contraction a! Do n't use computer-generated text for questions or answers on Physics rise to bracketed...
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curl of gradient is zero proof index notation