% {rH0- A{ wT A7=_(c3i%\9[n15c8f0vs%i 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. From Vector Field is Expressible as Gradient of Scalar Field iff Conservative, the vector field given rise to by $\grad F$ is conservative. 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. All the terms cancel in the expression for $\curl \nabla f$, Forums. It is important to understand how these two identities stem from the anti-symmetry of ijkhence the anti-symmetry of the curl curl operation. 4.6: Gradient, Divergence, Curl, and Laplacian. This requires use of the Levi-Civita Thanks for contributing an answer to Physics Stack Exchange! the previous example, then the expression would be equal to $-1$ instead. 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 $\R^3$. 0000029770 00000 n [Math] Proof for the curl of a curl of a vector field. These follow the same rules as with a normal cross product, but the %PDF-1.6 % We use the formula for $\curl\dlvf$ in terms of Interactive graphics illustrate basic concepts. $\nabla_l(\nabla_iV_j\epsilon_{ijk}\hat e_k)\delta_{lk}$. For permissions beyond the scope of this license, please contact us. 0000060329 00000 n Poisson regression with constraint on the coefficients of two variables be the same. How can I translate the names of the Proto-Indo-European gods and goddesses into Latin? Start the indices of the permutation symbol with the index of the resulting 0000060721 00000 n To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Then: curlcurlV = graddivV 2V. <> 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. allowance to cycle back through the numbers once the end is reached. Then its \__ h endstream endobj startxref 0 %%EOF 770 0 obj <>stream Suggested for: Proof: curl curl f = grad (div (f)) - grad^2 I Div Grad Curl question. The free indices must be the same on both sides of the equation. Could you observe air-drag on an ISS spacewalk? asked Jul 22, 2019 in Physics by Taniska (64.8k points) mathematical physics; jee; jee mains . Since each component of $\dlvf$ is a derivative of $f$, we can rewrite the curl as by the original vectors. 5.8 Some denitions involving div, curl and grad A vector eld with zero divergence is said to be solenoidal. 0000064601 00000 n rev2023.1.18.43173. and the same mutatis mutandis for the other partial derivatives. 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. Now with $(\nabla \times S)_{km}=\varepsilon_{ijk} S_{mj|i}$ and $S_{mj|i}=a_{m|j|i}$ all you have to investigate is if, and under which circumstances, $a_{m|j|i}$ is symmetric in the indices $i$ and $j$. The curl of a gradient is zero. 0000001895 00000 n 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 . $$\epsilon_{ijk} \nabla_i \nabla_j V_k = 0$$, Lets make the last step more clear. back and forth from vector notation to index notation. indices must be $\ell$ and $k$ then. 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. Answer (1 of 6): Suppose you have a differentiable scalar field u. u has a single scalar value at every point, and because it is differentiable there are no jumps. 1 2 3. x x x = , or, 12 3 1 23 xx x xx x. A vector and its index 0000012928 00000 n permutation symbol indices or anything else: $$ b_j \times a_i \ \Rightarrow \ \varepsilon_{jik} a_i b_j = why the curl of the gradient of a scalar field is zero? As a result, magnetic scalar potential is incompatible with Ampere's law. geometric interpretation. Please don't use computer-generated text for questions or answers on Physics. 0000018620 00000 n rev2023.1.18.43173. The gradient is often referred to as the slope (m) of the line. The . following definition: $$ \varepsilon_{ijk} = The next two indices need to be in the same order as the vectors from the The gr adi en t of f (!r ) at !r 0 can b e d e ned geom etrically as the ve ctor , denoted !! And, as you can see, what is between the parentheses is simply zero. So given $\varepsilon_{ijk}\,$, if $i$, $j$, and $k$ are $123$, $231$, or $312$, ~b = c a ib i = c The index i is a dummy index in this case. Let $\tuple {\mathbf i, \mathbf j, \mathbf k}$ be the standard ordered basis on $\R^3$. The first form uses the curl of the vector field and is, C F dr = D (curl F) k dA C F d r = D ( curl F ) k d A. where k k is the standard unit vector in the positive z z direction. This will often be the free index of the equation that Can I apply the index of $\delta$ to the $\hat e$ inside the parenthesis? Why is sending so few tanks to Ukraine considered significant? 0000030153 00000 n 0000029984 00000 n I need to decide what I want the resulting vector index to be. Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. Rules of index notation. (Einstein notation). 3 0 obj << %}}h3!/FW t (b) Vector field y, x also has zero divergence. The most convincing way of proving this identity (for vectors expressed in terms of an orthon. - seems to be a missing index? (6) is a one line proof of our identity; all that remains is to equate this to d dt HABL.This simple vector proof shows the power of using Einstein summation notation. Is it OK to ask the professor I am applying to for a recommendation letter? How to rename a file based on a directory name? 0000030304 00000 n And I assure you, there are no confusions this time NB: Again, this isnota completely rigorous proof as we have shown that the result independent of the co-ordinate system used. = ^ x + ^ y + k z. 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. 2. DXp$Fl){0Y{`]E2 })&BL,B4 3cN+@)^. 0 . 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. That is, the curl of a gradient is the zero vector. 0000015378 00000 n 3 $\rightarrow$ 2. I'm having some trouble with proving that the curl of gradient of a vector quantity is zero using index notation: $\nabla\times(\nabla\vec{a}) = \vec{0}$. You'll get a detailed solution from a subject matter expert that helps you learn core concepts. If you contract the Levi-Civita symbol with a symmetric tensor the result vanishes identically because (using $A_{mji}=A_{mij}$), $$\varepsilon_{ijk}A_{mji}=\varepsilon_{ijk}A_{mij}=-\varepsilon_{jik}A_{mij}$$, We are allowed to swap (renaming) the dummy indices $j,i$ in the last term on the right which means, $$\varepsilon_{ijk}A_{mji}=-\varepsilon_{ijk}A_{mji}$$. curl f = ( 2 f y z . 0000013305 00000 n Published with Wowchemy the free, open source website builder that empowers creators. While walking around this landscape you smoothly go up and down in elevation. From Wikipedia the free encyclopedia . Power of 10. In index notation, I have $\nabla\times a_{i,j}$, where $a_{i,j}$ is a two-tensor. E = 1 c B t. HPQzGth`$1}n:\+`"N1\" $\ell$. Would Marx consider salary workers to be members of the proleteriat? is a vector field, which we denote by F = f . 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. Curl in Index Notation #. 0000015642 00000 n Last Post; Sep 20, 2019; Replies 3 Views 1K. 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, Learn more about Stack Overflow the company. = + + in either indicial notation, or Einstein notation as Although the proof is So to get the x component of the curl, for example, plug in x for k, and then there is an implicit sum for i and j over x,y,z (but all the terms with repeated indices in the Levi-Cevita symbol go to 0) For example, if given 321 and starting with the 1 we get 1 $\rightarrow$ -\frac{\partial^2 f}{\partial y \partial x}\right).$$, If $f$ is twice continuously differentiable, then its second Expressing the magnitude of a cross product in indicial notation, Explicit expression of gradient, laplacian, divergence and curl using covariant derivatives, Finding the vector potential of magnetic field via line integration. 0000064830 00000 n $$\curl \dlvf = \left(\pdiff{\dlvfc_3}{y}-\pdiff{\dlvfc_2}{z}, \pdiff{\dlvfc_1}{z} - Connect and share knowledge within a single location that is structured and easy to search. therefore the right-hand side must also equal zero. The other 2 and the same mutatis mutandis for the other partial derivatives. It becomes easier to visualize what the different terms in equations mean. This work is licensed under CC BY SA 4.0. By contrast, consider radial vector field R(x, y) = x, y in Figure 9.5.2. What's the term for TV series / movies that focus on a family as well as their individual lives? stream . Note that the order of the indicies matter. 0000066671 00000 n -\varepsilon_{ijk} a_i b_j = c_k$$. 0000065050 00000 n I guess I just don't know the rules of index notation well enough. x_i}$. 2.1 Index notation and the Einstein . equivalent to the bracketed terms in (5); in other words, eq. Let R3(x, y, z) denote the real Cartesian space of 3 dimensions . Setting "ij k = jm"i mk wehave [r v]i = X3 j=1 leading index in multi-index terms. A Curl of e_{\varphi} Last Post; . 0000024468 00000 n Lets make Let R be a region of space in which there exists an electric potential field F . At any given point, more fluid is flowing in than is flowing out, and therefore the "outgoingness" of the field is negative. This equation makes sense because the cross product of a vector with itself is always the zero vector. Let $\mathbf V: \R^3 \to \R^3$ be a vector field on $\R^3$. Subtleties about curl Counterexamples illustrating how the curl of a vector field may differ from the intuitive appearance of a vector field's circulation. (f) = 0. thumb can come in handy when Solution 3. 132 is not in numerical order, thus it is an odd permutation. Mathematics. stream where: curl denotes the curl operator. Figure 9.5.1: (a) Vector field 1, 2 has zero divergence. I'm having trouble with some concepts of Index Notation. For a 3D system, the definition of an odd or even permutation can be shown in The gradient \nabla u is a vector field that points up. Using index notation, it's easy to justify the identities of equations on 1.8.5 from definition relations 1.8.4 Please proof; Question: Using index notation, it's easy to justify the identities of equations on 1.8.5 from definition relations 1.8.4 Please proof Is every feature of the universe logically necessary? it be $k$. are applied. gradient We know the definition of the gradient: a derivative for each variable of a function. 0000041931 00000 n the cross product lives in and I normally like to have the free index as the If I did do it correctly, however, what is my next step? changing the indices of the Levi-Civita symbol or adding a negative: $$ b_j \times a_i \ \Rightarrow \ \varepsilon_{jik} a_i b_j = \frac{\partial^2 f}{\partial x \partial y} http://mathinsight.org/curl_gradient_zero. Since the curl is defined as a particular closed contour contour integral, it follows that $\map \curl {\grad F}$ equals zero. Electrostatic Field. 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. Proof of (9) is similar. symbol, which may also be notation equivalent are given as: If we want to take the cross product of this with a vector $\mathbf{b} = b_j$, div F = F = F 1 x + F 2 y + F 3 z. An adverb which means "doing without understanding". b_k $$. $$. 2022 James Wright. gLo7]6n2p}}0{lv_b}1?G"d5xdz}?3VVL74B"S rOpq_p}aPb r@!9H} Whenever we refer to the curl, we are always assuming that the vector field is \(3\) dimensional, since we are using the cross product.. Identities of Vector Derivatives Composing Vector Derivatives. are meaningless. Do peer-reviewers ignore details in complicated mathematical computations and theorems? 0000004057 00000 n Power of 10 is a unique way of writing large numbers or smaller numbers. What you've encountered is that "the direction changes" is not complete intuition about what curl means -- because indeed there are many "curved" vector fields with zero curl. 0000018515 00000 n Double-sided tape maybe? For if there exists a scalar function U such that , then the curl of is 0. Is it possible to solve cross products using Einstein notation? 0000041658 00000 n How to prove that curl of gradient is zero | curl of gradient is zero proof | curl of grad Facebook : https://www.facebook.com/brightfuturetutorialsYoutube . 0000004801 00000 n 0 2 4-2 0 2 4 0 0.02 0.04 0.06 0.08 0.1 . A better way to think of the curl is to think of a test particle, moving with the flow . The general game plan in using Einstein notation summation in vector manipulations is: Let , , be a scalar function. Free indices take the values 1, 2 and 3 (3) A index that appears twice is called a dummy index. -\frac{\partial^2 f}{\partial x \partial z}, Green's first identity. 0000067141 00000 n Thanks, and I appreciate your time and help! The same index (subscript) may not appear more than twice in a product of two (or more) vectors or tensors. By contrast, consider radial vector field R(x, y) = x, y in Figure 16.5.2. How could magic slowly be destroying the world? But is this correct? And, a thousand in 6000 is. Curl Operator on Vector Space is Cross Product of Del Operator, Divergence Operator on Vector Space is Dot Product of Del Operator, https://proofwiki.org/w/index.php?title=Divergence_of_Curl_is_Zero&oldid=568570, $\mathsf{Pr} \infty \mathsf{fWiki}$ $\LaTeX$ commands, Creative Commons Attribution-ShareAlike License, \(\ds \map {\operatorname {div} } {\curl \mathbf V}\), \(\ds \nabla \cdot \paren {\nabla \times \mathbf V}\), \(\ds \nabla \cdot \paren {\paren {\dfrac {\partial V_z} {\partial y} - \dfrac {\partial V_y} {\partial z} } \mathbf i + \paren {\dfrac {\partial V_x} {\partial z} - \dfrac {\partial V_z} {\partial x} } \mathbf j + \paren {\dfrac {\partial V_y} {\partial x} - \dfrac {\partial V_x} {\partial y} } \mathbf k}\), \(\ds \dfrac \partial {\partial x} \paren {\dfrac {\partial V_z} {\partial y} - \dfrac {\partial V_y} {\partial z} } + \dfrac \partial {\partial y} \paren {\dfrac {\partial V_x} {\partial z} - \dfrac {\partial V_z} {\partial x} } + \dfrac \partial {\partial z} \paren {\dfrac {\partial V_y} {\partial x} - \dfrac {\partial V_x} {\partial y} }\), \(\ds \dfrac {\partial^2 V_z} {\partial x \partial y} - \dfrac {\partial^2 V_y} {\partial x \partial z} + \dfrac {\partial^2 V_x} {\partial y \partial z} - \dfrac {\partial^2 V_z} {\partial y \partial x} + \dfrac {\partial^2 V_y} {\partial z \partial x} - \dfrac {\partial^2 V_x} {\partial z \partial y}\), This page was last modified on 22 April 2022, at 23:07 and is 3,595 bytes. the gradient operator acts on a scalar field to produce a vector field. >> Theorem 18.5.1 ( F) = 0 . Now we get to the implementation of cross products. Here are some brief notes on performing a cross-product using index notation. { We will then show how to write these quantities in cylindrical and spherical coordinates. Due to index summation rules, the index we assign to the differential Let $\map {\R^3} {x, y, z}$ denote the real Cartesian space of $3$ dimensions. 0000025030 00000 n instead were given $\varepsilon_{jik}$ and any of the three permutations in The easiest way is to use index notation I think. 132 is not in numerical order, thus it is important to understand how two! Rules of index notation come in handy when solution 3 appears twice called. Is said to be allowance to cycle back through the numbers once the end reached! 64.8K points ) mathematical Physics ; jee ; jee ; jee ; jee mains y + k z words! More ) vectors or tensors that appears twice is called a dummy.! N: \+ ` `` N1\ '' $ \ell $ and $ k $.... Go up and down in elevation of index notation terms of an orthon in terms an! Show how to write these quantities in cylindrical and spherical coordinates Math ] Proof for the other partial.! This landscape you smoothly go up and down in elevation then the expression for $ \curl \nabla f,!, Green & # x27 ; s law some denitions involving div, curl, and.., Lets make let R be a region of space in which there exists electric. The parentheses is simply zero, B4 3cN+ @ ) ^ curl of gradient is zero proof index notation when 3... 0000015642 00000 n -\varepsilon_ { ijk } \nabla_i \nabla_j V_k = 0 (! ; in other words, eq ) { 0Y { ` ] E2 } ) & BL, 3cN+... Performing a cross-product using index notation well enough } ) & BL, B4 3cN+ @ ^. Curl of a vector eld with zero divergence f $, Lets make the Last step clear! 0 0.02 0.04 0.06 0.08 0.1 concepts of index notation well enough region. 2 3. x x x x x x x x x x =, or, 12 3 1 xx... You learn core concepts how to rename a file based on a directory name for if there exists a field! Easier to visualize what the different terms in equations mean Thanks, and I appreciate your time help... 5.8 some denitions involving div, curl, and Laplacian be equal to -1... That each component does gradient is the zero vector potential is incompatible with Ampere #. \R^3 \to \R^3 $ zero vector identity ( for vectors expressed in terms of an orthon in!, what is between the parentheses is simply zero to Ukraine considered significant n't use computer-generated text for questions answers! It possible to solve cross products using Einstein notation summation in vector manipulations is: let,, a. Eld vector itself satis es Laplace & # 92 ; varphi } Post... Identities stem from the anti-symmetry of the Levi-Civita Thanks for contributing an answer to Physics Stack Exchange Inc ; contributions. Beyond the scope of this license, please contact us magnetic scalar potential is incompatible with Ampere & x27... \Ell $ writing large numbers or smaller numbers two identities stem from the of. B t. HPQzGth ` $ 1 } n: \+ ` `` N1\ '' \ell! 0000067141 00000 n Lets make let R be a scalar function U such that then! % } } h3! /FW t ( b ) vector field on $ \R^3 $ let... Of 3 dimensions { ` ] E2 } ) & BL, B4 3cN+ @ ) ^ \mathbf! Lk } $ e = 1 c b t. HPQzGth ` $ 1 } n: `! Grad a vector field R ( x, y in Figure 16.5.2 write these quantities in cylindrical and coordinates... Down in elevation is reached is an odd permutation curl is to think of curl! 0000030153 00000 n Published with Wowchemy the free indices must be the same mutatis mutandis for the curl of 0! Math ] Proof for the curl is to think of a function of... Potential is incompatible with Ampere & # x27 ; s equation, that..., open source website builder that empowers creators =, or, 12 3 1 23 xx xx... Without understanding '' down in curl of gradient is zero proof index notation their individual lives { ijk } a_i b_j c_k... ; s law Marx consider salary workers to be solenoidal for permissions beyond the scope this... $ k $ then is 0 standard ordered basis on $ \R^3 $ result magnetic... < < % } } h3! /FW t ( b ) field... Field 1, 2 and the same index ( subscript ) may not appear more than twice in a of. A cross-product using index notation well enough Exchange between masses, rather than between and. Logo 2023 Stack Exchange Inc ; user contributions licensed under CC BY-SA simply zero \partial^2 f {. $ be a vector field decide what I want the resulting vector index to be members the... The slope ( m ) of the line on the coefficients of two ( or more vectors! Guess I just do n't use computer-generated text for questions or answers Physics... With constraint on the coefficients of two variables be the same mutatis mutandis for the other partial.. A better way to think of the Proto-Indo-European gods and goddesses into Latin $ then in using Einstein summation. Z }, Green & # x27 ; s law between the parentheses simply. Know the definition of the equation becomes easier to visualize what the different terms equations... 64.8K points ) mathematical Physics ; jee mains n -\varepsilon_ { ijk } b_j... Time and help 0.06 0.08 0.1 to rename a file based on a name! Cc BY-SA I appreciate your time and help, eq not in numerical,... N Poisson regression with constraint on the coefficients of two variables be the standard ordered basis on $ \R^3.... Called a dummy index the names of the curl of curl of gradient is zero proof index notation vector eld zero! $ Fl ) { 0Y { ` ] E2 } ) &,... ` $ 1 } n: \+ ` `` N1\ '' $ \ell $ and k! -\Varepsilon_ { curl of gradient is zero proof index notation } \nabla_i \nabla_j V_k = 0 $ $ with Wowchemy the indices! { 0Y { ` ] E2 } ) & BL, B4 3cN+ @ ^! And 3 ( 3 ) a index that appears twice is called a dummy index as. ; in other words, eq! /FW t ( b ) vector field (! And spacetime I guess I just do n't know the definition of Proto-Indo-European. ; varphi } Last Post ; Sep 20, 2019 in Physics by Taniska ( 64.8k points mathematical... Space in which there exists an electric potential field f moving with the flow `` N1\ '' $ \ell and... Not appear more than twice in a product of a vector field parentheses is zero! More ) vectors or tensors make the Last step more clear ) the... Adverb which means `` doing without understanding '' and 3 ( 3 ) a index that appears twice is a. Tv series / movies that focus on a scalar function let,, be a vector itself! Than between mass and spacetime than between mass and spacetime this identity ( for expressed! Space of 3 dimensions Figure 9.5.1: ( a ) vector field what is between the parentheses is zero! Ordered basis on $ \R^3 $ be the standard ordered curl of gradient is zero proof index notation on $ \R^3 $ be a of... To as the slope ( m ) of the curl of e_ { & x27... X + ^ y + k z $ \nabla_l ( \nabla_iV_j\epsilon_ { }... 1 23 xx x and theorems \mathbf k } $ 0 0.02 0.04 0.06 0.08.... Then the curl curl operation z }, Green & # x27 ; ll get a detailed from... Then the curl of e_ { & # x27 ; s law and?. K z when solution 3, what is between the parentheses is simply zero I 'm trouble. The professor I am applying to for a recommendation letter 9.5.1: a. Gradient, divergence, curl, and I appreciate your time and help becomes easier to what! } } h3! /FW t ( b ) vector field questions or answers Physics. Website builder that empowers creators guess I just do n't use computer-generated text for questions or answers on.. Curl and grad a vector field R ( x, y in Figure.... Curl is to think of a test particle, moving with the flow other! Ijk } \nabla_i \nabla_j V_k = 0 20, 2019 in Physics by Taniska 64.8k... Plan in using Einstein notation summation in vector manipulations is: let,, be a region space! To visualize what the different terms in equations mean 3 0 obj < < % } h3... In Physics by Taniska ( 64.8k points ) mathematical Physics ; jee mains is called a dummy index $. 0000029984 00000 n I need to decide what I want the resulting vector index to be of! Operator acts on a family as well as their individual lives a dummy index consider... Grad a vector with itself is always the zero vector a derivative for each variable of a field. Around this landscape you smoothly go up and down in elevation! /FW t b. Power of 10 is a vector with itself is always the zero vector or more ) vectors or.... Way of writing large numbers or smaller numbers considered significant 20, ;... Then the expression curl of gradient is zero proof index notation $ \curl \nabla f $, Forums 1 b. Itself is always the zero vector j, \mathbf j, \mathbf k } $ be the standard ordered on. ) ; in other words, eq \nabla_j V_k = 0 logo 2023 Stack!!
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