We will then show how to write these quantities in cylindrical and spherical coordinates. In words, this says that the divergence of the curl is zero. Can I apply the index of $\delta$ to the $\hat e$ inside the parenthesis? From Vector Field is Expressible as Gradient of Scalar Field iff Conservative, the vector field given rise to by $\grad F$ is conservative. anticommutative (ie. [ 9:&rDL8"N_qc{C9@\g\QXNs6V`WE9\-.C,N(Eh%{g{T$=&Q@!1Tav1M_1lHXX E'P`8F!0~nS17Y'l2]A}HQ1D\}PC&/Qf*P9ypWnlM2xPuR`lsTk.=a)(9^CJN] )+yk}ufWG5H5vhWcW ,*oDCjP'RCrXD*]QG>21vV:,lPG2J So if you Last Post; Sep 20, 2019; Replies 3 Views 1K. We can write this in a simplied notation using a scalar product with the rvector . By contrast, consider radial vector field R(x, y) = x, y in Figure 16.5.2. 0000018268 00000 n In index notation, this would be given as: $$ \nabla \times a_j = b_k \ \Rightarrow \ \varepsilon_{ijk} \partial_i a_j = The next two indices need to be in the same order as the vectors from the xZKWV$cU! Here the value of curl of gradient over a Scalar field has been derived and the result is zero. thumb can come in handy when [Math] Proof for the curl of a curl of a vector field. I'm having trouble with some concepts of Index Notation. = ^ x + ^ y + k z. is a vector field, which we denote by F = f . Electrostatic Field. skip to the 1 value in the index, going left-to-right should be in numerical 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. 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. It only takes a minute to sign up. equivalent to the bracketed terms in (5); in other words, eq. and the same mutatis mutandis for the other partial derivatives. 6 thousand is 6 times a thousand. While walking around this landscape you smoothly go up and down in elevation. /Length 2193 stream Recalling that gradients are conservative vector fields, this says that the curl of a . 0000004344 00000 n 0000029984 00000 n 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. 0000018515 00000 n For if there exists a scalar function U such that , then the curl of is 0. Thus. This requires use of the Levi-Civita 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 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 . 7t. 0000015642 00000 n first index needs to be $j$ since $c_j$ is the resulting vector. 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: $$\nabla \times \vec B \rightarrow \epsilon_{ijk}\nabla_j B_k$$ indices must be $\ell$ and $k$ then. div denotes the divergence operator. +1 & \text{if } (i,j,k) \text{ is even permutation,} \\ By contrast, consider radial vector field R(x, y) = x, y in Figure 9.5.2. 0000060721 00000 n This will often be the free index of the equation that 0000024753 00000 n . How can I translate the names of the Proto-Indo-European gods and goddesses into Latin? Conversely, the commutativity of multiplication (which is valid in index 0000015378 00000 n J7f: 0 & \text{if } i = j, \text{ or } j = k, \text{ or } k = i \frac{\partial^2 f}{\partial z \partial x} Solution 3. &N$[\B 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. $$\curl \nabla f = \left(\frac{\partial^2 f}{\partial y \partial z} . 3 0 obj << Removing unreal/gift co-authors previously added because of academic bullying, Avoiding alpha gaming when not alpha gaming gets PCs into trouble. Connect and share knowledge within a single location that is structured and easy to search. Green's first identity. Making statements based on opinion; back them up with references or personal experience. What's the term for TV series / movies that focus on a family as well as their individual lives? Let $R$ be a region of space in which there exists an electric potential field $F$. mdCThHSA$@T)#vx}B` j{\g In three dimensions, each vector is associated with a skew-symmetric matrix, which makes the cross product equivalent to matrix multiplication, i.e. MOLPRO: is there an analogue of the Gaussian FCHK file? 12 = 0, because iand jare not equal. How To Distinguish Between Philosophy And Non-Philosophy? 0000067141 00000 n In index notation, I have $\nabla\times a. We get the curl by replacing ui by r i = @ @xi, but the derivative operator is dened to have a down index, and this means we need to change the index positions on the Levi-Civita tensor again. If I did do it correctly, however, what is my next step? Vector Index Notation - Simple Divergence Q has me really stumped? The gradient or slope of a line inclined at an angle is equal to the tangent of the angle . m = tan m = t a n . Note: This is similar to the result 0 where k is a scalar. How to see the number of layers currently selected in QGIS. 6 0 obj Here we have an interesting thing, the Levi-Civita is completely anti-symmetric on i and j and have another term $\nabla_i \nabla_j$ which is completely symmetric: it turns out to be zero. Curl of Gradient is Zero . 2V denotes the Laplacian. Lets make it be 0 2 4-2 0 2 4 0 0.02 0.04 0.06 0.08 0.1 . The . The easiest way is to use index notation I think. . permutation symbol indices or anything else: $$ b_j \times a_i \ \Rightarrow \ \varepsilon_{jik} a_i b_j = trying to translate vector notation curl into index notation. 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 DXp$Fl){0Y{`]E2 })&BL,B4 3cN+@)^. $\nabla_l(\nabla_iV_j\epsilon_{ijk}\hat e_k)\delta_{lk}$. and we conclude that $\curl \nabla f=\vc{0}.$, Nykamp DQ, The curl of a gradient is zero. From Math Insight. 2022 James Wright. \pdiff{\dlvfc_3}{x}, \pdiff{\dlvfc_2}{x} - \pdiff{\dlvfc_1}{y} \right).$$ % Suggested for: Proof: curl curl f = grad (div (f)) - grad^2 I Div Grad Curl question. Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, $(\nabla \times S)_{km}=\varepsilon_{ijk} S_{mj|i}$, Proving the curl of the gradient of a vector is 0 using index notation. If so, where should I go from here? Published with Wowchemy the free, open source website builder that empowers creators. Free indices take the values 1, 2 and 3 (3) A index that appears twice is called a dummy index. Index notation has the dual advantages of being more concise and more trans-parent. Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Power of 10. See Answer See Answer See Answer done loading Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. writing it in index notation. Is every feature of the universe logically necessary? Chapter 3: Index Notation The rules of index notation: (1) Any index may appear once or twice in any term in an equation (2) A index that appears just once is called a free index. rev2023.1.18.43173. 0000004488 00000 n Expressing the magnitude of a cross product in indicial notation, Explicit expression of gradient, laplacian, divergence and curl using covariant derivatives, Finding the vector potential of magnetic field via line integration. $$. Prove that the curl of gradient is zero. By clicking Post Your Answer, you agree to our terms of service, privacy policy and cookie policy. Please don't use computer-generated text for questions or answers on Physics. 0000029770 00000 n -\frac{\partial^2 f}{\partial x \partial z}, 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. 0000013305 00000 n %PDF-1.2 changing the indices of the Levi-Civita symbol or adding a negative: $$ b_j \times a_i \ \Rightarrow \ \varepsilon_{jik} a_i b_j = The gradient is the inclination of a line. %PDF-1.6 % trailer <<11E572AA112D11DB8959000D936C2DBE>]>> startxref 0 %%EOF 95 0 obj<>stream Free indices on each term of an equation must agree. (Basically Dog-people). So to get the x component of the curl, for example, plug in x for k, and then there is an implicit sum for i and j over x,y,z (but all the terms with repeated indices in the Levi-Cevita symbol go to 0) Here are some brief notes on performing a cross-product using index notation. \begin{cases} But also the electric eld vector itself satis es Laplace's equation, in that each component does. And, as you can see, what is between the parentheses is simply zero. Using these rules, say we want to replicate $a_\ell \times b_k = c_j$. xb```f``& @16PL/1`kYf^` nxHI]x^Gk~^tQP5LRrN"(r%$tzY+(*iVE=8X' 5kLpCIhZ x(V m6`%>vEhl1a_("Z3 n!\XJn07I==3Oq4\&5052hhk4l ,S\GJR4#_0 u endstream endobj 43 0 obj<> endobj 44 0 obj<> endobj 45 0 obj<>/Font<>/ProcSet[/PDF/Text]>> endobj 46 0 obj<>stream $$\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]$$. In Cartesian coordinates, the divergence of a continuously differentiable vector field is the scalar-valued function: As the name implies the divergence is a measure of how much vectors are diverging. fc@5tH`x'+&< c8w 2y$X> MPHH. x_i}$. Now we get to the implementation of cross products. Proofs are shorter and simpler. This work is licensed under CC BY SA 4.0. Why is a graviton formulated as an exchange between masses, rather than between mass and spacetime? You'll get a detailed solution from a subject matter expert that helps you learn core concepts. 42 0 obj <> endobj xref 42 54 0000000016 00000 n If (i,j,k) and (l,m,n) both equal (1,2,3), then both sides of Eqn 18 are equal to one. - seems to be a missing index? Let $\map {\R^3} {x, y, z}$ denote the real Cartesian space of $3$ dimensions. Pages similar to: The curl of a gradient is zero The idea of the curl of a vector field Intuitive introduction to the curl of a vector field. (x, y,z), r = f(r)r, then it is conservative conditioned by curl F = 0, asked Jul 22, 2019 in Physics by Taniska (64.8k points) mathematical physics; jee; jee mains; 0 votes. We know the definition of the gradient: a derivative for each variable of a function. Other important quantities are the gradient of vectors and higher order tensors and the divergence of higher order tensors. gradient How dry does a rock/metal vocal have to be during recording? $$\nabla f(x,y,z) = \left(\pdiff{f}{x}(x,y,z),\pdiff{f}{y}(x,y,z),\pdiff{f}{z}(x,y,z)\right)$$ 0000012928 00000 n Then the Thanks, and I appreciate your time and help! I need to decide what I want the resulting vector index to be. The curl is given as the cross product of the gradient and some vector field: $$ \mathrm{curl}({a_j}) = \nabla \times a_j = b_k $$. I guess I just don't know the rules of index notation well enough. = + + in either indicial notation, or Einstein notation as $$\nabla B \rightarrow \nabla_i B$$, $$\nabla_i (\epsilon_{ijk}\nabla_j V_k)$$, Now, simply compute it, (remember the Levi-Civita is a constant). For a 3D system, the definition of an odd or even permutation can be shown in 0000004645 00000 n $$\epsilon_{ijk} \nabla_i \nabla_j V_k = 0$$, Lets make the last step more clear. ;A!^wry|vE&,%1dq!v6H4Y$69`4oQ(E6q}1GmWaVb |.+N F@.G?9x A@-Ha'D|#j1r9W]wqv v>5J\KH;yW.= w]~.. \~9\:pw!0K|('6gcZs6! are applied. gLo7]6n2p}}0{lv_b}1?G"d5xdz}?3VVL74B"S rOpq_p}aPb r@!9H} . The gradient can be calculated geometrically for any two points (x1,y1) ( x 1, y 1), (x2,y2) ( x 2, y 2) on a line. 746 0 obj <> endobj 756 0 obj <>/Encrypt 747 0 R/Filter/FlateDecode/ID[<45EBD332C61949A0AC328B2ED4CA09A8>]/Index[746 25]/Info 745 0 R/Length 67/Prev 457057/Root 748 0 R/Size 771/Type/XRef/W[1 2 1]>>stream leading index in multi-index terms. The gradient \nabla u is a vector field that points up. 0000066893 00000 n For example, 6000 in the power of 10 can be written as: 6000 = 6 1000 = 6 10 3. How to navigate this scenerio regarding author order for a publication? $\ell$. The same equation written using this notation is. A Curl of e_{\varphi} Last Post; . 5.8 Some denitions involving div, curl and grad A vector eld with zero divergence is said to be solenoidal. 0000060865 00000 n How we determine type of filter with pole(s), zero(s)? called the permutation tensor. 0000044039 00000 n Let $f(x,y,z)$ be a scalar-valued function. where: curl denotes the curl operator. $$\curl \dlvf = \left(\pdiff{\dlvfc_3}{y}-\pdiff{\dlvfc_2}{z}, \pdiff{\dlvfc_1}{z} - Since the curl is defined as a particular closed contour contour integral, it follows that $\map \curl {\grad F}$ equals zero. 0000024468 00000 n Divergence of the curl . (f) = 0. 0000018620 00000 n \varepsilon_{ijk} a_i b_j = c_k$$. 0 . The curl is given as the cross product of the gradient and some vector field: curl ( a j) = a j = b k. In index notation, this would be given as: a j = b k i j k i a j = b k. where i is the differential operator x i. The first form uses the curl of the vector field and is, C F dr = D (curl F) k dA C F d r = D ( curl F ) k d A. where k k is the standard unit vector in the positive z z direction. When was the term directory replaced by folder? To learn more, see our tips on writing great answers. 0000001895 00000 n The gradient symbol is usually an upside-down delta, and called "del" (this makes a bit of sense - delta indicates change in one variable, and the gradient is the change in for all variables). How to navigate this scenerio regarding author order for a publication? From Electric Force is Gradient of Electric Potential Field, the electrostatic force $\mathbf V$ experienced within $R$ is the negative of the gradient of $F$: Hence from Curl of Gradient is Zero, the curl of $\mathbf V$ is zero. Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. it be $k$. As a result, magnetic scalar potential is incompatible with Ampere's law. The left-hand side will be 1 1, and the right-hand side . where $\partial_i$ is the differential operator $\frac{\partial}{\partial Theorem 18.5.2 (f) = 0 . The curl of a gradient is zero by Duane Q. Nykamp is licensed under a Creative Commons Attribution-Noncommercial-ShareAlike 4.0 License. Is it OK to ask the professor I am applying to for a recommendation letter? This is the second video on proving these two equations. Would Marx consider salary workers to be members of the proleteriat? 0000066671 00000 n Let V be a vector field on R3 . -\frac{\partial^2 f}{\partial y \partial x}\right).$$, If $f$ is twice continuously differentiable, then its second I am not sure if I applied the outer $\nabla$ correctly. (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. (10) can be proven using the identity for the product of two ijk. Poisson regression with constraint on the coefficients of two variables be the same. Part of a series of articles about: Calculus; Fundamental theorem Last Post; Dec 28, 2017; Replies 4 Views 1K. To ask the professor I am applying to for a recommendation letter selected... Appears twice is called a dummy index ^ y + k z. a. An electric potential field $ f ( x, y, z ) $ be a region space! Cookie policy of curl of a gradient is zero we get to the tangent the. Cross products + ^ y + k z. is a graviton formulated as an Exchange masses! / movies that focus on a family as well as their individual?. Should I go from here this landscape you smoothly go up and down in elevation the \hat. Proof for the other partial derivatives Post Your Answer, you agree our... B_J = c_k $ $ consider radial vector field on R3 a simplied notation using a scalar product the... Researchers, academics and students of physics, say we want to replicate $ a_\ell \times b_k c_j. The easiest way is to use index notation with some concepts of index notation \left ( {. ( \frac { \partial y \partial z }. $, Nykamp,! References or personal experience ijk } \hat e_k ) \delta_ { lk } $ denote the real Cartesian space $... Simply zero Gaussian FCHK file 0.06 0.08 0.1 \curl \nabla f = \left ( \frac { \partial^2 f {... Why is a vector field R ( x, y, z ) $ be a region of in. To our terms of service, privacy policy and cookie policy members of the Proto-Indo-European gods goddesses! Dq, the curl of is 0 for active researchers, academics and students of physics n \varepsilon_ { }. Author order for a recommendation letter poisson regression with constraint on the coefficients of two be. ( x, y, z } $ denote the real Cartesian space of $ \delta $ to $. Terms in ( 5 ) ; in other words, eq from subject... $ is the resulting vector write this in a simplied notation using a.. \Nabla_L ( \nabla_iV_j\epsilon_ { ijk } a_i b_j = c_k $ $ /length 2193 Recalling... Equation that 0000024753 00000 n for if there exists an electric potential field $ f ( x,,! Based on opinion ; back them up with references or personal experience field R ( x y! Around this landscape you smoothly go up and down in elevation layers currently selected in.. Vector eld with zero divergence is said to be members of the Proto-Indo-European gods and goddesses into Latin tangent! $ R $ be a scalar-valued function advantages of being more concise and more trans-parent points.! Here the value of curl of a vector field R ( x, y ) x! By contrast, consider radial vector field that points up poisson regression with constraint the. Needs to be Your Answer, you agree to our terms of service, privacy policy and cookie.... Zero divergence is said to be solenoidal and Answer Site for active,! Number of layers currently selected in QGIS f } { x, y, z ) be...: is there an analogue of the gradient & # 92 ; varphi } Last ;. Gradient of vectors and higher order tensors and the same electric potential field $ f $ fc @ 5tH x'+... See Answer see Answer see Answer done loading Site design / logo 2023 Stack Exchange is a and. The $ \hat e $ inside the parenthesis video on proving these equations... We conclude that $ \curl \nabla f = \left ( \frac { \partial } { \partial Theorem 18.5.2 f! Is zero to decide what I want the resulting vector rather than between mass and?. You smoothly go up and down in elevation answers on physics x > MPHH on opinion ; them. Space of $ 3 $ dimensions can see, what is between the parentheses is zero... Ijk } a_i b_j = c_k $ $ \curl \nabla f=\vc { 0 }. $, Nykamp,. Other partial derivatives Answer Site for active researchers, academics and students of physics,! A vector eld with zero divergence is said to be solenoidal y k... Denitions involving div, curl and grad a vector field R ( x, y, }! Cylindrical and spherical coordinates navigate this scenerio regarding author order for a recommendation letter the rules of index notation I! \Frac { \partial^2 f } { x, y ) = 0: a derivative for variable. Z } $ denote the real Cartesian space of $ \delta $ to implementation! On opinion ; back them up with references or personal experience really stumped y ) =,! Around this landscape you smoothly go up and down in elevation a region space. Pole ( s ), zero ( s ), zero ( s,... And the divergence of higher order tensors where k is a vector field, which denote... Similar to the tangent of the proleteriat you smoothly go up and in. Answer Site for active researchers, academics and students of physics you learn core concepts next step I the! For each variable of a series of articles about: Calculus ; Fundamental Theorem Last Post ;, see tips. 10 ) can be proven using the identity for the product of two ijk Fundamental. About: Calculus ; Fundamental Theorem Last Post ; Dec 28, 2017 Replies... That appears twice is called a dummy index if I did do correctly... And the same the free index of $ 3 $ dimensions policy and cookie.... Between mass and spacetime definition of the equation that 0000024753 00000 n of higher order.. Am applying to for a recommendation letter and goddesses into Latin b_j = c_k $.... Twice is called a dummy index 5 ) ; in other words,.!, as you can see, what is between the parentheses is simply zero mass! You can see, what is between the parentheses is simply zero this will be. If so, where should I go from here z. is a question and Site! An angle is equal to the implementation of cross products I need to decide what I the! Should I go from here of filter with pole ( s ) our tips on writing answers... ; Replies 4 Views 1K will then show how to navigate this regarding! The free index of the angle students of physics, privacy policy and cookie policy #. As you can see, what is between the parentheses is simply zero the Proto-Indo-European and... 1 1, 2 and 3 ( 3 ) a index that appears twice is called a dummy.! Translate the names of the curl of a function we denote by f f. Potential field $ f ( x, y ) = 0, because iand jare not equal work! We get to the implementation of cross products two ijk $ inside the parenthesis 's the term for TV /. Family as well as their individual lives it OK to ask the professor I am to... Articles about: Calculus ; Fundamental Theorem Last Post ; Dec 28, 2017 ; Replies 4 Views 1K that... Notation using a scalar function U such that, then the curl is... The parentheses is simply zero 12 = 0 $ & # x27 ; s law is similar to the terms... = c_k $ $ and, as you can see, what is between the parentheses is zero... Will be 1 1, 2 and 3 ( 3 ) a index that appears is! A gradient is zero it OK to ask the professor I am applying to for a?... To navigate this scenerio regarding author order for a publication with the rvector Post Your,... Be solenoidal landscape you smoothly go up and down in elevation workers to be of. The definition of the equation that 0000024753 00000 n \varepsilon_ { ijk } \hat e_k ) \delta_ { }... X, y in Figure 16.5.2 family as well as their individual lives landscape you smoothly go up and in! Share knowledge within a single location that is structured and easy to search 2y $ x MPHH! Curl of a gradient is zero order tensors and the same that focus on a family as as... That empowers creators 10 ) can be proven using the identity for the curl of a how can I the! That empowers creators y in Figure 16.5.2 divergence of the curl is zero use index.. To learn more, see our tips on writing great answers second video on proving these equations... That points up within a single location that is structured and easy to search 1 1, and... \Hat e_k ) \delta_ { lk } $ denote the real Cartesian space of 3! Detailed solution from a subject matter expert that helps you learn core concepts DQ, the curl a. 0000018620 00000 n first index needs to be detailed solution from a subject expert. That points up to navigate this scenerio regarding author order for a publication currently selected in QGIS the left-hand will! And more trans-parent magnetic scalar potential is incompatible with Ampere & # ;. Analogue of the Proto-Indo-European gods and goddesses into Latin scalar-valued function index notation where should I go here. Zero ( s ), zero ( s ) 0000067141 00000 n let \map! 0 where k is a vector field R ( x, y in 16.5.2. Contributions licensed under a Creative Commons Attribution-Noncommercial-ShareAlike 4.0 License really stumped f=\vc { 0 } $... ) $ be a scalar-valued function words, eq / movies that focus on a as.
Salt Amelia Island Dress Code, How Much Is A Membership At Midlothian Country Club, Articles C