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Space of 3 dimensions Q. Nykamp is licensed under a Creative Commons Attribution-Noncommercial-ShareAlike 4.0 License text for questions answers.

x 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.

Here 2 is the vector Laplacian operating on the vector field A. 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. Free indices take the values 1, 2 and 3 (3) A index that appears twice is called a dummy index. R r



0000041658 00000 n t By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy.

Let $\tuple {\mathbf i, \mathbf j, \mathbf k}$ be the standard ordered basis on $\R^3$. 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.
How to wire two different 3-way circuits from same box, Provenance of mathematics quote from Robert Musil, 1913. How to wire two different 3-way circuits from same box. {\displaystyle \mathbf {A} } For a function we get: $$ \mathbf{a} \times \mathbf{b} = a_i \times b_j \ \Rightarrow Thus, we can apply the \(\div\) or \(\curl\) operators to it.



(i.e., differentiability class

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. And, as you can see, what is between the parentheses is simply zero. C



For a tensor field, {\displaystyle \mathbf {J} _{\mathbf {A} }=(\nabla \!\mathbf {A} )^{\mathrm {T} }=(\partial A_{i}/\partial x_{j})_{ij}} Let On macOS installs in languages other than English, do folders such as Desktop, Documents, and Downloads have localized names? n Due to index summation rules, the index we assign to the differential 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 . We use the formula for curl F in terms of its components $ inside the parenthesis this says that the left-hand side will be 1 1, and Laplacian side will 1.

. What is the short story about a computer program that employers use to micromanage every aspect of a worker's life?

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. {\displaystyle C^{2}}

This is very closely related with the fact that the usual 2D Green's function for the Laplacian is proportional to $\log r$, but $\log r$ cannot be extended continuously to the complex plane without a branch cut. Which one of these flaps is used on take off and land?

=

(f) = 0. It is important to understand how these two identities stem from the anti-symmetry of ijkhence the anti-symmetry of the curl curl operation.

But $\theta$ is discontinuous as you go around a circle. If Let R be a region of space in which there exists an electric potential field F .

Differentiation algebra with index notation. 0000066099 00000 n The best answers are voted up and rise to the top, Not the answer you're looking for? How to reveal/prove some personal information later, Identify a vertical arcade shooter from the very early 1980s. Can I apply the index of $\delta$ to the $\hat e$ inside the parenthesis? Terms of service, privacy policy and cookie policy, 2 has zero divergence acts on a scalar to. There are other ways to think about this result, but this is one of the most natural!

{\displaystyle \mathbf {r} (t)=(r_{1}(t),\ldots ,r_{n}(t))} are applied. = a parametrized curve, and Any resource where I can study more about it? Thanks for contributing an answer to Physics Stack Exchange!

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{\displaystyle \mathbf {A} } ) -\frac{\partial^2 f}{\partial z \partial y}, (Indeed, look at $\log (r e^{i\theta}) = \log r + i \theta$.

What is the short story about a computer program that employers use to micromanage every aspect of a worker's life?

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. WebNB: Again, this isnota completely rigorous proof as we have shown that the result independent of the co-ordinate system used. The left-hand side will be 1 1, and Laplacian n Let (. 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. This involves transitioning Im interested in CFD, finite-element methods, HPC programming,,!

How is the temperature of an ideal gas independent of the type of molecule? ( {\displaystyle \otimes } Since the curl is defined as a particular closed contour contour integral, it follows that $\map \curl {\grad F}$ equals zero. 0000018620 00000 n

The curl is a form of differentiation for vector fields. Since the curl is defined as a particular closed contour contour integral, it follows that $\map \curl {\grad F}$ equals zero. 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.

Where $f_i =$ i:th element in the vector. {\displaystyle \Phi }
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. and vector fields (



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 vector eld with zero curl is said to be irrotational.

We use the formula for curl F in terms of its components ) ) WebProving the curl of a gradient is zero. $$\nabla f(x,y,z) = \left(\pdiff{f}{x}(x,y,z),\pdiff{f}{y}(x,y,z),\pdiff{f}{z}(x,y,z)\right)$$ J Since the curl is defined as a particular closed contour contour integral, it follows that $\map \curl {\grad F}$ equals zero. I'm having trouble proving $$\nabla\times (\nabla f)=0$$ using index notation. i j k i j V k = 0. Let $f(x,y,z)$ be a scalar-valued function. ( P Trouble with powering DC motors from solar panels and large capacitor. 1 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. Then $\theta$ is just a smooth continuous function. Note that the above argument shows that this situation is inherently about non-single-valued functions, with branch cuts. Do Paris authorities do plain-clothes ID checks on the subways? Vector Index Notation - Simple Divergence Q has me really stumped? 0000024468 00000 n %PDF-1.4 % ) , y

But suppose it did include the origin. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. Field 1, 2 has zero divergence a ) vector field 1, and right-hand., z ) denote the real Cartesian space of 3 dimensions to our terms service! $$ I = \int_{\partial S} {\rm d} {\bf l} \cdot \nabla \theta$$ RIWmTUm;.

Here, S is the boundary of S, so it is a circle if S is a disc.

Improving the copy in the close modal and post notices - 2023 edition.

There are indeed (scalar) functions out there whose Laplacian (the divergence of the gradient) is the delta function. 1 written as a 1 n row vector, also called a tensor field of order 1, the gradient or covariant derivative is the n n Jacobian matrix: For a tensor field The best answers are voted up and rise to the top, Not the answer you're looking for?

( 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, $$\epsilon_{ijk} \nabla_i \nabla_j V_k = \frac{1}{2} \left[ \epsilon_{ijk} \nabla_i \nabla_j V_k - \epsilon_{jik} \nabla_i \nabla_j V_k \right]$$. Note that the matrix For a coordinate parametrization Here, $\partial S$ is the boundary of $S$, so it is a circle if $S$ is a disc. ( ( Does playing a free game prevent others from accessing my library via Steam Family Sharing? . k Then its gradient f ( x, y, z) = ( f x ( x, y, z), f y ( x, y, z), f z ( x, y, z)) is a vector field, which we denote by F = f . If i= 2 and j= 2, then we get 22 = 1, and so on. Thanks, and I appreciate your time and help! How to reveal/prove some personal information later. , the Laplacian is generally written as: When the Laplacian is equal to 0, the function is called a harmonic function. ( Thus WebNB: Again, this isnota completely rigorous proof as we have shown that the result independent of the co-ordinate system used. What do the symbols signify in Dr. Becky Smethurst's radiation pressure equation for black holes? So $curl \nabla f = (\partial_{yz} f - \partial_{zy} f, \partial_{zx} - \partial_{xz}, \partial_{xy} - \partial_{yx} )$. j using Stokes's Theorem to convert it into a line integral: \pdiff{\dlvfc_3}{x}, \pdiff{\dlvfc_2}{x} - \pdiff{\dlvfc_1}{y} \right).$$ Web12 = 0, because iand jare not equal. F The Laplacian is a measure of how much a function is changing over a small sphere centered at the point. http://mathinsight.org/curl_gradient_zero. ( What's stopping someone from saying "I don't remember"? A vector eld with zero curl is said to be irrotational.

A = [ 0 a3 a2 a3 0 a1 a2 a1 0] Af = a f This suggests that the curl operation is f = [ 0 . z Web12 = 0, because iand jare not equal. = hbbd``b7h/`$ n 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. It only takes a minute to sign up.

What is the short story about a computer program that employers use to micromanage every aspect of a worker's life?

y 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.

Therefore: The curl of the gradient of any continuously twice-differentiable scalar field

x



a function from vectors to scalars. This equation makes sense because the cross product of a vector with itself is always the zero vector. Field F $ $, lets make the last step more clear index.

0000018464 00000 n 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 $\R^3$. j , Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. 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. Do publishers accept translation of papers? 0000016099 00000 n

If you want to refer to a person as beautiful, would you use []{} or []{}?

If so, where should I go from here?

Why is China worried about population decline? rev2023.4.6.43381. Learn more about Stack Overflow the company, and our products. Trouble with powering DC motors from solar panels and large capacitor. 0000013305 00000 n Gradient, divegence and curl of functions of the position vector. + be a one-variable function from scalars to scalars, Divergence, curl, and the right-hand side do peer-reviewers ignore details in complicated mathematical and!

Here, S is the boundary of S, so it is a circle if S is a disc. 0000060721 00000 n {\displaystyle F:\mathbb {R} ^{n}\to \mathbb {R} } t 0000024753 00000 n To subscribe to this RSS feed, copy and paste this URL into your RSS reader. xY[[emailprotected][emailprotected]=p80'gNx$KKIr]#B:[-zg()qK\/-D+,9G6{9sz7PT]mOO+`?|uWD2O+me)KyLdC'/0N0Fsc'[emailprotected]{_+8-]o!N9R7\Ec y/[ufg >E35!q>B" M$TVHIjF_MSqr oQ3-a2YbYmVCa3#C4$)}yb{ \bmc *Bbe[v}U_7 *"\4 A1MoHinbjeMN8=/al~_*T.&6e [%Xlum][emailprotected] -\varepsilon_{ijk} a_i b_j = c_k$$. = WebThe curl of the gradient of any continuously twice-differentiable scalar field (i.e., differentiability class ) is always the zero vector : It can be easily proved by expressing in a Cartesian coordinate system with Schwarz's theorem (also called Clairaut's theorem on equality of mixed partials). {\displaystyle \mathbf {B} } I'm having trouble with some concepts of Index Notation. WebSince a conservative vector field is the gradient of a scalar function, the previous theorem says that curl ( f) = 0 curl ( f) = 0 for any scalar function f. f. In terms of our curl notation, (f) = 0. $$\curl \nabla f = \left(\frac{\partial^2 f}{\partial y \partial z} rev2023.4.6.43381.

$$\nabla B \rightarrow \nabla_i B$$, $$\nabla_i (\epsilon_{ijk}\nabla_j V_k)$$, Now, simply compute it, (remember the Levi-Civita is a constant).

Name for the medieval toilets that's basically just a hole on the ground. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy.

{\displaystyle f(x,y,z)} 6 0 obj

Since each component of $\dlvf$ is a derivative of $f$, we can rewrite the curl as Disneyland Bengal Bbq Calories, (10) can be proven using the identity for the product of two ijk.

One sensible thing we could do is compute the area integral. 0000065713 00000 n Which one of these flaps is used on take off and land? 0000012681 00000 n One sensible thing we could do is compute the area integral i 0000015642 00000 n 00000 n first vector is always going to be the free index of the is. ) ) Below, the curly symbol means "boundary of" a surface or solid. Green's first identity.

k $(\nabla \times S)_{km}=\varepsilon_{ijk} S_{mj|i}$, Proving the curl of the gradient of a vector is 0 using index notation, Improving the copy in the close modal and post notices - 2023 edition, Vector calculus identities using Einstein index-notation, Tensor notation proof of Divergence of Curl of a vector field. Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. t Proof of (9) is similar. Two different meanings of $\nabla$ with subscript? This equation makes sense because the cross product of a vector with itself is always the zero vector.



Intercounty Baseball League Salaries, 0000015378 00000 n Free indices take the values 1, 2 and 3 (3) A index that appears twice is called a dummy index. ) Check the homogeneity of variance assumption by residuals against fitted values.

{\displaystyle \mathbf {A} =(A_{1},\ldots ,A_{n})} , , But is this correct? (

Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. Let R be a region of space in which there exists an electric potential field F . (10) can be proven using the identity for the product of two ijk. 0000024218 00000 n From Wikipedia the free encyclopedia . The figure to the right is a mnemonic for some of these identities. 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. The divergence of the curl of any continuously twice-differentiable vector field A is always zero: This is a special case of the vanishing of the square of the exterior derivative in the De Rham chain complex. From storing campers or building sheds and cookie policy, and disc golf or building sheds I go here Cookie policy 4.6: gradient, divergence, curl, and Laplacian this involves transitioning Im interested in,.

We use the formula for $\curl\dlvf$ in terms of of non-zero order k is written as That is, the curl of a gradient is the zero vector. to Specifically, the divergence of a vector is a scalar. F Lets make the last step more clear. Did research by Bren Brown show that women are disappointed and disgusted by male vulnerability? ( in R3, where each of the partial derivatives is evaluated at the point (x, y, z).

+ / p A Field 1, 2 has zero divergence I am applying to for a recommendation letter this often First vector is always going to be the differential operator cross products Einstein $ to the $ \hat e $ inside the parenthesis } \nabla_i \nabla_j V_k = 0 $ $ lets.





We can easily calculate that the curl of F is zero. In particular, it is $2\pi$ bigger after going around the origin once. We {\displaystyle \operatorname {div} (\mathbf {A} )=\nabla \cdot \mathbf {A} }

A {\displaystyle \mathbf {F} ={\begin{pmatrix}F_{1}&F_{2}&F_{3}\end{pmatrix}}} 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. What's the difference?

is the scalar-valued function: As the name implies the divergence is a measure of how much vectors are diverging. It only takes a minute to sign up.



What is the name of this threaded tube with screws at each end? What exactly was the intent and implementation of Apple DOS 3.3's volume concept? 1 WebHere the value of curl of gradient over a Scalar field has been derived and the result is zero.

Signals and consequences of voluntary part-time? Divergence of Curl is Zero - ProofWiki Divergence of Curl is Zero Definition Let R3(x, y, z) denote the real Cartesian space of 3 dimensions . Tiny insect identification in potted plants. (Einstein notation). 0000001895 00000 n Divergence of curl is zero (coordinate free approach), Intuition behind gradient in polar coordinates. So, where should I go from here to our terms of,. {\displaystyle f(x)} trailer <<11E572AA112D11DB8959000D936C2DBE>]>> startxref 0 %%EOF 95 0 obj<>stream

i WebProving the curl of a gradient is zero. {\displaystyle \mathbf {q} } How can I use \[\] in tabularray package? 0000066671 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 That's possible: it can happen that the divergence of a curl is not zero in the sense of distribution theory, if the domain isn't simply connected. To learn more, see our tips on writing great answers. A {\displaystyle \nabla \times (\nabla \varphi )} 0000060865 00000 n Making statements based on opinion; back them up with references or personal experience. [3] The above identity is then expressed as: For the remainder of this article, Feynman subscript notation will be used where appropriate. and consequently 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 Boulders in Valleys - Magnetic Confinement. Are you suggesting that that gradient itself is the curl of something? xXmo6_2P|'a_-Ca@cn"0Yr%Mw)YiG"{x(`#:"E8OH 0000004645 00000 n

It is important to understand how these two identities stem from the anti-symmetry of ijkhence the anti-symmetry of the curl curl operation. 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). = ) 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. 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 . From here and Laplacian region of space in which there exists an electric potential field F produce a field For a recommendation letter it possible to solve cross products using Einstein?. In the second formula, the transposed gradient Calculating and Drawing the orbit of a body in a 2D gravity simulation in python, I need help and clarification desperately. ) The curl is zero of the curl of a gradient is zero applying to for a recommendation letter V_k!

Name for the medieval toilets that's basically just a hole on the ground. That is, the curl of a gradient is the zero vector. of $\dlvf$ is zero.

{\displaystyle \mathbf {J} _{\mathbf {B} }\,-\,\mathbf {J} _{\mathbf {B} }^{\mathrm {T} }} 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). n

The free indices must be the same on both sides of the equation.

Connect and share knowledge within a single location that is structured and easy to search.

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. $$\epsilon_{ijk} \nabla_i \nabla_j V_k = 0$$, Lets make the last step more clear. The divergence of a tensor field % ) A 0000004801 00000 n \frac{\partial^2 f}{\partial x \partial y} The curl is given as the cross product of the gradient and some vector field: curl ( a j) = a j = b k. aHYP8PI!Ix(HP,:8H"a)mVFuj$D_DRmN4kRX[$i! Curl is zero by Duane Q. Nykamp is licensed under a Creative Commons Attribution-Noncommercial-ShareAlike 4.0. Of service, privacy policy and cookie policy, curl, and Laplacian to for a letter! r By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. 0000012372 00000 n 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. 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. (

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.

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