G i 1 r x , ( + ( The best answers are voted up and rise to the top, Not the answer you're looking for? G G are the contravariant basis vectors in a curvilinear coordinate system, with coordinates of points denoted by ( 2 ( W ( + + = = 2 ^ j LECTURE 6: THE RIEMANN CURVATURE TENSOR 1. T ) Some of the identities have been proved using Levi-Civita Symbols by other mathematicians and Physicists. x $$= \partial_i \sum_j\partial_j A_j - \sum_j\partial_j^2 A_i = \partial_i (\nabla\cdot\mathbf{A}) - (\nabla\cdot\nabla) A_i,$$ ( ) x r 1 j i i ( x G i ( + + 1 {\displaystyle \operatorname {div} (\mathbf {A} )=\nabla \cdot \mathbf {A} } W f ( 2 F = ) ^ F ( x = ( ( y i and an = G i i y ^ i ) v is antisymmetric. ) v , i V i {\displaystyle \nabla \times (\mathbf {F} \times \mathbf {G} )=((\nabla \cdot \mathbf {G} )\mathbf {F} +(\mathbf {G} \cdot \nabla )\mathbf {F} )-((\nabla \cdot \mathbf {F} )\mathbf {G} +(\mathbf {F} \cdot \nabla )\mathbf {G} )}, G i r v x x be two second order tensors, then, Since f 1 + ) G 1 f + of two vectors, or of a covector and a vector. , and , i + ( be a real valued function of the second order tensor V G i ) + G i i ( G i ) x 0 = ( x S x i i H i {\displaystyle \mathbf {F} } F 3 2 g ( x F , ^ $$\operatorname{grad} \operatorname{div}\textbf{A}=\operatorname{rot}\operatorname{rot}\textbf{A}+\operatorname{div}\operatorname{grad}\mathbf{A}$$ x 0 are the basis vectors in a Cartesian coordinate system, with coordinates of points denoted by ( x ) ( input function ( Let $\delta _{ij}$ denote the Kronecker delta. ( f ( f + ) 1 {\displaystyle =(\nabla ^{2}\mathbf {F} )+(\nabla ^{2}\mathbf {G} )}, Given scalar fields matrices which can be written as a tensor product always have rank 1. , then G G The Ricci tensor represents how a volume in a curved space differs from a volume in Euclidean space. ( + {\displaystyle =(i,G_{i}(F_{i}H_{i}+F_{i+1}H_{i+1}+F_{i+2}H_{i+2})-(F_{i}G_{i}+F_{i+1}G_{i+1}+F_{i+2}G_{i+2})H_{i})}, = x x V n cos i i + ( ) i f [1], The directional derivative provides a systematic way of finding these derivatives.[2]. 2 Answers. + {\displaystyle =(i,((\mathbf {V} \cdot \nabla )f)G_{i}+f((\mathbf {V} \cdot \nabla )G_{i}))} 2 , ( We have already encountered two such tensors: namely, the second-order identity tensor, , and the third-order permutation tensor, . i S + x + The derivatives of scalars, vectors, and second-order tensors with respect to second-order tensors are of considerable use in continuum mechanics. 1 i ) + , g The last equation is equivalent to the alternative definition / interpretation[4], In curvilinear coordinates, the divergences of a vector field v and a second-order tensor field , H x g F F V ( 1 F = + {\displaystyle =v_{\rho }\mathbf {0} +{\frac {v_{\phi }}{\rho }}{\hat {\mathbf {\phi } }}+v_{z}\mathbf {0} } {\displaystyle g} 1 x ) 1 Tensor-vector identity 2 [edit | edit source] Let v {\displaystyle \mathbf {v} } be a vector field and let S {\displaystyle {\boldsymbol {S}}} be a second-order tensor field. , then V i i + ) i ( F , and F y {\displaystyle \mathbf {G} } ( v + {\displaystyle f} i Of course, all scalars are isotropic. = x We have that (S T)(e i . ) i ^ + ) , 1 {\displaystyle =(i,({\frac {\partial f}{\partial x_{i+1}}}G_{i+2}+f{\frac {\partial G_{i+2}}{\partial x_{i+1}}})-({\frac {\partial f}{\partial x_{i+2}}}G_{i+1}+f{\frac {\partial G_{i+1}}{\partial x_{i+2}}}))} F A tensor-valued function of the position vector is called a tensor field, Tij k (x). 2 1 1 G ( ) ) F 1 ( \implies \nabla \cdot (T \cdot v) = \frac{\partial (T_{im}v_m e_i)}{\partial x_n}\cdot e_n = \frac{\partial (T_{im}v_m)}{\partial x_n} \delta_{in}=\frac{\partial (T_{pm}v_m)}{\partial x_p} = \frac{\partial (T_{pq}v_q)}{\partial x_p} $$. 2 ( x $$ \nabla\times\nabla\times\mathbf{A} = \nabla(\nabla\cdot\mathbf{A}) - (\nabla\cdot\nabla)\cdot\mathbf{A}.$$. x g , then , i 1 = ) ) 1 i + ) ) A i Why is there a limit on how many principal components we can compute in PCA? ) ( + v , then ( ) + ) f ( Do school zone knife exclusions violate the 14th Amendment? {\displaystyle \nabla \times (\nabla f)=\nabla \times (i,{\frac {\partial f}{\partial x_{i}}})} rev2022.12.7.43084. r ) only changes with respect to i I + i i = + ^ ( ) A , ) {\displaystyle =(i,{\frac {\partial }{\partial x_{i+1}}}({\frac {\partial f}{\partial x_{i+2}}})-{\frac {\partial }{\partial x_{i+2}}}({\frac {\partial f}{\partial x_{i+1}}}))} ( ) i + r As you can see, $$v_q\frac{\partial T_{qp}}{\partial x_{p}} + T_{pq}\frac{\partial v_{q}}{\partial x_{p}} \neq \frac{\partial T_{pq}v_q}{x_p}$$. i ) i i 1-forms, and X j vector elds, by using that Lie derivatives satises Leibniz . . ( g ( ( G {\displaystyle \phi } i F C 0 , = {\displaystyle \mathbf {V} } = ) + + H x ) ) j F i {\displaystyle =(i,F_{i}(\nabla \cdot \mathbf {G} )-(\nabla \cdot \mathbf {F} )G_{i}-(\mathbf {F} \cdot \nabla )G_{i}+(\mathbf {G} \cdot \nabla )F_{i})} F ( ) ) ), divergence ( f + . {\displaystyle {\frac {\partial {\hat {\mathbf {r} }}}{\partial \theta }}={\hat {\mathbf {\theta } }}} Did they forget to add the layout to the USB keyboard standard? 1 F {\displaystyle \mathbf {G} } {\displaystyle \mathbf {G} } G x x ) x 1 where $e_i$ are unit vectors in an orthogonal basis. f Let x rev2022.12.7.43084. y ) 2 {\displaystyle \nabla \cdot (\nabla \times \mathbf {F} )=\nabla \cdot (i,{\frac {\partial F_{i+2}}{\partial x_{i+1}}}-{\frac {\partial F_{i+1}}{\partial x_{i+2}}})} + V {\displaystyle f} cos = 2 : Product rule for multiplication by a scalar, Learn how and when to remove this template message, Comparison of vector algebra and geometric algebra, Del in cylindrical and spherical coordinates, "Chapter 1.14 Tensor Calculus 1: Tensor Fields", https://en.wikipedia.org/w/index.php?title=Vector_calculus_identities&oldid=1125973323, Articles lacking in-text citations from August 2017, Pages using sidebar with the child parameter, Creative Commons Attribution-ShareAlike License 3.0, This page was last edited on 6 December 2022, at 21:35. j ( i ) ^ 1 {\displaystyle \phi } V ( G For example, i f + ^ cos x Making statements based on opinion; back them up with references or personal experience. F ) + 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. j f + F + + ) j For example: {\displaystyle =v_{\rho }{\frac {\partial {\hat {\mathbf {\rho } }}}{\partial \phi }}+{\frac {v_{\phi }}{\rho }}{\frac {\partial {\hat {\mathbf {\phi } }}}{\partial \phi }}+v_{z}{\frac {\partial {\hat {\mathbf {\phi } }}}{\partial z}}} sin V f F Important vector identities with the help of Levi-Civita symbols and Kronecker delta tensorare proved and presented in this paper. ( ( ) + ) ( W 1 Proof: For the first identity, using index notation, we have [ . v ( i i f ) + F ) i {\displaystyle ={\frac {1}{r}}(-v_{\theta }{\hat {\mathbf {r} }}+\cot \theta v_{\phi }{\hat {\mathbf {\phi } }})}, i i F ( i ( ( Less general but similar is the Hestenes overdot notation in geometric algebra. Kolda, Multilinear Operators for Higher-order Decompositions, Tech. + Site design / logo 2022 Stack Exchange Inc; user contributions licensed under CC BY-SA. ( ) z + i ( The alternating tensor can be used to write down the vector equation z = x y in sux notation: z i = [xy] i = ijkx jy k. (Check this: e.g., z 1 = 123x 2y 3 + 132x 3y 2 = x 2y 3 x 3y 2, as required.) 2 ( f f + F When does money become money? i 2 1 H F i the curl is the vector field: As the name implies the curl is a measure of how much nearby vectors tend in a circular direction. ^ ( F i i G H g 1 {\displaystyle \nabla ^{2}(fg)=(\nabla ^{2}f)g+2(\nabla f)\cdot (\nabla g)+f(\nabla ^{2}g)}, When H = i sin i A g {\displaystyle \mathbf {V} _{\perp }} + ) = and ^ = , then r Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. sin G Why do American universities cost so much? F ). = A ) y f + F ( i ) i The rests are presented for the first time. ) i + i ^ = F f , ) ( 1 v ) ) {\displaystyle =(i,v((\mathbf {V} \cdot \nabla )F_{i}))} i ( + 2 = ( ( i f = 1 G is a 1 n row vector, and their product is an n n matrix (or more precisely, a dyad); This may also be considered as the tensor product i ^ f + ( , {\displaystyle \mathbf {g} ^{1},\mathbf {g} ^{2},\mathbf {g} ^{3}} F + 1 = {\displaystyle {\hat {\mathbf {\phi } }}=(-\sin \phi )\mathbf {i} +(\cos \phi )\mathbf {j} } i i x G A ) ( Let i j denote the Kronecker delta. ( $$v\cdot(\nabla \cdot T)=(v_ie_i)\cdot\left(\frac{\partial}{\partial x_j}(T_{kl} e_k e_l)\cdot e_j\right)=(v_ie_i)\cdot\left(\frac{\partial T_{kl}}{\partial x_j} e_k\delta _{lj}\right) = (v_ie_i)\cdot\left(\frac{\partial T_{km}}{\partial x_m} e_k\right)=v_i\frac{\partial T_{km}}{\partial x_m} \delta _{ik}=v_q\frac{\partial T_{qp}}{\partial x_p}$$ i f F ( ( sin , ( i {\displaystyle =(\mathbf {V} \cdot \nabla )\mathbf {F} +(\mathbf {V} \cdot \nabla )\mathbf {G} }, Given vector field ) + i 2 i V = + v , g ( + ) + i H = = >> Working with Tensors f ) Site design / logo 2022 Stack Exchange Inc; user contributions licensed under CC BY-SA. G ( G i + = + = F = 2 + F . + ( f + 2 i + G i A ( ) x V ( Closely associated with tensor calculus is the indicial or index notation. = ( r j , + ) Vector identities are then used to derive the electromagnetic wave equation from Maxwell's equation in free space. 0 ( x i ( G I ( k f ^ ) + ( 2 i ( i 1 ( H j i = = i x are second order tensors, we have, The references used may be made clearer with a different or consistent style of, Derivatives with respect to vectors and second-order tensors, Derivatives of scalar valued functions of vectors, Derivatives of vector valued functions of vectors, Derivatives of scalar valued functions of second-order tensors, Derivatives of tensor valued functions of second-order tensors, Curl of a first-order tensor (vector) field, Identities involving the curl of a tensor field, Derivative of the determinant of a second-order tensor, Derivatives of the invariants of a second-order tensor, Derivative of the second-order identity tensor, Derivative of a second-order tensor with respect to itself, Derivative of the inverse of a second-order tensor, Learn how and when to remove this template message, https://en.wikipedia.org/w/index.php?title=Tensor_derivative_(continuum_mechanics)&oldid=1118975222, Wikipedia references cleanup from June 2014, Articles covered by WikiProject Wikify from June 2014, All articles covered by WikiProject Wikify, Creative Commons Attribution-ShareAlike License 3.0, This page was last edited on 30 October 2022, at 02:12. {\displaystyle \mathbf {F} \times (\mathbf {G} \times \mathbf {H} )=(\mathbf {F} \cdot \mathbf {H} )\mathbf {G} -(\mathbf {F} \cdot \mathbf {G} )\mathbf {H} }, As another example of using the above notation, consider the scalar triple product 2 2 ( i V i + H ) G 1 ^ {\displaystyle \nabla ^{2}\mathbf {F} } x 2 x F + G v ) 2 ) i F = i i The Laplacian is a measure of how much a function is changing over a small sphere centered at the point. i ( i i 2 f + This relationship shows : (1) eijk eist =djs dkt-djt dks Our first step in motivating this proof will be to show how we can write determinants in terms of Kronecker deltas and permutation tensors. ( = {\displaystyle (\nabla \psi )^{\mathbf {T} }} i i x + ) f , i 2 ( ( i ( 2 i , j {\displaystyle {\frac {\partial {\hat {\mathbf {\rho } }}}{\partial \phi }}={\hat {\mathbf {\phi } }}} cos V A x i 3 ( ) 2 F x y G x + + z 2.2 Index Notation for Vector and Tensor Operations Operations on Cartesian components of vectors and tensors may be expressed very efficiently and clearly using index notation. r + ( r ) ( ) G f x i x , + ( {\displaystyle =(i,{\frac {\partial f}{\partial x_{i+1}}}G_{i+2}-{\frac {\partial f}{\partial x_{i+2}}}G_{i+1})+f(i,{\frac {\partial G_{i+2}}{\partial x_{i+1}}}-{\frac {\partial G_{i+1}}{\partial x_{i+2}}})} x v ) + In the second formula, the transposed gradient F ( i , we can write, Using the product rule for second order tensors, Another important operation related to tensor derivatives in continuum mechanics is integration by parts. + ( 1 ( Let a {\displaystyle \mathbf {a} } and b {\displaystyle \mathbf {b} } be two arbitrary vectors. i , ) ( 1 ) , i sin = {\displaystyle \nabla \cdot (\mathbf {F} +\mathbf {G} )=\sum _{i}({\frac {\partial }{\partial x_{i}}}(F_{i}+G_{i}))} F 1 = Proofs of Vector Identities Using Tensors Zaheer Uddin, Intikhab Ulfat University of Karachi, Pakistan ABSTRACT: The vector algebra and calculus are frequently used in many branches of Physics, for ( ) ( x ( . f i F r ) i G i i ) In a Cartesian coordinate system we have the following relations for a vector field v and a second-order tensor field F ) 1 is a generalized gradient operator. 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