02%20Vector%20Calculus%20and%20Indicial%20Notation6812%20F11

02%20Vector%20Calculus%20and%20Indicial%20Notation6812%20F11...

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EGM 6812, F11, University of Florida, M. Sheplak 1/15 Section 2, Vector Calculus and Indicial Notation 2 Vector Calculus and Indicial Notation 1 To analyze fluid dynamics, we need to formally define and discuss some properties of scalar, vectors, and tensors. The goal is to gain familiarity and comfort indicial notation and vector calculus. 2.1 Definitions & Notation 2 Advanced notation can compact the governing equations for various phenomena. Two forms of notation are reviewed/introduced here. Symbolic: A (benefit of symbolic notation is that you don’t have to choose a coordinate system) Symbolic (Gibbs) notation (such as the following) is NOT a function of a coordinate system. b DV pf Dt      Examples of symbolic: ,, A o Matrix representation: 1 2 3 A AA A    , with the Transpose,   1 2 3 T A A A A Indicial: i A where i is any index (limited to the Cartesian coordinate system) Indicial notation (such as the following) is limited to Cartesian coordinate systems i ji i b ij Du p f Dt x x     o Free Index Any index that appears only once in an additive term “for 1,2,3 j ” is implied for a free index free indices must balance for each additive term in an equation except for constants o Dummy Index (or Summation Index) Any index that appears twice in the same additive term 3 1 n ” or “from 1 to 3” are implied o Order of the expression “Rank of the Tensor”= # of free indices 1 Panton Chapter 3 2 Panton pg 33-41
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EGM 6812, F11, University of Florida, M. Sheplak 2/15 Section 2, Vector Calculus and Indicial Notation Zero order: Scalar ( T , p , ) First Order: “vector” ,, i i i j j u u x u x , where j is a dummy index and i is a free index Second Order: “Tensor” 1 2 j i ij ji u u xx      where i and j are free indices Third Order ijk Note: o Dimension : number of independent spatial variables o Steady or unsteady : function of time-unsteady; not function of time-steady Note: Do not confuse dimensionality with rank Example:       , V u x y i v x j w x k is 2-D but Rank 1 , U Ci C is a constant is 0-D but still Rank 1 o Alternative forms of indicial notation Comma notation       , 2 2 , 2 , , : free :free, :dummy , : free : free i ij j i i jj jj ij i i u u V i j x u u V i j u u V t i t  Subscripted del notation (Panton) 2 2 i j i j j i i i u u x u u u u t       
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EGM 6812, F11, University of Florida, M. Sheplak 3/15 Section 2, Vector Calculus and Indicial Notation 2.2 Vector Transformation Law 2.2.1 “Defines a Vector” Consider a right-handed Cartesian coordinate system (RHCCS) where p has the position (a vector) coordinates 1 2 3 , , ( , , ) x x x x y z that can be generally expressed as for 1,2,3 i xi If RHCCS is rotated to a new direction the coordinates of p will change to ' for ' 1,2,3 j xj 2 x 2' x 3
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02%20Vector%20Calculus%20and%20Indicial%20Notation6812%20F11...

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