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Rewrite in index notation

  1. Sep 30, 2009 #1
    w=∇×u Is this correct? w_i=ε_ijk ∂/(∂x_j ) u_k
    w and u are the vectors

    C=(x∙y)z Is this correct? C_i= ∑_i〖(x_i y_j)∙z_i 〗
    C, x, y, z are vectors

    A^T∙A ∙x=A^T∙b Is this correct? A_ij^T∙A_ij∙x_j=A_ij^T∙b_i
    A is tensor and x and b are vectors

    A^T is A transpose
     
  2. jcsd
  3. Sep 30, 2009 #2

    HallsofIvy

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    Assuming you are not using subscripts and superscripts to distinguish between a vector and its transpose, yes, that is correct.
     
  4. Sep 30, 2009 #3
    C=(x∙y)z Is this correct? C_i= ∑_i〖(x_i y_j)∙z_i 〗
    C, x, y, z are vectors

    Hall is usually right on, though this one should be (assuming orthonormal coordinates)
    C=(x∙y)z C_j = z_i ∑_j (x_j y_j)
     
  5. Sep 30, 2009 #4
    I want to make sure the way I am writing this A^T dot A dot x = A^T dot b in index notation correctly.

    Would you mind to do that one time for me so that I can match up with my answer?

    My answer was

    A^T dot A dot x = A^T dot b >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>

    >>>>>>>>>>>>>>> A_ij^T dot A_ij dot x_j = A_ij^T dot b_i

    A_ij^T mean A_ij is a tensor with ij components and A^T mean the transpose of A
    x and b are both vector

    This problem was asking me to rewrite the A^T dot A dot x = A^T dot b in index notation. Thank you!
     
  6. Sep 30, 2009 #5
    Just a quick hint before I knock off. The transpose of a matrix interchanges rows with columns. (A_ij)^T = A_ji.

    To make the conversion to index notation it's convenient to think of the first index as indexing rows and the second as indexing columns:

    C dot D = ∑_j (C_ij D_jk)
     
  7. Sep 30, 2009 #6
    Is this correct? ∑_ j A_ji dot A_ij dot x_j = ∑_ i A_ij^T dot b_i ?
     
  8. Oct 1, 2009 #7
    The summation should occur between only two tensors. As you've written it, you are summing over all three on the left hand side.

    I should have noticed this earlier, but you should place the dot operator only between two vectors.

    U·V = UVT = ∑i Ui Vi

    Vector U is multiplied with the transpose of vector V.

    So we should write the expression AT·Ax as

    ATAxT, where A is a matrix, ATA is a matrix, and x is a row vector.

    ATA is equal to AijT multiplied by Aij, which means we are going to multiply the rows of the first matrix by the rows of the second

    ATA = ∑i Aij Aij

    ------------------------------------------------------------------
    For cut and paste.

    α β γ δ ε ζ η θ ι κ λ μ ν ξ ο π ρ ς σ τ υ φ χ ψ ω . . . . . Γ Δ Θ Λ Ξ Π Σ Φ Ψ Ω
    ∂ ∫ ∏ ∑ . . . . . ← → ↓ ↑ ↔ . . . . . ± − · × ÷ √ . . . . . ¼ ½ ¾ ⅛ ⅜ ⅝ ⅞
    ∞ ° ² ³ ⁿ Å . . . . . ~ ≈ ≠ ≡ ≤ ≥ « » . . . . . † ‼
     
    Last edited: Oct 1, 2009
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