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Math Help - What is the gradient of u(Ox) when O is orthogonal.

  1. #1
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    What is the gradient of u(Ox) when O is orthogonal.

    Suppose we know \nabla{u(x)}. Let O be an orthogonal matrix.

    What is \nabla{u(Ox)}?

    Thank you!
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  2. #2
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    See here.
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  3. #3
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    Yea, I don't understand exactly what that means though.

    Is it [LaTeX ERROR: Convert failed] ?
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  4. #4
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    I don't think there's a dot product there (except implicitly, in the definition of the gradient.) Looking at the equation applied to your case, we'd have

    \nabla(u(O\mathbf{x}))=O^{T}(\nabla u)(O\mathbf{x}).

    As I see it, the way to interpret the LHS of this equation is to take the vector \mathbf{x}, rotate by left-multiplying by O, then you stuff the result of the rotation into the function u, and finally, you take the gradient.

    I interpret the RHS of this equation as follows: first, you take the gradient of u, and then you evaluate that function at the point Ox, and finally you left-multiply by O^{T}.

    If I were you, I'd work out a simple example in, say, 2 dimensions. See how it works out.
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