Recent content by Dafe

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    Vector calculus identities navigation

    Homework Statement I'm reading in a fluid dynamics book and in it the author shortens an equation using identities my rusty vector calculus brain cannot reproduce. Homework Equations \vec{e} \cdot \frac{\partial}{\partial t}(\rho \vec{u}) = -\nabla\cdot (\rho\vec{u})\cdot\vec{e} -...
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    Continuity equation - mass

    That makes sense. Thank you very much.
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    Continuity equation - mass

    Homework Statement I am having problems understanding the differential form of the conservation of mass. Say we have a small box with sides \Delta x_1, \Delta x_2, \Delta x_3. The conservation of mass says that the rate of accumulated mass in a control volume equals the rate of mass going in...
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    Prove that the dual norm is in fact a norm

    Ah, didn't think of it that way. Thank you very much.
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    Prove that the dual norm is in fact a norm

    Homework Statement Let ||\cdot || denote any norm on \mathbb{C}^m. The corresponding dual norm ||\cdot ||' is defined by the formula ||x||^=sup_{||y||=1}|y^*x|. Prove that ||\cdot ||' is a norm. Homework Equations I think the Hölder inequality is relevant: |x^*y|\leq ||x||_p ||y||_q...
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    Total energy required

    Here's my try. I'm using wolframalpha for the differentiation and integration... \frac{dh}{dT}=\frac{d}{dT}\left(\frac{f(T)}{g(T)}\right) =\frac{d}{dT}\left(\frac{1.28T}{378-3.16T}\right) =\frac{4.04T}{(378-3.16T)^2}+\frac{1.28}{378-3.16T} So now I have the change in time...
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    Total energy required

    Hi, a question at work popped up and it's been too long since I went to school :p The total energy [Wh] required to heat the system to temperature T is given by f(T)=1.28T. The effect [W] applied to the system is given by g(T)=378-3.16T. How long does it take to heat the material to say 80...
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    Academic Searches using Google

    Great stuff, thanks!
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    If m<n prove that y_1, ,y_m are linear functionals

    Homework Statement Prove that if m<n, and if y_1,\cdots,y_m are linear functionals on an n-dimensional vector space V, then there exists a non-zero vector x in V such that [x,y_j]=0 for j=1,\cdots,m. What does this result say about the solutions of linear equations? Homework Equations...
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    Span of subspace

    Ah, I certainly missed the point of the question! Let v be in the subspace spanned by y and z. Then v=ay+bz for some numbers a and b. But x+y+z=0 so z=-x-y. v=(a-b)y+(-bx), that is, v is a linear combination of y and x and so is in the span of y and x. This proves that the subpsace spanned by...
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    Span of subspace

    Homework Statement Here's a statement, and I am supposed to show that it holds. If x,y, and z are vectors such that x+y+z=0, then x and y span the same subspace as y and z. Homework Equations N/A The Attempt at a Solution If x+y+z=0 it means that the set {x,y,z} of vectors...
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    Proof with intersection of subspaces

    That makes sense and looks prettier. Thanks.
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    Proof with intersection of subspaces

    Thank you very much :)
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    Proof with intersection of subspaces

    Since m+n \in L,\;n\in L and L is a subspace (closed under vector addition), we know that m \in L? From (L \cap N) I know that n \in L, and from (L \cap M) I know that m \in L. m+n must also be in L since it is a subspace. Now, m+n \in (M+(L \cap N)) and m+n \in L and so x is an element of...
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