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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

    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

    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

    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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    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

    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

    Homework Statement Suppose L, M, and N are subspaces of a vector space. (a) Show that the equation L \cap (M+N) = (L \cap M)+(L \cap N) is not necessarily true. (b) Prove that L \cap (M+(L \cap N))=(L \cap M) + (L \cap N) Homework Equations N/A The Attempt at a Solution...
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    Partitioned Orthogonal Matrix

    "Partitioned Orthogonal Matrix" Hi, I was reading the following theorem in the Matrix Computations book by Golub and Van Loan: If V_1 \in R^{n\times r} has orthonormal columns, then there exists V_2 \in R^{n\times (n-r)} such that, V = [V_1V_2] is orthogonal. Note that...
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    Poly function of degree n with no roots

    Homework Statement (a) If n is even find a polynomial function of degree n with n roots. (b) If n is odd find one with only one root. Homework Equations N/A The Attempt at a Solution If by no roots, they mean no real roots then I guess: f(x) = x^n+1 would work for both even...
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    Questions about proof of the division article.

    (Thread should be named: Question about proof of the division algorithm, sorry about that) Hi, I am reading this proof of the division article: http://xmlearning.maths.ed.ac.uk/lecture_notes/polynomials/division_algorithm/division_algorithm.php" [Broken] I will write some of it here in case...
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    Coupled mass problem

    Homework Statement Suppose masses m_{1}, m_{2}, m_{3}, m_{4} are located at positions x_{1}, x_{2}, x_{3}, x_{4} in a line and connected by springs with constants k_{12}, k_{23}, k_{34} whose natural lengths of extension are l_{12}, l_{23}, l_{34}. Let f_{1}, f_{2}, f_{3}, f_{4} denote the...
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    Drawing math figures

    Hi, What do you guys use for creating graphs and other figures for use in math/physics papers? Is there some industry standard being used by science book writers? Thanks.
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    [Linear Algebra] solution to A^TCAx=f

    Homework Statement With conductances c_{1}=1, c_{2}=c_{3}=2, multiply matrices to find A^TCAx = f . For f = (1,0,-1) find a solution to A^TCAx = f . Write the potentials x and currents y = -CAx on the triangle graph, when the current source f goes into node 1 and out from node 3...
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    Use Schwarz inequality to prove triangle inequality

    Homework Statement Use Schwarz inequality on \bar{v} \bullet \bar{w} to prove: ||\bar{v} + \bar{w}||^2 \leq (||\bar{v}|| + ||\bar{w}||)^2 Homework Equations Schwarz inequality: |\bar{v} \bullet \bar{w}| \leq ||\bar{v}|| ||\bar{w}|| The Attempt at a Solution The way I...
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    Prove that the matrices have the same rank.

    Homework Statement Prove that the three matrices have the same rank. \left[ \begin{array}{c} A\\ \end{array} \right] \left[ \begin{array}{c} A & A\\ \end{array} \right] \left[ \begin{array}{cc} A & A\\ A & A\\ \end{array} \right] Homework...
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    Giving math a shot (again).

    Hi, I am a working mechanical engineer (Bachelor), and am trying to learn math (again). My grades from virtually every math class I've taken are great, but I now know that's the case because my school was rubbish. The last two months I've done a bit of math daily. I'm currently going...
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    [Thermodynamics] Temperature gradient around a warm sphere.

    Hi, Say I have a sphere of radius r that has a constant surface temperature of T_s. The sphere is surrounded by air at a constant temperature T_amb. I am interested in the temperature gradient surrounding the sphere. From the little I know, I think i have to look at the natural...
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    [Linear Algebra] Nullspace equals Column space

    Homework Statement Why does no 3 by 3 matrix have a nullspace that equals its column space? Homework Equations NA The Attempt at a Solution A = \begin{bmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \] C(A) = \begin{bmatrix} 1 \\ 0 \\ 0...
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    Powers of a permutation matrix.

    Homework Statement If you take powers of a permutation matrix, why is some P^k eventually equal to I ? Homework Equations - The Attempt at a Solution From the solutions manual of the book: There are n! permutation matrices of order n. Eventually, two powers of P must...
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    Explain why cosine formula is always -0,5.

    Homework Statement Pick any numbers that add to: x + y + z = 0 Find the angle between your vector \textbf{v} = (x, y, z) and the vecor \textbf{w} = (z, x, y) Explain why \textbf{v}\bullet\textbf{w} / ||\textbf{v}||||\textbf{w}|| is always -\frac{1}{2} Homework Equations Cosine...
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    Five linearly independent 3X3 matrices

    Homework Statement In the space of 2 by 2 matrices, find a basis for the subspace of matrices whose row sums and column sums are all equal. (Extra credit: Find five linearly independent 3 by 3 matrices with this property) The Attempt at a Solution The first one is ok. The matrix is...
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    Permutation matrix and PA = LDU

    Homework Statement Find the PA = LDU factorizations for: A = \left[ \begin{array}{ccc} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 2 & 3 & 4 \end{array} \right] The author chooses a permutation matrix : P = \left[ \begin{array}{ccc} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{array} \right] If I do...
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    Points on a line

    Homework Statement Problem 1.2.10 from Linear Algebra and Its Applications by Gilbert Strang: Under what condition on y_{1}, y_{2}, y_{3} do the points (0,y_{1}), (1,y_{2}), (2,y_{3}) lie on a straight line? Homework Equations y = ax + b The Attempt at a Solution If y_{1}=0...
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    Which rules/theorems should you be able to prove?

    Hi, I was thinking back at the days when I took calculus and how we had to prove things like the product rule etc. Of course, most of that has magically disappeared from my mind. I've decided to try and take a masters degree in applied mathematics, and have about a year to get my head...
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    Subsets of two-dimensional space

    Homework Statement a) construct a subset of two-dimensional space closed under vector addition and even subtraction, but not under scalar multiplication. b) construct a subset of two-dimensional space (other than two opposite quadrants) closed under scalar multiplication but not under...
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    Multiple electrodes - capacitance.

    Hey, I have a length of pipe. Inside the pipe walls I have placed four electrodes. If one is giving out x volts, and I know the dielectric constant of the material between them, is there an easy way of finding the capacitance on the other three electrodes? I'm looking for hints as well as...
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    Linear Algebra - number of entries

    Homework Statement a) How many entries can be chosen independently, in a symmetric matrix of order n? b) How many entries can be chosen independently, in a skew-symmetric matrix of order n? Homework Equations The Attempt at a Solution All I know are the definitions of a...
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    Permutation symbol - indicial notation

    1. The problem statement and attempt at solution Hey, I'm still trying to get my head around indicial notation. I'm finding it quite hard.. I think this is somewhat right, but I don't know if the answer is clear enough.. Any hints/comments are greatly appretiated! Thank you
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    Tensor - the indicial notation, beginners problem

    1. The problem statement and attempt at solution Given the matrix S_ij and a_i evaluate a),b),c),d) and e) For a) I think i use Einsteins convention. b) I just first sum on i, and then on j giving me 9 terms. The answer I get is 24. d) can i change m with i since they are both dummy...
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    Continuum mechanics

    Hey, I've picked up a book on continuum mechanics to do a bit of studying by my self. As this topic is somewhat odd (difficult) I wonder if anyone knows of a place where students of continuum mechanics meet for discussion? I'm not sure if there is a subforum here at physicsforums which deals...
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