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1. ### 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} -...
2. ### Continuity equation - mass

That makes sense. Thank you very much.
3. ### 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...
4. ### Prove that the dual norm is in fact a norm

Ah, didn't think of it that way. Thank you very much.
5. ### 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...
6. ### 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...
7. ### 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...

Great stuff, thanks!

Gentle bump.
10. ### 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...
11. ### 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...
12. ### 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...
13. ### Proof with intersection of subspaces

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

Thank you very much :)
15. ### 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...
16. ### Proof with intersection of subspaces

From our assumption that x\in L \cap (M+(L\cap N)) , we have that x\in L. Since x=m+n we have that m\in L and n \in L , so L \cap (M+(L\cap N)) \subset (L\cap M)+(L\cap N) . Is that it or do I have to show that (L\cap M)+(L\cap N)\subset L\cap (M+(L\cap N)) ? Suppose x \in (L\cap...
17. ### 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...
18. ### 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...
19. ### Does my proof make sense?

You are too kind, thanks. I feel comfortable with this now.
20. ### Poly function of degree n with no roots

Ah, I should have stated that, thank you. You are usually the one that answers all questions I post around here. It's incredible that you do that for free..You should set up a paypal account :) Thanks again.
21. ### 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...
22. ### Does my proof make sense?

Hi, Here's my try at the n=3 case: \begin{tabular}{ r c l } $$f(x)$$ & $$=$$ & c_3x^3+c_2x^2+c_1x+c_0\) \\ & $$=$$ & c_3x^3-c_3x^2a+c_3x^2a+c_2x^2+c_1x+c_0\) \\ & $$=$$ & (x-a)c_3x^2+c_3x^2a+c_2x^2+c_1x+c_0\) \end{tabular} c_3x^2a+c_2x^2+c_1x+c_0 is a polynomial of...
23. ### Does my proof make sense?

Since we show that f(x) can be written as (x-a)g(x)+b for the n=1 case, we can assume that it holds for k \leq n, just as you wrote in an earlier post. Then we check to see if the statement holds for the n+1 case. I understand that (to the extent I can understand anything). f(x)=c_1x+c_0 if...
24. ### Does my proof make sense?

Hi, could someone explain how we go from 1. to 2. in the expressions below? I fail to see how c_{n+1}x^na=(x-a)r(x)+k_1 Thanks. \begin{align*} 1.f(x) &= (x-a)(c_{n+1}x^n) + c_{n+1}x^na + (x-a)q(x) + k_0\\ 2.f(x) &= (x-a)(c_{n+1}x^n) + (x-a)r(x) + k_1 + (x-a)q(x) + k_0\\ \end{align*}
25. ### Questions about proof of the division article.

Thank you :blushing: One more question (kind of the same): The article goes on with the induction step: Now we assume that this is true whenever d<k and let d=k, so that m=n+k. Let f_1=f-(\frac{a_m}{b_n}x^{m-n}g). I do not understand this last step. Since \frac{a_m}{b_n}x^{m-n}...
26. ### 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...
27. ### Spivak's Calculus polynomial question

Ah, now I get it :) Thank you both very much!
28. ### Spivak's Calculus polynomial question

Ok, so my new function is, f(x) = \sum^n_{i=1}a_if_i(x) if I put x_i into this new function, I would get: f(x_i) = a_1+a_2+...+a_n The problem asks for a function where f(x_1)=a_i . Does this imply a sum over the a_i's? By the way, I am in no way saying your answer is wrong...
29. ### Spivak's Calculus polynomial question

First of all, I know that this thread is very old, but since I am working on this exact problem I assume it is better not to create a new thread. (+ it shows that I did a search :) ) Here's my attempt: f_i(x) = \prod^n_{\frac{j=1}{j\neq i}} \frac{x-x_j}{x_i-x_j} The next part of this...
30. ### Coupled mass problem

Here's my complete solution. I expand and simplify the equations given in my first post. Then I put together expressions for the x's and separate the constants. I'm sorry about the formatting. Does this look ok? \[ f_{1} = k_{12}(x_{2}-x_{1}-l_{12}) = k_{12}x_{2} - k_{12}x_{1} -...
31. ### Coupled mass problem

By the way, here is a figure showing how I am visualizing this problem: I am assuming m_{1} to be constrained. Just throwing out ideas :)
32. ### 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...
33. ### Nullspace of mxn matrix

We know that the row space is in R^{n} and that it is orthogonal to the null space. Imagine that we have a 2x3 matrix with rank 2. It's row space would be a plane in R^{3}, and it's null space a line perpendicular to that plane. If we pick a point x in that plane, wouldn't the point closest to...
34. ### Drawing math figures

That looks sweet trambolin! Thanks
35. ### Drawing math figures

TikZ looks interesting. I really do not mind a steep learning curve :) Thanks!
36. ### Drawing math figures

CompuChip: Does mathematica have a drawing tool? Say you want to explain how projection matrices work, and would like to draw a plane and some vectores. Is this easily done in Mathematica? Asymptote looks interesting and quite hard to learn.. Thanks for the reply.
37. ### 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.
38. ### Best Beginner Programming Language?

I've heard people saying that Java is not good as a first programming language. I fail to see why though, care to explain? Thanks
39. ### Best Beginner Programming Language?

I notice that many universities use Java as an introduction to programming and CS. Stanford has a free online Java class, but I would not recommend it as they use custom libraries. This makes it harder to get help at various discussion forums. MIT uses Python and they have an entire class...
40. ### Can explain to me what is mean of span?

I think it helps to think of span just as Mark44 explains it. In R^3 a vector such as <1, 0, 0> would span a line, in this case the x-axis. You can find any point on the x-axis by multiplying the vector with a scalar; 3<1, 0, 0> gives you a point (3,0,0) on the x-axis. Then if you have two...
41. ### Subspace of Polynomials of degree 2

Maybe it would help if you see it like this: (I'm no good with latex, sorry). \left[ \begin{array}{ccc} a_{11} & a_{12} & 0 \\ a_{21} & a_{22} & -1 \\ a_{31} & a_{32} & 1\end{array} \right] [ \begin{array}{ccc} a_{0} \\ a_{1} \\ a_{2}\end{array} ] = 0
42. ### Why would f(x) not be included in the determinant?

It follows from the fact that det I = 1. By eliminating entries above the pivots in your upper triangular matrix, it can be made into a diagonal matrix. Since we also know that the determinant is a linear function of each row/column separately, we may factor out: a_{i,j} for i = j you get...
43. ### I need a better Linear Algebra book

Hi, I am currently half-way through Strangs Introduction to Linear Algebra, Second Edition. Together with his online lectures, assignments, exams and other goodies, I feel it's a nice way to learn the subject. After looking through a couple of other books, I get the idea that Strangs...
44. ### [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...
45. ### Linear Algebra Exam Papers

Do a search for Linear Algebra MIT OCW.
46. ### Use Schwarz inequality to prove triangle inequality

||\bar{v} + \bar{w}||^2 = \bar{v} \bullet \bar{v} + 2\bar{v} \bullet \bar{w} + \bar{w} \bullet \bar{w} \leq ||\bar{v}||^2 + 2||\bar{v}|| ||\bar{w}|| + ||\bar{w}||^2 = (||\bar{v}|| + ||\bar{w}||)^2 I think I see the Schwarz in there :) Thank you lanedance.
47. ### 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...
48. ### Prove that the matrices have the same rank.

Hi Billy Bob, thanks for the reply. Here's the way I think you would do it: (just showing one matrix) \left[ \begin{array}{c} A & A\\ \end{array} \right] \left[ \begin{array}{c} R & 0\\ \end{array} \right]...
49. ### 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...
50. ### Giving math a shot (again).

Great post lurflurf! I will get a copy of that book, thanks!