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} -...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
"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...
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.
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...
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...
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...
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*}
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}...
(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...
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...
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...
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} -...