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...