Hey! :o
"Let the polynomials
$$f(x)=1+\sum_{k=1}^{8}{(2k)x^{2k}} \text{ and } g(x)=1+ \sum_{k=1}^{8}{(3k)x^{3k}}$$
of $\mathbb{Q}$. Without calculating $f(x)g(x)$, find the coefficient of $x^{21}$ at $f(x)g(x)$."
Let's consider $f(x)=\sum_{k=0}^{\infty}{a_kx^k}$ and...
Is there an expression, in general, for the product of two matrix exponentials, for non-commuting matrices?
i.e. something of the form,
e^Ae^B = e^{( * )}
where the ( * ) would, I assume, depend in some way on the commutator [A,B] ?
I can only find examples online when [A,B] = 0...
Homework Statement
This makes intuitive sense to me, but I am getting stuck when trying to read the Dirac notation proof.
Anyway, the author (Shankar) is just demonstrating that the product of two operators is equal to the product of the matrices representing the factors.
Homework Equations...
Hey everyone,
This has been bugging me for a bit. I think I'm probably missing something pretty easy.
A dot B= ABcos(θ), where θ is the angle between A and B.
There is the little shortcut that says where B is the derivative of A, A dot B= AB. Clearly then cos(θ) = 1, and the angle between a...
suppose we consider the measurement operator A=diag(1,-1).
Then the tensor product of A by itself is in components : A\otimes A=a_{ij}a_{kl}=c_{ijkl}
giving c_{1111}=c_{2222}=1, c_{1122}=c_{2211}=-1 and all other component 0.
to diagonalize a tensor of order 4, we write ...
The interpretation of the vector product is the area of the parallelogram with sides made up of a and b and the scalar triple product is the volume of the parallelpiped with sides a, b, and c, but what is the interpretation of the vector triple product. Is it just simply the area of the...
Homework Statement
{a_n} and {b_n} are two sequences given by a_n = (x)^{1/2^n}+(y)^{1/2^n} and
b_n = (x)^{1/2^n}-(y)^{1/2^n}. Then find a_1 a_2 a_3 ... a_n in terms of x,y and b_n
Homework Equations
The Attempt at a Solution
I tried substituting n=1,2,3 and so on for a few...
"Projection Using Dot Product" "Finding a Force" (Boat Problem)
Homework Statement
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A 600 pound boat sits on a ramp inclined at 30 degrees. What force is required to keep the boat from rolling down...
Given pq where p < q are prime, but either (p \equiv 1 \mod 4 and q \equiv 3 \mod 4) or (p \equiv 3 \mod 4 and q \equiv 1 \mod 4).
Is there a ppt algorithm that will distinguish the two possibilities?
So the complex exponential Fourier series form an orthonormal basis for the space of functions. A periodic function can be represented with countably many elements from the basis, and an aperiodic function requires uncountably many elements.
Given a signal, we can find the coefficients of the...
Cartesian product of indexed family of sets
The definition of a Cartesian product of an indexed family of sets (X_i)_{i\in I} is \Pi_{i\in I}X_i=\left\{f:I \rightarrow \bigcup_{i \in I} \right\}
So if I understand correctly, it's a function that maps every index i to an element f(i) such...
Hi all,
I'm trying to work out how I can get the set at specific index in a cross product without creating the whole product.
For instance:
I have array A of length 512
I also have a number that specifies how many times that array needs to be 'cross-producted' against itself.
This gives me...
Two days ago, I was absolutely certain to have proved the theorem hereafter. But then, micromass pointed out that another theorem the truth of which I was also certain was in fact false. It seems that in mathematics, never be certain until other mathematicians are.
This is the reason why I...
Homework Statement
Prove that (A×B) . [(B×C)×(C×A)]=(A,B,C)^2
where A, B, C are vectors in R3.
Homework Equations
W×(U×V)=(W . V) U - (W × U) V
The Attempt at a Solution
Assuming K=(A×B), M=(B×C):
K . [M×(C×A)]
K . [(M . A) C - (M . C) A]
[(M . A)(K . C) - (M . C)(K . A)]...
I understand that the cross product, in lay mans terms doesn't exist unless we're in 3 dimensions.
When you multiply two matrices together I have been told you get something similar. I hear that this is because a matrix can be treated as a vector.
So if we are talking about measurable...
It is known that:\mathcal{F}\{f\ast g\}=\mathcal{F}\{f\}\mathcal{F}\{g\}\mathcal{F}\{f g\}=\mathcal{F}\{f\}\mathcal\ast{F}\{g\}
But this property is valid for inverse tranform too?\mathcal{F}^{-1}\{F\ast G\}=\mathcal{F}^{-1}\{F\} \mathcal{F}^{-1}\{G\}\mathcal{F}^{-1}\{F...
Can someone check my working. I don't understand why i am getting different answers?
u(x,t)=\frac{{e}^{-\frac{x^2}{4Dt}}}{\sqrt{4Dt}}
Differentiate w.r.t 't' by quotient rule:
\frac{\partial u}{\partial t}=\left[ \frac{1}{\sqrt{4Dt}}\cdot \frac{x^2}{4Dt^2}\cdot...
First, thanks to both Deveno and ThePerfectHacker for helping me to gain a basic understanding of tensor products of modules.
In a chat room discussion ThePerfectHacker suggested I show that {\mathbb{Z}}_a \otimes_\mathbb{Z} {\mathbb{Z}}_b where a and b are relatively prime integers - that is...
Hello,
The product of a 2x5 matrix P and a 5x3 matrix B shall be a 2x3 zero matrix. P and B are all matrices of integers.
P = [6 2 -5 -6 1;3 6 1 -6 -5]
One possible B is [0 -4 0;3 0 0;-1 -1 3;2 -3 -2;1 1 3]
This solution B has a property: det(PPt) = det(BtB) = 7778
The question is: What...
Homework Statement
Find an expression equivalent for the derivative of the scalar triple product
a(t) . (b(t) x c(t))The Attempt at a Solution
Initially I figured since whatever comes out of B X C is being dotted with A, I can use the derivative rules of a dot product:
(a(t)' . (b(t) x...
Homework Statement
Let A and B be 4-vectors. Show that the dot product of A and B is Lorentz invariant.
The Attempt at a Solution
Should I be trying to show that A.B=\gamma(A.B)?
Thanks
mathematical "connection" in the cartesian product
What is the mathematical connection between elements of a cartesian product ##A\times{}B## and the elements of the sets ##A## and ##B##?
In other words, what is the difference between the set ##A\times{}B## and just any set ##Z## with...
Homework Statement
You are given that with x = (x1,x2), y = (y1,y2), the formula
(x,y) = [x1 x2] [2 1;1 2] [y1;y2] (where ; represents a new row).
is a inner product for the vectors in R2
Using this inner product, find a unit vector perpendicular to the vector (1,1)
Homework Equations
The...
Hi everyone,
What does \int \vec{f} × \vec{dl} mean? does it mean "The Line Integral of Vector Function on a positive curve L"?
And are the following named correctly?
\int \vec{f} . dl : Scalar Line Integral of a Vector Function
\int udl : Line Integral of a Scalar Function
I'm not sure which section is best to post this question in.
I was wondering if the expression (u $ ∇) is the same as (∇ $ u).
Here $ represents the dot product (I couldn't find this symbol.
∇=del, the vector differentiation operator
and u is the velocity vector or any other vector
Homework Statement
The value of 3i(dot)(jxk)
Homework Equations
The Attempt at a Solution
I know the answer is 3, but could someone please explain how to work this problem?
Problem 1: Let S1 be a sphere centered at(0, 1, -3) with radius 1 and let S2 be a sphere centered at (3, 5, -9) with radius 2. Find the distance between the two spheres.
problem 2: Given three non-zero vectors v1, v2, v3 we say that they are mutually orthogonal when v1 dot v2= 0, v1 dot v3=0 ...
Homework Statement
The dot product of.two vectors is -1which of the following statements is true
A. They must be unit vectors pointing in opposite directions.
B. They must be unit vectors pointing j. The same direction.
C. They must be more than 90( and less than 270 )degrees from each...
I'm trying to solve the product log of (- (3/2) * e ^(3/2) ) as shown in the link below.
http://www.wolframalpha.com/input/?i=product+log+of+%28-+%283%2F2%29e^%283%2F2%29%29
This is a complex number of about 1 + 2i.
Should I ignore the complex part of the value and is the value (of...
Homework Statement
Show that the dot product in two-dimensional space is linear:
<u|(|v> + |w>) = <u|v> + <u|w>
The Attempt at a Solution
I feel like I'm missing some grasp of the concept here ...
I would think to just distribute the <u| and be done in that one step,
but I'm being...
hey pf!
can someone please explain physically what the dyadic product represents: \vec{a} \vec{b}
i know its a matrix (tensor), but that's all i know physically.
thanks!
Homework Statement
So I am not sure how to multiply these two (A*R^2) together.
Homework Equations
A=( x^2 + y^2 + z^2 ) (xe + y e + z e )
Where x represents the three vector compones
I also have R^2=x^2+y^2+z^2
The Attempt at a Solution
Is the product of A (x^3e + y^3 e + z^3...
I have been asked to solve the equation for x - All letters bar λ are vectors.
λx + (a cross x) = b
I have worked it down as far as
x.(λb + (b cross a)) = |b|^2
by taking the dot product with b of both sides.
But is there any way I can now solve this equation for x?
I was developing a pythagorean theorem proof based on the cross product of two vectors..Below is my final solution...My problem is I had to get around using the distance/magnitude formula because that is using the pythagorean theorem to prove the pythagorean theorem. But after searching, it may...
Homework Statement
In 1993 the radius of Hurricane Emily was about 350 km. The wind speed near the center ("eye") of the hurricane, whose radius was about 30 km, reached about 200 km/h. As air swirled in from the rim of the hurricane toward the eye, its angular momentum remained roughly...
Hi guys, I'm not sure how to evaluate this inner product at step (3.8)
I know that:
##\hat {H} |\phi> = E |\phi>##
<E_n|\frac{\hat H}{\hbar \omega} + \frac{1}{2}|E_n>
<E_n| \frac{\hat H}{\hbar \omega}|E_n> + <E_n|\frac{1}{2}|E_n>
I also know that ##<\psi|\hat Q | \psi>## gives the...
Homework Statement
That is prove that |a•c|≤|a||c| for any vector a=<a1,a2,a3> & c=<c1,c2,c3>
Homework Equations
The Attempt at a Solution
I really don't have much of an attempt at the solution. I am not sure where to start. I can kind of justify it in my mind by saying the...