Product Definition and 1000 Threads

  1. Saitama

    MHB Find Infinite Product: $\sqrt{\frac{1}{2}}$

    Find : $$ \sqrt{ \frac{1}{2}}\sqrt{ \frac{1}{2} + \frac{1}{2} \sqrt{\frac{1}{2}}} \sqrt{ \frac{1}{2} + \frac{1}{2} \sqrt{ \frac{1}{2} + \frac{1}{2} \sqrt{\frac{1}{2}}}}\sqrt{ \frac{1}{2} + \frac{1}{2} \sqrt{ \frac{1}{2} + \frac{1}{2} \sqrt{\frac{1}{2} +\frac{1}{2}\sqrt{\frac{1}{2}}}}} \cdots...
  2. O

    Product Detector vs Envelope Detector

    What will happen to product detector and envelope detector when modulation depth is increasing?
  3. M

    MHB Find the coefficient of x^{21} without calculating the product

    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...
  4. D

    What is the General Expression for the Product of Two Matrix Exponentials?

    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...
  5. T

    Understanding Dirac notation - Product of ops. is product of matrices

    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...
  6. J

    Sum and product of coefficient binomials

    Given two coefficient binomials \binom{a}{b} and \binom{c}{d} is possbile to express the sum and product those coefficient binomials as one other?
  7. M

    Dot Product of a Vector and its Derivative- Reality

    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...
  8. jk22

    Use of component tensor product in quantum mechanics?

    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 ...
  9. Sudharaka

    MHB Difference Between Tensor Product and Outer Product

    Hi everyone, :) Xristos Lymperopoulos on Facebook writes (>>link<<);
  10. E

    What is the geometric interpretation of the vector triple product?

    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...
  11. J

    Is the Cross Product Cancellative?

    If u × v = u × w, so v = w ?
  12. anthonyk2013

    How Can the Derivative of y=3x²ln(x) Be Simplified Further?

    y=3x2Inx u=3x2 v=Inx du/dx=6x dv/dx=I/x y=u*v dy/dx=(u)(dv/dx)+(v)(du/dx) dy/dx=(3x2)(I/x)+(Inx)(6x) Can I simplify this further
  13. U

    Finding the Product of Sequences a_n and b_n in Terms of x, y, and b_n

    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...
  14. Lebombo

    Projection Using Dot Product Finding a Force (Boat Problem)

    "Projection Using Dot Product" "Finding a Force" (Boat Problem) Homework Statement ------------------------------------------------------------------------------------------- A 600 pound boat sits on a ramp inclined at 30 degrees. What force is required to keep the boat from rolling down...
  15. D

    Distinguish pq Prime Cases: Algorithm for ppt

    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?
  16. B

    Fourier transform vs Inner product

    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...
  17. C

    Cartesian product of index family of sets

    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...
  18. S

    Calculate item from n-size cross product without creating product?

    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...
  19. C

    Product of the conjugates of a polynomial

    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...
  20. JasonHathaway

    Proof for Vectors Product: (A×B) . (B×A) + (A . B)^2 = A^2 . B^2"

    Homework Statement Proof that (A×B) . (B×A) + (A . B)^2= A^2 . B^2 Homework Equations A×(B×C)=(A . C)B - (A . B)C The Attempt at a Solution Assuming K=(A×B) K . (B×A) + (A . B)^2 = A^2 . B^2 B . (A×K) + (A . B)^2 = A^2 . B^2 B . [A×(A×B)] + (A . B)^2 = A^2 . B^2 B . [(A . B)A...
  21. JasonHathaway

    Proving the Mixed Product Formula for Vectors in R3

    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)]...
  22. J

    The Integration-By-Parts Formula for Vector Products

    We know that: \frac{d}{dx}(\vec{f} \cdot \vec{g}) = \frac{d\vec{f}}{dx} \cdot \vec{g} + \vec{f} \cdot \frac{d\vec{g}}{dt} and: \frac{d}{dx}(\vec{f} \times \vec{g}) = \frac{d\vec{f}}{dx} \times \vec{g} + \vec{f} \times \frac{d\vec{g}}{dt} But, exist some formula (some expansion) for: \int \vec{f}...
  23. MarkFL

    MHB Hp1357's question at Yahoo Answers regarding an extended product rule

    Here is the question: I have posted a link there to this thread so that the OP may view my work.
  24. R

    A question about the cross product as related to matrix multiplication

    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...
  25. anemone

    MHB How can $5^{1985}-1$ be factored into three integers greater than $5^{100}$?

    Factor $5^{1985}-1$ into a product of three integers, each of which is greater than $5^{100}$.
  26. J

    Property of the convolution product

    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...
  27. anemone

    MHB Find the floor of the product of a series

    Evaluate $\left|\dfrac{2017}{2013}\times \dfrac{2009}{2005}\times \cdots \times \dfrac{49}{45}\times \dfrac{41}{37}\times\dfrac{33}{29} \right|$.
  28. AntSC

    Quotient & Product Rule: Same funtion, different answers?

    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...
  29. Math Amateur

    MHB Is There Only One Possible Z-Linear Map from T to T'?

    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...
  30. S

    Why this property of the product of two matrices

    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...
  31. S

    Scalar Triple Product Derivative

    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...
  32. C

    Proving the dor product of 4-vectors is Lorentz invariant

    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
  33. V

    Mathematical connection in the cartesian product

    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...
  34. Z

    Finding perpendicular unit vector using inner product

    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...
  35. JasonHathaway

    What is the meaning of line integrals?

    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
  36. P

    Dot product of vector and del.

    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
  37. jdawg

    How do I solve the dot and cross product of 3i and jxk?

    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?
  38. S

    MHB These two problems are based on Vectors, dot product and distance for sphere.

    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 ...
  39. T

    Find the True Statement About Dot Product of Two Vectors

    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...
  40. Mechatron

    Product Log Function - Lambert W Function

    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...
  41. L

    Show that the dot product is linear: Bra-ket notation

    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...
  42. M

    What is the physical meaning of the dyadic product \vec{a} \vec{b}?

    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!
  43. T

    Product Rule Shortcut for Complicated Derivatives

    Find y' y=(x2+1)7(x9+2)5(x3+1)3(x8+7)3 Is there a shortcut to doing this problem? Or do I have to actually use the product rule more than 3 times?
  44. C

    The product of a vector and the length of a polar coordinate

    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...
  45. J

    Is There a Geometric Interpretation for the Multiplication or Power of Angles?

    Exist an geometric interpretation for the multiplication of 2 angles? Or exist an geometric interpretation for the square/cube of an angle?
  46. E

    Can the Vector Scalar Product Equation be Solved for x Using Cross Product?

    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?
  47. T

    Pythagorean theorem based on cross product.

    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...
  48. F

    Hurricane Fluid flow, torque, cross product

    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...
  49. U

    Inner Product in this step of the working

    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...
  50. S

    Prove that the absolute value of a dot product is less than or equal t

    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...
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