Form Definition and 1000 Threads

Homework Statement Complete the square using the symmetric matrix that defines the given quadratic form: ##x^2 - 4xy + 6xz + 2xt + 4y^2 + 2yz + 4yt + 5z^2 - 6zt - t^2## and write this quadratic as the sum and difference of squares after completing the square using the matrix. The Attempt at...

Closed form for "geometricish" series (index squared in the exponent)? Hi all, Is there a nice closed form for the following series? \sum_{k=0}^n x^{k^2} Even a decently tight upper bound and lower bound would be nice (obviously it is bounded by the corresponding geometric series \sum...

If I am given an E field E = uniti Asin[w(t-z/c)+pi/4]. Is the phasor form = unit i Aexp(j pi/4) or unit i Aexp(-j(wz/c - pi/4) ?

Homework Statement I need to find i1+i2 Homework Equations for i1=0.092<-98.86 i2=-4i1-j10/25=-4i1-0.4 i1=0.092<-98.86 x=rcosθ=0.092cos(-98.86)=-0.0142 y=rsinθ=0.092sin(-98.86)=-0.0909 i1=-0.0142-j0.0909 Therefor...

I know y is a function of x [i.e. y=f\left(x\right)] with two known boundary conditions, that is f\left(x=A\right)=C and f\left(x=B\right)=D where C and D are known constants (please see figure). I do not know the form of this function and therefore I am trying to find the form of y as a...

Hi everyone, :) Recently I encountered the following problem. Hope you can confirm whether my method is correct. My answer seems so trivial and I have doubts whether it is correct. Problem: Find the Jordan normal form of a unitary linear transformation. My Solution: Now if we take the...

Homework Statement Find all functions for which any tangent in first quadrant "forms" a triangle with constant surface P. (You can assume that y'<0) Homework Equations The Attempt at a Solution Now, I know I should somehow get to differential equation and then solve it but, I...

If $p$ is an odd prime expressible as $$p=x^2+5y^2$$ where $x,y$ are integers, then prove that $p \equiv 1 \text{ or }9\text{ (mod 20)}$.

if the greater of two numbers is one more than twice the smaller and the smaller is x, what is the bigger? if the bigger is x, what is the smaller number? my answer to the first part is 2x+1. but i don't know how to represent the second condition in a mathematical form. and can you also give...

Homework Statement Curious about this ...I have to find the sum. Homework Equations The Attempt at a Solution Ʃ (1/4)(-1/3)^n from 1 to infinity I want to know the proper form and why. Is it (1/4) Ʃ (-1/3)(1/3)^n-1 or (1/4) Ʃ (1/3)(-1/3)^(n-1) You get different answers

Homework Statement Hello, I am trying to put the following equations into matrix form in order to solve the system. If anyone could please explain to me how to do it or show me an example it would be awesome. All material given in question: For the system of inhomogeneous differential...

Hi, quite new in chemistry.. I would like to know the beginning of the atoms in the universe. Therefore I must understand the process from Hydrogen to Helium. But to create Helium you need the isotope(#2) "Deuterium" from Hydrogen. But how did the Deuterium form? From two Protons...

Homework Statement {U_0 = 9, U_1 = -3} U_(n+2) = -(5/4) U_(n+1) + (3/8) U_(n) Homework Equations The Attempt at a Solution First step was to attempt to find the common difference by trying to find the 3rd term: U_(2) = -(5/4) u_(1) + 3/8 U_(0) = -(57/8) This does not...

I am solving a question that asks me to find an H field in phasor form from the given E field in phasor form Es = j30(beta)(I)(dl)sin(theta)e^(-j(beta)r) a(theta) V/m Given that the EM wave propagates in free space. Why do I get different answers if I : 1) Divide Es by the magnitude...

Hello, I'm a second year physics student and I'm interested in new approaches to spacecraft propulsion. I apologize if this has been asked many times - while this interest may be partially inspired by science fiction, I am well aware that the real-life situation is not at all as glamorous...

Hi everyone, :) Take a look at the following question. Problem: Determine which of the following quadratic functions \(q_1,\,q_2:\,V\rightarrow\Re\) is positive definite and find a basis of \(V\) where one of \(q_1,\,q_2\) has normal form and the other canonical...

Hi everyone, :) Here's a question with the summary of my method of how to solve it. I would really appreciate if you could go through it and let me know if there are any mistakes with my approach. Also are there any easier methods? Problem: Find an orthogonal transformation that reduces the...

Hi! I have encountered a little problem. I want to show that the explicit form of the Feynman propagator for massless scalar fields is given by: \begin{align} G_F(x) & = - \lim_{\epsilon \to +0} \int \dfrac{\mathrm{d}^{4}k}{(2 \pi)^{4}} \dfrac{1}{k^{2} + i \epsilon} \mathrm{e}^{- i k...

Hello, I've been looking for the form of the solution(s) to following differential equations: \frac{\partial^2}{\partial x \partial y}f(x,y) = a \cdot g(x,y) \frac{\partial^2}{\partial x \partial y}g(x,y) = b \cdot f(x,y) Where a and b are unrelated constants, and f,g are of the same...

If this is in the wrong section, please let me know. I will gladly re-post to the correct area. Here is a real problem we face at work, and I would like some help quantifying it. We have a round form, whose diameter is adjustable. Air cylinders are used to expand or contract the overall...

Hi! I have a question about Kähler manifolds. Of course there are many books (I prefer Nakahara) and lecture notes on this topic, but as a physicist I need a very hands-on way of dealing with metrics, etc. Given a metric, what is the simplest way to find the Kähler form and to prove the...

another question: convert $|\frac{1-i}{3}|$ to polar form i am getting $\frac{\sqrt{2}}{3} e^{\frac{i\pi}{4}}$ but the solutions say: $e^{\frac{-i\pi}{4}}$ i did $ x = r\cos(\theta)$ and $y=r\sin(\theta)$ so $\frac{1}{3} = {\frac{\sqrt{2}}{3}}\cos(\theta)$ $\frac{1}{3} = \cos(\theta)$ And...

Hi everyone, :) Here's a question I found on a Wiki book. Question: The matrix \(T\) is \(5\times 5\) with the single eigenvalue 3. The nullities of the powers are: \((T-3I)\) has nullity two, \((T-3I)^2\) has nullity three, \((T-3I)^3\) has nullity four, and \((T-3I)^4\) has nullity five...

Hi everyone, :) Here's a question that I encountered recently. I would appreciate if you could go through my solution and let me know if you see any mistakes or have any comments. Question: Given a linear transformation \(f:\,\mathbb{C}\rightarrow \mathbb{C}\) with matrix...

y dx-4(x+y^6)dy=0 so is it legal to add 4(x+y^6)dy to both sides of the equation? I thought it's bad to add integrands (i.e. dx or dy). Or is it legal to divide both sides of the equation by dx? Would this result in y-4(x+y^6)\frac{dy}{dx}=0 or y-\frac{4(x+y^6)}{dx}\frac{dy}{dx}=0

I started of with attempting to convert the numerator first $ | 1 + i | = \sqrt{1^2+i^2}$ $= \sqrt{1-1} = 0$ ? this is wrong obviously, i don't see why its $\sqrt{2}$ for the second part $ |\sqrt{3} - i|= \sqrt{3+1} = 2$ $ x = r \cos\theta$ $ y = r\sin\theta$ $x = 2\cos\theta$ $...

A committee of 12 is to be selected from 10 men and 10 women. In how many ways can the selection be carried out if there must be an even number women? I answered {10 \choose 2}{10 \choose 10}+{10 \choose 4}{10 \choose 8}+{10 \choose 6}{10 \choose 6}+{10 \choose 8}{10 \choose 4}+{10 \choose...

Sorry. I know I must sound like a hick. Time is ... the 4th dimension of our conventional reference? If so, are one or more of the other dimensions posited by String Theory considered possibly to be an additional time dimension(s)?

Let ##f(x,y)=(x,y,h(x,y))## be a parametrization of the graph ##T_h## of ##h:\mathbb{R}^2\to \mathbb{R}##. Compute the first fundamental forms for ##T_h## and also compute the second fundamental form. For the first fundamental form. I got that ##f_u = \langle 1, 0, f_u \rangle## and ##f_v...

Find the standard form of the equation of the parabola with the given characteristics and vertex at the origin. Passes through the point (-5, 1/8); vertical axis. I know that there is no focus of the parabola or equation given for this problem, so how would I solve this problem? Is the correct...

Find the standard form of the equation of the parabola with the given characteristics and vertex at the origin. Passes through the point (-1, 1/8); vertical axis. I know that there is no focus of the parabola or equation given for this problem, so how would I solve this problem? Is the correct...

Hi everyone, :) I have very limited knowledge on linear algebra and things like Jordan Normal form of matrices. However I am currently doing an Advanced Linear Algebra course which is compulsory and I am trying hard to understand the content which is quite difficult for me. One of the things...

how do you write a polynomial in three variables say x,y,z in standard form?

Homework Statement Use vectors to demonstrate that on a circle any two diametrically opposed points along with an arbitrary third point(on the circle) form a right triangle Homework Equations Hint: assume without a loss of generality that the circle is centered at the origin and let v...

Hi there, During a recent tutorial, I asked my tutor about the notation he uses for vectors - he draws the little half arrow above them and I was curious whether that was significant, as opposed to just underlining vectors. He said it was a mathematical technicality and suggested I look up...

Homework Statement Using that \frac{1}{1-x} = \sum_{n=0}^{\infty} x^n for |x|<1 and that f'(x) =\sum_{n=0}^{\infty} (n+1)a_{n+1}(x-x_0)^n , write \sum_{n=0}^{\infty} n^2x^n in closed form. Homework Equations The Attempt at a Solution In this series, a_n = n^2 and x_0 = 0 ...

Here are the questions: I have posted a link there to this topic so the OP can see my work.

I just got a new Ti-nspire Cx CAS calculator and I am having trouble with being able to express a Radical in simplified form when there are exponents and variables of x and y. My problem is that this calculator will not show the simplified form correctly. I have taken others advice in setting...

calculations and boxed answers are mine my question is with (iii) (provided the (i) and (ii) are correct. from the diagram $cos \theta$ would be $\frac{OC}{OA}$ or $\frac{11.2}{13}$ but would need the the length of $OC$ to do so, what would be the best approach to get point $C$ we could...

So we just started finding general solutions for homogenous&linear two-variabled PDEs by using separation of variables in my engineering-math class. There the professor tells us to assume the solution of a PDE is in the form of F(x)*G(t). But when is the solution in the form of F(x)*G(t)? When...

The problem is that the Earth has lost all velocity and begins plummeting toward the sun. I need to find the time it takes for it to hit the sun. Note: Primes indicate "dummy variables" This solution begins with the Work K.E. Theorem...

Homework Statement I have this problem where the Earth immediately loses all orbital velocity and begins to fall towards the sun, and I need to find the time it takes for the Earth to hit it. Seemed straight forward enough. Homework Equations Started with the work k.e. theorem...

Dark Energy is said to be present in order to drive the observed acceleration of the expansion of the universe and Dark Matter is hypothesised to be present to drive observed gravitational effects. Would a large mass of dark matter distributed around the outer parts of the universe have the...

Homework Statement For my waves class, I have to do this problem. I've previously completed a question like this except there was no phase constant (∏/4) in that question. Express the following in the form x = Re [Ae^i\alphae^iwt x=cos(wt + ∏/4) - sin(wt) Homework Equations euler's...

Sorry if i post this in the wrong spot. I am trying to form the curve of the half quadrant of a circle. And i wonder that how do we know which or where is our control point? For cubic bezier, the 2nd control point should be on the tangent line of the starting point and the 3rd control point...

Homework Statement 1. Create a table of values for the dimensions of a cylinder with a volume of 49.54 cubic inches. Does it appear that the cleanser container minimizes surface area? 2. Suppose you are designing a coffee creamer container that has a volume of 48.42 cubic inches. Use the...

I am reading R.Y Sharp's book: "Steps in Commutative Algebra". On page 6 in 1.11 Lemma, we have the following: [see attachment] "Let S be a subring of the ring R, and let \Gamma be a subset of R. Then S[ \Gamma ] is defined as the intersection of all subrings of R which contain S and...

Homework Statement I want to find the transition matrix for the rational canonical form of the matrix A below. Homework Equations The Attempt at a Solution Let ##A## be the 3x3 matrix ##\begin{bmatrix} 3 & 4 & 0 \\-1 & -3 & -2 \\ 1 & 2 & 1 \end{bmatrix}## The...

I read that a point relative to origin in 3D coordinate system is conveniently specified by the vector form, why?

Find a vector equation of the line passing through $(–1, 4)$ and $(3, –1)$. Give answer in the form $$r = p + td$$, where $$t \in {R}$$ position vector would be $$(3, -1) = p$$ direction vector would be (3+1,-1-4) = (4,-5) = d so $$r=(3,-1)+t(4,-5)$$