Geometric Definition and 790 Threads

I am a beginner in theory of GR and am trying to understand it better. I have a problem with understanding tensors. I got the algebriac idea, incliding covariance, contravariance and transformations etc of tensors. But not the geometric. Tensors are abstract but can I not have geometric...

Homework Statement Find y component of vector C from its length and the angle it makes with the x axis, that is, from geometry. Express the y component of vector C in terms of C and \phi. Homework Equations Vector addition using geometry: 1) C = \sqrt{A^{2}+B^{2}-2ABcos(c)}...

Hello everyone on these forums. :) If you would, please consider the 3-vector function r(t) = <f(t),g(t),h(t)>. What sort of geometric meaning can be assigned to the following integral? \int_a^b \vec{r}(t) dt = \left\langle \int_a^b f(t) dt, \int_a^b g(t) dt, \int_a^b h(t) dt\right\rangle...

Until today I learned in geometric optics that Object distance +ve for real object else -ve Image distance +ve for real image else -ve Radius of curvature +ve for if light comes to the surcace from the side lying center of curvature else -ve On the basis of this the lens formula...

Homework Statement we had a a function on a graph of f(x)=1/x and then we are suposed to find the area of a triangle where the tangent line is the hypontenuse, and the x and y-axis are the base and height...i found f'(x)= -1/x^2 from here i used the formula y-yo=x0(x-x0) and got that the x...

Homework Statement I already counted V_{0}=-1 and q=\frac{1}{3} given: V_{n}=1-\frac{2}{U_{n}} Homework Equations count: \sum_{k=0}^{n}V_{k} The Attempt at a Solution i counted the sum and i got : ((\frac{1}{3})^{n+1}-1)(\frac{2}{3}) is that correct?

i guys, I'm stuck on wording of a homework assignment and thought you might be able to help me. There are several questions... Consider the geometric series: (Sum from k=0 to infinity) of ar^k and consider the repeating decimal .717171717171 for these problems: Question 1: Find a formula...

Hi, I'm trying to come up with a probability for a game I play with a friend of mine. In the game, units "attack" by rolling six-sided dice; either 2 or 4 sides of the die count as a "hit" when rolled, depending on certain circumstances. The specific situation I am trying to figure out the...

Homework Statement I have a question with asks to solve a differential equation via power series and I've done everything up to finding the recurrence relation which is a_{n+2} = -\frac{a_{n}}{n+2} Given the initial conditions a_{o} = 1 and a_{1} = 0 I'm trying to simplify the series into a...

Hello everyone! My question is twofold. Firstly, how do I solve for term numbers in a geometric sequence and secondly, how do I algebraically solve for variables that are exponents? Homework Statement Given the following geometric sequences, determine the number of terms, n. t1=5 r...

Homework Statement x,y, z are vectors in R^n. We have the equation: ax +by +cz, where a,b,c are constants such that a+b+c=1, and a,b,c>=0 What is the geometric interpretation of the equation? Homework Equations sv + tu, where u,v are vectors in R^n and s,t are constants such that...

Homework Statement I am trying to prove the sum of a geometric series, but one of the steps involves deriving this result: \lim_{n\to\infty}r^{n}=0 so that you can simplify the sum of a geometric series, where I have got to this stage: S_{\infty} = \frac{a(1-r^{\infty})}{1-r}...

Homework Statement The sum of an infinite geometric sequence is 131/2, and the sum of the first three terms is 13. Find the first term. Homework Equations S∞ = a/(1-r) Sn = a-arn/(1-r) The Attempt at a Solution a/(1-r) = 131/2 a-ar3/(1-r) = 13 2a = 27-27r ...... 1 a-ar3 =...

I've seen a number of books and articles touting Geometric Algebra as an important new area of math that will have large application to physics. Is there anything to these claims? Is it worth studying for a physics student?

Homework Statement Find the sum of Ʃ(3→∞) 3(.4)^(n+2) Homework Equations Sum of Geometric Series = ao/(1 - r), ao=3, r = .4 = 2/5 The Attempt at a Solution I thought that I could use the definition of the sum of a geometric series (above) to determine the sum of this equation...

Say we have an eigenvalue \lambda and corresponding eigenvectors of the form (x,x,2x)^T. What is the geometric multiplicity?

Dot product proof question? Hi, I'm having trouble understanding the proof of the dot product in three dimensions (not using the cosine rule approach). Here's what I have for the 2D proof: u = u1 i + u2 j v = v1 i + v2 j u.v = u1v1 + u2v2 u.v = |u| |v| cos(θ) => u1v1 + u2v2 = |u| |v|...

Homework Statement A 20 × 20 matrix C has characteristic polynomial (λ^2 − 4)^10. It is given that ker(C−2I), ker (C − 2I)^2, ker (C −2I)^3 and ker (C −2I)^4 have dimensions 3,6,8,10 respectively. It is given that ker (C + 2I), ker (C +2I)^2, ker (C +2I)^3 and ker (C +2I)^4 have di- mensions...

I've been browsing through MTW recently and I found something that puzzles me: They claim that if you have two form, call it \mathbf{T}, it's value, say \mathbf{T}(\mathbf{u} , \mathbf{v} ) can be represented geometrically as follows: take two vectors \mathbf{u} and \mathbf{v}; the surface...

Bill Alsept started a thread raising the general question---do cosmic models with regularly repeating big bangs conflict with thermodynamics' 2nd Law? (The law to the effect that, where it can be defined, entropy does not decrease, or does so only by rare accident, at irregular intervals if at...

Homework Statement I feel bad asking another question after I just asked one yesterday, but I'm really close this time, I think! I have: \sum_{n=2}^{\infty}\frac{n^2-n}{2^n} And need to find the sum. Homework Equations \sum_{n=1}^{\infty}nx^{n-1}=\frac{1}{(1-x)^2} The Attempt at a...

Consider the following experiment: a coin that lands heads with probability p is flipped once; if on this first flip it came up H, it is then repeatedly flipped until a T occurs; else, if on the first...

This has been a curiosity of mine lately. I am wondering about what makes an algebra person an algebra person. I know geometers(at least it seems like it) seem to have a keen ability of spatial visualization. What characterizes the abilities of an algebra person? To clarify, I'm not just talking...

Can anyone help me answer this question? " Every exponential function is a geometric progression but not every geometric progression is an exponential function. Explain."

Geometric series problem urgent Homework Statement Calculate the geometric series of Ʃfrom n=1 to infinity of 1/n Homework Equations The Attempt at a Solution I don't know how to start solving, how can I solve this? I have test about this tomorrow I really need some help please.

Statistics: geometric distribution "proof" problem Homework Statement If Y has a geometric distribution with success probability p, show that: P(Y = an odd integer) = \frac{p}{1-q^{2}} Homework Equations p(y)=p(q)^{2} The Attempt at a Solution p(1)=pq^0 p(3)=pq^2 p(5)=pq^4...

Hi, All: Just curious to know if there is an interpretation for lower cohomology that is as "nice", as that of the lower fundamental groups, i.e., Pi_0(X) =0 if X is path-connected (continuous maps from S^0:={-1,1} into a space X are constant), and Pi_1(X)=0 if X is...

Hello, I have the following equation in x and y: xy - \sqrt{(x^2+a^2)(y^2+c^2)} = -\frac{1}{a^2}-\frac{1}{c^2} where the quantities a2 and c2 are given real constants, and I have to find real values for x, and y such that the equation above is always satisfied. Actually, I know that the...

For a homogeneous system of 3 equations in 3 unknowns (geometrically this is 3 planes in space all containing the origin) describe the relationship between the (three) geometric possibilities for the solution set and the number of free variables (non pivots) in RREF(A) where A is the...

Homework Statement If O is any point inside a triangle ABC, prove that BA + AC > BO + OC. Homework Equations The Attempt at a Solution Any hints? Thanks...

I need to find the solution to the geometric series expansion of the form... \sumn^2*x^n , for n=0,1,2,... most resources I've found only have answers for n*x^n or n*x^(n-1). I have no idea how to calculate this, so I was wondering if there's a book out there that has massive lists of...

I am currently doing a course on Computer Graphics Algorithms. This involves lot of matrix transformations i.e. for eg - rotating co-ordinates, translating, reflecting etc. I am solving the problems on paper using a calculator, but I need some software which will help me verify the solution...

We're covering this in my History of Mathematics class. I'm not entirely sure what they're asking.

Homework Statement Q.: Show that if log a, log b and log c are three consecutive terms of an arithmetic sequence, then a, b and c are in geomtric sequence. Homework Equations Un = a + (n - 1)d and Sn = \frac{a(r^n - 1)}{r - 1} The Attempt at a Solution Attempt: Consider...

Homework Statement Q.: The sum of the first five terms of a geometric series is 5 and the sum of the next five terms is 1215. Find the common ratio of this series. Homework Equations Sn = \frac{a(r^n - 1)}{r - 1} The Attempt at a Solution a + ar + ar^2 + ar^3 + ar^4 = 5 ar^5 +...

Homework Statement Q.: The numbers \frac{1}{t}, \frac{1}{t - 1}, \frac{1}{t + 2} are the first, second and third terms of a geometric sequence. Find (i) the value of t, (ii) the sum to infinity of the series. Homework Equations S\infty = \frac{a}{1 - r} The Attempt at a...

Homework Statement Q.: A geometric series has first term 1 and common ratio \frac{1}{2}sin2\theta. Find the sum of the first 10 terms when \theta = \frac{\pi}{4}, giving your answer in the form h - \frac{1}{2^k}, where h, k \in N. Homework Equations Sn = \frac{a(1 - r^n)}{1 - r}, when...

Homework Statement (1) \frac{1}{(1+x^{2})}+\frac{1}{(1+x^{2})^{2}}+...+\frac{1}{(1+x^{2})^{n}} The Attempt at a Solution (2)...

Homework Statement Hi, I'm trying to solve the problem in the attachment. I was asked to evaluate the left hand side equation of the equal sign. I was unsure how to go about evaluating it so I consulted my solutions manual to look up the first step. The right hand side equation of the...

Homework Statement Q. Find the range of values of x for which the sum to infinity exists for each of these series: (i) 1 + \frac{1}{x} + \frac{1}{x^2} + \frac{1}{x^3} + ... (ii) \frac{1}{3} + \frac{2x}{9} + \frac{4x^2}{27} + \frac{8x^3}{81} + ... Homework Equations S\infty =...

Homework Statement Q. Find, in terms of x, the sum to infinity of the series... 1 + (\frac{2x}{x + 1}) + (\frac{2x}{x + 1})^2 + ... Homework Equations S\infty = \frac{a}{1 - r} The Attempt at a Solution S\infty = \frac{a}{1 - r} a = 1 r = U2/ U1 = (\frac{2x}{x + 1})/ 1...

Homework Statement Q.: A geometric series has first term a and common ratio r. Its sum to infinity is 12. The sum to infinity of the squares of the terms of this geometric series is 48. Find the values of a and r. Ans.: From textbook: a = 6, r = 1/ 2 Homework Equations...

Hi there. I have this exercise in my practice for differential equations, and it asks me to find the curve that satisfice for every point (on the xy plane) the distance from (x,y) to the points of intersection for the tangent line and the x axis, and the normal with the x-axis remains constant...

Homework Statement 3,6,12...1536 determine the number of terms in the progression Homework Equations The Attempt at a Solution a=3 r=2 n= ar^n-1 1536= (3) (2)^n-1

Homework Statement A 35mm slide(picture size is actually 24 by 36 mm)is to be projected on a screen1.80m by 2.70 m placed 7.50m from the projector. What focal length lens should be used if the image is to cover the screen?Homework Equations the only equation i can think of is the lens equation...

Hi all, I would like to ask for the geometric interpretation of the riemann-stieltjies integral. Suppose we have an integral, (integrate f dg) over the interval [a,b], where g is monotonically increasing. Can i interpret it as the area between f and the g function? Moreover, i am a...

Homework Statement Point A is on a circle whose center is O, AB is a tangent to the circle, AB = 6, D is inside of the circle, OD = 2, DB intersects the circle at C, and BC = DC = 3. Find the radius of the circle. Homework Equations Power of a point theorem (several cases found online...

So, suppose for visualization there are only two dimensions: ct and x. Now if the metric where Euclidean, we could visualize this space is a simple plane. What would be the shape of the "plane" when the metric is +1, -1 (Minkowski)? Is it somehow hyperbolic?

I have a quick question. First let me give a definition. Let a_1, a_2, ..., a_k be independent vectors in R^n. We define the k-dimensional parallelopiped \mathbb{P}(a_1, ..., a_k) to be the set of all x in R^n such that x = c_1a_1 + \cdots + c_k a_k for scalars c_i such that 0 <= c_i...

i know how the basic geometric sequence system works, but what if i want to subtract a fixed amount every For example if i start with $5000 (a1) and is multiplied by 1.05 (5% / r) every day for 20 days (n) I would have $13,267. But what would I have if $20 dollars was subtracted from the...