Curve Definition and 1000 Threads

Given the domain ℂ\[-1,1] and the function, f(z)=\frac{z}{(z-1)(z+1)}, defined on this domain, the Residue Theorem shows that for \alpha a positive parametrization of the circle of radius two centered at the origin, that: \int_{\alpha}f(z)=\int_{\alpha}\frac{z}{(z-1)(z+1)} = 2\pi i Can I...

Please consider the following scenario: Assume we have an object (o) at rest with a mass = m. Hence, we can calculate by general relativity the gravitational force (g) on this mass. Now, assume we remain stationary, at the origin, and a velocity (v) is imparted to the mass along the...

1. A pitcher throws a curveball that reaches the catcher in .60s. The ball is spinning at an average angular velocity of 330 rev/min, (assumed constant) on the path to the catcher's mitt. Find angular displacement. 2. θ=ωt where t is time, ω is angular velocity, and θ is angular displacement...

Homework Statement Find the Exact length of the Polar Curve for r=2(1+cosθ) No limits of Integration were given which I found to be odd. Homework Equations L= ∫√(r^2+(dr/dθ)^2)dθ The Attempt at a Solution r=2(1+cosθ) dr/dθ=-2sinθ L=∫√((2+2cosθ)^2+(-2sinθ)^2)dθ...

Consider the function F(x) where F(x) > 0 for all x. If we know that \int^{x_{2}}_{x_{1}}F(x)dx = 0 can we prove that x_{1}=x_{2} ? I can visually imagine that they are equal since the function is always positive, its integral must be monotically increasing, but I can't imagine how I...

How do you determine the equation of all possible tangents to a curve (say, a parabola) that pass through a given point that is not on said curve? This is more of a conceptual question, and it's not homework, so I thought it fit in this forum. I think there might be a question like this on the...

Homework Statement A curve is defined by the parametric equations: x = 2t^3 y = 2t^2 t =/ 0 1)Prove that the equation of the tangent at the point with parameter t is 2x - 3ty + 2t^3 = 0. Proven, and I've no problem with this part. 2.)The tangent at the point t = 2 meets the curve...

Hello guys, I have a quick question: Would a leftward shift of the supply curve result in a shortage?

sketch the level curve z=(x^2-2y+6)/(3x^2+y) at heights z=0 and z=1 i have already compute the 2 equations for the 2 z values and drawn it in 2d but when it comes to plotting it with the extra z axis i don't know what to do. please help...

Can someone please help me on this question. I tried to solve it by integrating 0.5*(1-3sin(θ)^2 from -Pi/2 to 0 but I didnt get the answer.

Hello. I had always been annoyed when dealing with parametric equations but it finally dawned on me how awesome the concept it. I want to make sure I understand things properly. So say I have a regular straight line given by rectangular equation y=2x+1 Now I can graph this. Now say...

I tried to find the length of a sine curve using calculus.I got stuck in the integral of integral(sqrt(cos(x)^2+1), x, 0, a). Limits are from 0 to point a,i.e. length of curve from 0 to any point a.With some approximations I found out the length of sine curve as a result.I have attached the...

Homework Statement Find the area of the surface generated by revolving the curve Homework Equations x = 5(cos3t), y = 5(sin3t), 0 ≤ t ≤ \pi, about the y axis. The Attempt at a Solution x' = -15(cos2t)(sin t) y' = 15(sin2t)(cos t) (I think this forms the top part of an astroid)...

Homework Statement This is not homework, just a personal project. I do not know if this is the correct section to post, sorry if it's not. I'm using a motion analysis software to track the position of a bouncing ball (y vs t). I imported the data into Excel and the graph is attached. What type...

Homework Statement z=1-x^2-2y^2 find volume under curve bounded by the xy plane. is the answer sheet wrong? (see below) why am i struggling so much with this?! how do i do it? Homework Equations according to other answer sheet, it is pi/sqrt 2 The Attempt at a Solution i did...

If the total nuclear binding energy of a nucleus increases whenever the number of nucleons increase, why does the nuclear binding energy curve rise and then fall? Don't protons and neutrons 'bring in' the same amount of binding energy every time one of them is added? Please explain. In addition...

Homework Statement The curve x=y^(2) and the line x=4 is rotated about the x axis. Homework Equations pi* integral from a to b of Radius^(2) The Attempt at a Solution pi* integral from 0 to 4 of (square root of x)^(2) dx. My teacher has this answer as 8pi but I think that that...

The relation between two terminology "cusp" (group & algebraic curve) Dear Folks: I come across the word "cusp" in two different fields and I think they are related. Could anyone specify their relationship for me?? Many thanks! the cusp of an algebraic curve: for example: (0,0)...

I've heard of curves being described as one dimensional but I don't understand how anything other than a straight line can be one dimensional as surely once the curve becomes, well, curved it is now in two dimensions? I have illustrated this in the above diagram with the curve and...

Homework Statement At what point on the curve y=2(x-cosx) is the tangent parallel to the line 3x-y=5. The Attempt at a Solution 1. rewrite 3x-y=5 as y-3x-5 2. equate 2(x-cosx) = y-3x-5 3. differentiate: 2+2sinx = 3 4. solve for x: sin^-1(,5) = 0.524 5. plug into y=2(x-cosx) to get...

Hello, Hoping this can be cleared up quickly as I have my final exam in 1 hour. My question is for the attached image. I know for pnp transistors that ic=0, and Vce=0 at top right hand corner. However, one thing I don't understand is why the base current starts at 20uA at the bottom, rather...

Hello all I was hoping someone could help me with the following problem. I work on the railroads and my task is to improve the alignment of a curve. I have surveyed the existing geometry by going along the track and recording x, y, z co-ordinates at approx 1m intervals. I have...

If given radius=r meters weight=mg Friction=μ Banking angle=θ For the y direction, NCosθ-mg-µNSinθ=0 My question. What is the x-direction equation? I know its equal to mv^2/r. Always mixed up between "real" force like mg and acquired force like N and friction. If the car at rest on the...

What is the measure of the sin(x) wave for x=0 to 2∏?

Homework Statement Given that near (1,1,1) the curve of intersection of the surfaces x^4 + y^2 + z^6 - 3xyz = 0 and xy + yz + zx - 3z^8 = 0 has the parametric equations x = f(t), y = g(t), z = t with f, g differentiable. (a) What are the values of the derivatives f'(1), g'(1)? (b)...

Homework Statement \int_{|z-2i|=2} = \frac{dz}{z^2-9} 2. The attempt at a solution I know that the contour described by |z-2i|=2 is a circle with a center of (0,2) (on the complex plane) with a radius of 2. The singularities of the integral fall outside of the contour (z+3 and...

I am trying to analyse a past series of numbers that flucuates between 107&210 with a normal frequency distribution of mean 162. What is the way to model and project short term future range for trendless but cyclical type of time series?

EDIT: Figured it out. Stupid me. I should have solved in terms of x, giving me x=1-(y+3)^2 as my answer. Homework Statement x= 1−t^{2}, y= t−3, −2 ≤ t ≤ 2 Eliminate the parameter to find a Cartesian equation of the curve for −5 ≤ y ≤ −1 Homework Equations N/A The Attempt at a Solution...

Hello all I work in railway transport. I am trying to re-design an existing railway curve; the existing curvature has several irregularities which result in the train being laterally displaced as it traverses the curve. What I have done is taken x,y and z co-ordinates along the...

Homework Statement At what angle should the roadway on a curve with a 50m radius be banked to allow cars to make the curve at 12 m/s even if friction is 0? Homework Equations The Attempt at a Solution All of the centripetal acceleration comes from normal force from the road on...

Hello there, I have a Figure from a book and some text explaining the figure and I was hoping that somebody could explain/clarify what it means. Here is the Figure http://dl.dropbox.com/u/54057365/All/pic.JPG Here is the text explaining the Figure: "The data are divided into ten bins...

Homework Statement The given curve is r(t) = <t2, 2t, -3> Write an equation for the length of the curve from <0,0,-3> to <1, 2, -3> 2. The attempt at a solution I take the derivative of r(t) for r'(t), then plug it into the length formula. L = ∫ of √( (2t)2 + 22 ) For...

Newton's laws of motion state that objects in motion/rest will remain in motion/rest unless acted on by a force (yank, pull, jerk and any other such force/derivative of force) but then the question I beg to ask is why? It makes sense that in an abstract empty block of space with no forces...

I have a problem with the determination of the correct focal curve of an achromatic double. I'm considering here a doublet formed by a biconvex Flint Glass lens attached to a plano-concave crown glass lens. ( I know that the typical design of the doublet is a biconvex crown lens attach to a...

Hi, I just registered to this forum, I'm working on the following problem. In the picture you see a family of linear functions. I need to find a function that is tangent/'just touches' (to) the lines. I immediately thought of a tractrix, but it seems to be a little different. I'd like some...

Find the area of the loop of the curve y^2=x^3(1-x)^2 using integral calculus. y=√x^3(1-x)^2 y=√x^3/2 (1-x) To sketch the curve, I assigned values for x and then solved the corresponding values of y. x= -1, y= -2 x= -0.5, y= -0.53 x=0, y= 0 x= 0.5, y= 0.177 x=1, y=0 how can i...

Consider a system of coupled differential equations x'=5x-y where x(0) = 6 y'=-x+5y where y(0)=-4 a) Show that the parametrised curve (x,y)= r(t)=(exp(4t) + 5exp(6t), exp(4t) - 5exp(6t)) How would you go about showing this?

Hi We attempted to trace the B-H curve of soft magnetic material by using principles of electromagnetic induction. Attached with this is the curve obtained. I am unable to figure out why am I getting the two loops at the end? Please help

http://img849.imageshack.us/img849/9018/curvess.jpg So pump curve, as flow increases, pressure drops. For the system curve, it's the other way around. How come? How come the pump disobeys the way the system is supposed to behave? The pump is a part of the system, and in the physical world...

Homework Statement Consider the plane curve \overrightarrow{r(t)}=e^tcost(t)\hat{i}+e^tsin(t) \hat{j} Find the following when t= ∏/2 Part A: \hat{T}(t) Part B: \hat{B}(t) Part C: \hat{N}(t) Homework Equations \hat{N}(t)=\frac{\hat{T}(t)}{||\hat{T}(t)||}...

If a curve with a radius of 89.0m is perfectly banked for a car traveling 71.0km/hr, what is the minimum coefficient of static friction for a car not to skid when traveling at 91.8km/hr? I figured out theta = 24. 03degs from the equations F(normal)*cos(theta) = mg and F(normal)*sin(theta) =...

Homework Statement Find the length of the curve r(t)=<e^(t) , e^(t)sin(t) , e^(t)cos(t)> between points (1,0,1) and (e^(2pi) , 0 , e^(2pi)) Homework Equations Length of curve=∫(llv(t)ll Where the limits of integration are the distance between the given points. The Attempt at a...

Homework Statement If t = pi/2, then that would equal 2, but they don't say that t = pi/2 so how do they get 2? Maybe since it's circular motion t = pi, that would work too, but I want to be sure before i move on.

Homework Statement "Given that near (1,1,1) the curve of intersection of the surfaces x^4 + y^2 + z^6 - 3xyz = 0 and xy + yz + zx - 3z^8 = 0 has the parametric equations x = f(t), y = g(t), z = t with f, g, differentiable. (a) What are the derivatives f'(1), g'(1)? (b) What is the...

Homework Statement Find the equation of the line with slope -1 that is tangent to the curve y=1/x-1 Homework Equations y=1/x-1 The Attempt at a Solution Slope of -1 means y=-1x+k So... -1x+k = 1/x-1 I don't know how to rearrange this into a quadratic equation so that I...

I have a curve of this form (as attached) drawn in excel. Now, the point P upto which the red curve is (almost) a straight line is called the proportional limit. OK. Now, I need to find the proportional limit as hown by the point p on the curve. It is given in literature that; As a...

Does any of our quantum gravity theories like String Theory or Loop Quantum Gravity (what else?) answer "why" mass/stress/energy curve Spacetime? Or do they just describe it a priori? Note I'm not asking why mass/stress/energy curve Spacetime. I just want to know if there is a Quantum Gravity...

Homework Statement a racecourse is designed with curves with a radius of 200m and a 10degree banking. What is the maximum speed a car can negotiate the curve without friction? Homework Equations Newtons 3 laws The Attempt at a Solution tanTheta = v^2/gR tan10 = v^2/(9.8 * 200m) v...

I have attached both the question and the solution. I just have questions as to why the solution is the way it is (sorry if they seem stupid but, while I get how to do it mechanically, I don't understand the fundamental reasoning as to why anything is being done): 1) Why are we taking the...

Here is the situation I am concerned with - Consider a smooth curve g:[0,1] \to M where M is a topological manifold (I'd be happy to assume M smooth/finite dimensional if that helps). Let Im(g) be the image of [0,1] under the map g . Give Im(g) the subspace topology induced by...