Curve Definition and 1000 Threads

In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that appeared more than 2000 years ago in Euclid's Elements: "The [curved] line is […] the first species of quantity, which has only one dimension, namely length, without any width nor depth, and is nothing else than the flow or run of the point which […] will leave from its imaginary moving some vestige in length, exempt of any width."This definition of a curve has been formalized in modern mathematics as: A curve is the image of an interval to a topological space by a continuous function. In some contexts, the function that defines the curve is called a parametrization, and the curve is a parametric curve. In this article, these curves are sometimes called topological curves to distinguish them from more constrained curves such as differentiable curves. This definition encompasses most curves that are studied in mathematics; notable exceptions are level curves (which are unions of curves and isolated points), and algebraic curves (see below). Level curves and algebraic curves are sometimes called implicit curves, since they are generally defined by implicit equations.
Nevertheless, the class of topological curves is very broad, and contains some curves that do not look as one may expect for a curve, or even cannot be drawn. This is the case of space-filling curves and fractal curves. For ensuring more regularity, the function that defines a curve is often supposed to be differentiable, and the curve is then said to be a differentiable curve.
A plane algebraic curve is the zero set of a polynomial in two indeterminates. More generally, an algebraic curve is the zero set of a finite set of polynomials, which satisfies the further condition of being an algebraic variety of dimension one. If the coefficients of the polynomials belong to a field k, the curve is said to be defined over k. In the common case of a real algebraic curve, where k is the field of real numbers, an algebraic curve is a finite union of topological curves. When complex zeros are considered, one has a complex algebraic curve, which, from the topological point of view, is not a curve, but a surface, and is often called a Riemann surface. Although not being curves in the common sense, algebraic curves defined over other fields have been widely studied. In particular, algebraic curves over a finite field are widely used in modern cryptography.

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  1. B

    Ln sigma vs 1/T curve for p-type semiconductor

    Homework Statement a) Sketch a schematic band structure diagram for a p-type semiconductor. Mark the energy of the top of the valence band, the bottom of the conduction band and the impurity level on the diagram. b) Sketch the ln (σ) versus 1/T curve for a p-type semiconductor, where σ is...
  2. F

    Finding point where slope of line equals curve

    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...
  3. S

    Why Does Base Current Start at 20uA at the Bottom in NPN Curve Tracers?

    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...
  4. K

    Help with a proof of the yield curve.

    Homework Statement I need to show that the yield curve defined by r(t) = 1/t integral r(s) ds from 0 to t is a nondecreasing function iff: P(αt) ≥ (P(t))^α, for all 0<=α<=1 , t>= 0 and P(t) is defined as: P(t) = exp{-integral r(s) ds from 0 to t} and r(s) is the spot rate...
  5. T

    How Can Railroads Implement Design Alignment with Tangential Track Movements?

    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...
  6. azizlwl

    Max speed of a car at a curve banked road.

    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...
  7. Loren Booda

    Determine the length of the curve sin(x)

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

    Solving Intersection Curve at (1,1,1): Derivatives & Tangent Line

    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)...
  9. S

    Complex Integrals - Poles of Integration Outside the Curve

    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...
  10. S

    How to Model and Project a Trendless Cyclical Time Series?

    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?
  11. G

    Eliminate the parameter to find a Cartesian equation of the curve.

    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...
  12. T

    Help Re-Designing A Curve Using X Y Z Co-ordinates

    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...
  13. 1

    Banked curve angle w/no friction - teacher's work differs

    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...
  14. T

    Is a gradient perpendicular to the osculating plane of a regular curve?

    Homework Statement Prove the following or disprove with a counterexample: Let f be a differentiable function in an open set U in R^3 and (a, b, c) be a point in U where the gradient of the function f isn't zero. If r: I -> U is a regular curve with a regular derivative on an open interval I...
  15. B

    A question about a cumulative distribution curve

    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...
  16. C

    The equation for length of a curve: what are the integral ends?

    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...
  17. F

    Momentums and Curve Our Space is clearly a little weirder than expected

    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...
  18. J

    Focal Curve of an Achromatic Doublet

    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...
  19. B

    Finding a formula for this curve

    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...
  20. S

    Finding the area of the loop of the curve y^2=x^3(1-x)^2

    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...
  21. S

    Solving System of Coupled DEs: Parametrized Curve Solution

    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?
  22. P

    Interpreting the BH curve obtained experimentally

    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
  23. Femme_physics

    Hydraulics - System curve VS pump curve

    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...
  24. M

    Given the plane curve, find tangent vector

    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)||}...
  25. R

    Static friction on banked curve

    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) =...
  26. T

    Finding The Length of a Curve

    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...
  27. B

    What is the trig identity for sin^2x + cos^2x = 1?

    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.
  28. S

    Curve of intersection of surfaces problem (Answer included).

    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...
  29. G

    Find the equation of the line with slope -1 that is tangent to the curve y=1/x-1

    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...
  30. S

    Drawing a parallel line to the straight portion of the curve

    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...
  31. W

    Why mass/stress/energy curve Spacetime?

    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...
  32. C

    Car going around banked curve with no friction

    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...
  33. S

    Find a vector tangent to the curve of intersection of two cylinders

    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...
  34. R

    Topological dimension of the image of a smooth curve in a manifold

    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...
  35. S

    Equation of a curve in 3 dimensions

    Homework Statement A heat-seeking missile is located at (2,-3) on a plane. The temperature function is given by T(x; y) = 20-4x^2-y^2. Find the equation of the curve along which the missile travels, if it continuously moves in the direction of maximum temperature increase. Express your...
  36. A

    Help to make a Unit Hydrograph of Reservoir Level - Storage Curve for a Dam

    Hello! I'm just asking for a clue on how to make a unit hydrograph Reservoir Level - Storage Curve for a Dam. I have been searching information on internet, but I could not find anything about that. I would appreciate any idea or suggestion. Thanks!
  37. F

    How can a car maintain traction while speeding around a banked curve?

    Homework Statement A curve with a 200-m radius on a level road is banked at the correct angle for a speed of 60 km/h, i.e. a car traveling at this speed would remain on the road, even if the surface were frictionless. a) If a car travels around this curve at a speed of 90 km/h, what is the...
  38. A

    How to proof that a curve has no rational points

    Hello, I'm trying to do exercise number 20 from chapter 6 of this http://www.people.vcu.edu/~rhammack/BookOfProof/index.html, it asks to show that the curve x2 + y2 - 3 = 0 has no rational points. In the answer it has this tip: first show that a2 + b2 = 3c has no solutions, other than the...
  39. C

    Fitted curve to measured data - statistical properties of the fit error

    Dear all, I have a set of measurements {xm(Ti,mi)=x(Ti)+e(Ti,mi)}, where: _xm is the measured value _x is the actual value _e is a random measurement error for the measurement mi _Ti is a parameter I need to fit a curve to this data by some method. For example, if I use least squares...
  40. G

    What is the Length of a Parabolic Segment Using Calculus 3?

    Homework Statement Find the length of the parabolic segment r = \frac{12}{1+cos(θ))}, 0\leqθ\leq\frac{∏}{2}. Homework Equations L = ∫\sqrt{f(θ)^{2}+f^{'}(θ)^{2}}dθ The Attempt at a Solution I got up to here and didn't know whta to do L = 12 ∫^{0}_{\frac{∏}{2}}...
  41. A

    Integral of plane over a sine curve

    Hi, I want to verify if my answer to this problem I made up is correct? Suppose we have a plane z=x+y Lets find the magnitude of the volume of the space underneath the plane and over the REGION in the xy plane defined by y=sin(x), from 0<=x<=2pi. So for example, my definition of the...
  42. V

    Optimisation along a curve on a surface

    Hi everyone, First of all I am not sure if I have chosen the right category for this posting but this looked the most reasonable out. I have a problem that I would like to solve but I am not sure where to look for answers. It seems like something other people might have worked with before...
  43. J

    PDE Characteristic Curve Method

    Homework Statement Solve u_x^2+u_y^2=1 subject to u(x, ax)=1 Homework Equations The Attempt at a Solution I let u_x=p andu_y=q and F=p^2 +q^2 -1=0 Then x'=2p, y'=2q, u'=p.2p+q.2q=2, and p'=0=q'. So p=p_0, q=q_0 are constants. I got x'=2p_0, y'=2q_0 and integrating the...
  44. M

    Curve of intersection of 2 functions

    Homework Statement A particle moves along the curve of intersection of shapes y = -x2 and z = x2 in the direction in which x increases. At the instant when the particle is at the point P(1,-1,1), its speed is 9cm/s and that speed is increasing at a rate of 3cm/s2. Find the velocity and...
  45. A

    Area of Polar Curve: Find r = 1 + 2cos(θ)

    Homework Statement Find the area inside the inner loop of the limacon curve : r = 1 + 2cos(θ) Homework Equations A = ∫\stackrel{α}{β}(\frac{1}{2}r2)dθ The Attempt at a Solution i have the solution, my question is : how do you find α and β ? here α = 2π/3 and β = π A =...
  46. M

    Reparametrize the curve in terms of arc length

    Reparametrize the curve R(t) in terms of arc length measured from the point where t = 0 R(t) is defined by x = et, y = \sqrt{2}t, z = -e-t Arc length S = ∫ ||R'(t)||dt ||R'(t)||= sqrt{\dot{x}2 + \dot{y}2 + \dot{z}2}The attempt at a solution Getting R'(t) ==> x = et, y = \sqrt{2}, z = e-t...
  47. B

    Calculating Area of Curve Using Greens Theorem

    Homework Statement Find area of curve using area formula of Greens theorem Homework Equations r(t)=(t-sin t) i +(1- cos t ) j for 0 <= t <= 2 pi. The curve is y = sin x The Attempt at a Solution Do i let x(t)=t...?
  48. J

    Finding the total area between the curve and x axis

    Homework Statement find the total area of the region between the curve and the x- axis Homework Equations 1) y=2-x, 0≤x≤3 2)y=3x^2-3,-2≤x≤2 3)y=x^3-3x^2+2x, 0≤x≤2 4)y= x^3-4x, -2≤x≤2 The Attempt at a Solution I've tried using my graphic calculator to see what the graphs looked...
  49. R

    MATLAB Programming B-spline curve in Matlab from scratch

    I decided to write my problem and attach as a separate document so that everything was included and would leave room here to discuss my problem. From what I understand from the equations is that for each value of u I calculate the value of the basis function for each basis function i. Then...
  50. K

    Assumption of Diminishing Marginal Rate of Substitution in Indifference Curves

    Homework Statement What assumptions do we make to rule out vertical or horizontal indifference curves when slope is zero? Homework Equations There are several assumptions: completeness reflexivity transitivity being continuous strong monotonicity diminishing marginal rates of...
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