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. T

    Volume generated by revolving curve around axis

    Homework Statement Find the volume generated by revolving the regions bounded by the given curves about the x-axis. Use indicated method in each case. Question 11: y = x^3, y = 8, x = 0 Question 15: x = 4y - y^2 - 3, x = 0 Homework Equations for question 11: Shells for Question 15...
  2. M

    Why is the salt solubility curve flat?

    I know most salts' have increased solubility in 100g of water with an increase in temperature, a few have an inverse relationship, but why does NaCl flatline regardless of temperature? Like is there a mechanism that explains this phenomenon? Thanks in advance.
  3. D

    Elemental Abundance in Stars - The Curve of Growth

    I am studying for my undergraduate Astrophysics module and the lectures notes say that that all abundances are measured relative, in terms of H = 12.00 by mass or number of atoms. Is this correct? I thought it was all based off of Carbon = 12. Am I missing something?
  4. S

    Lorentzian Curve: Is It Normalized?

    Is a lorentzian curve by definition normalized? As far as I can tell it is such that ∫L(x) = 1.
  5. S

    Path integral(Parametric curve)

    Homework Statement Find $$\int_{C} z^3 ds $$ where C is the part of the curve $$ x^2+y^2+z^2=1,x+y=1$$ where$$ z ≥ 0 $$ then I let $$ x=t , y=1-t , z= \sqrt{2t-2t^2}$$ . Is it correct? Or there are some better idea? Homework Equations The Attempt at a Solution
  6. M

    Line integral of a vector field over a square curve

    Homework Statement Please evaluate the line integral \oint dr\cdot\vec{v}, where \vec{v} = (y, 0, 0) along the curve C that is a square in the xy-plane of side length a center at \vec{r} = 0 a) by direct integration b) by Stokes' theoremHomework Equations Stokes' theorem: \oint V \cdot dr =...
  7. R

    Line Integral of Scalar Field Along a Curve

    Homework Statement For some scalar field f : U ⊆ Rn → R, the line integral along a piecewise smooth curve C ⊂ U is defined as \int_C f\, ds = \int_a^b f(\mathbf{r}(t)) |\mathbf{r}'(t)|\, dt where r: [a, b] → C is an arbitrary bijective parametrization of the curve C such that r(a) and r(b)...
  8. R

    Finding the equation of a parametric curve

    [b]1. if y(t)= (a/t, b/t, c/t) [b]2. Prove that this curve is a straight line. Find the equation of the line [b]3. i found the first part without a problem, i just am not sure how to find the equation f the line.
  9. M

    Flux Equations for a Solid Surface and a Curve

    are both of the equations i posted flux equations. one of them is for a surface of a solid and the other is for a curve?
  10. M

    Find the area bewteen the curve

    http://store3.up-00.com/Nov12/fWJ65017.jpg http://store3.up-00.com/Nov12/kpS65017.jpg
  11. J

    Equation to graph a 180 degree curve comprised of a radius and an ellipse

    Hi All, I'm not a math guy so I am coming to you for help. I am trying to come up with an equation to graph any 180 degree curve that is comprised of: a 135 degree radius, and a 45 degree ellipse (135 + 45 = 180). The two curves being the same curvature (slope?) where they meet. The portion...
  12. M

    Proving Stoke's Theorem for a Plane Curve

    Homework Statement Let C be a simple closed plane curve in space. Let n = ai+bj+ck be a unit vector normal to the plane of C and let the direction on C match that of n. Prove that (1/2)∫[(bz-cy)dx+(cx-az)dy+(ay-bx)dz] equals the plane area enclosed by C. What does the integral reduce...
  13. I

    General simple closed curve equation

    I have a list of points on a 2D simple closed curve and I'd like to approximate that curve using a polynomial such that the approximation will be given by: Ʃai,jxiyj = 0 However, I still need to limit the ai,j to make sure the approximation is also a simple closed curve, while still keeping...
  14. beyondlight

    Engineering The torque curve of a indcution motor (laboration task)

    Homework Statement I have to measure the entire torque curve and stator-current of a inductive motor from idle running speed to 0 rpm. This will bee done for 130 V instead of 400 V because the energy loses will be too high if it is done with a stator voltage of 400V. Question: How does the...
  15. M

    Elasticity (understanding elasticity from stress strain curve)

    Hello .. I have problem understanding how to decide which material is more elastic based on stress strain curve.. my understanding is as follows 1)if a material has big youngs modulus.. then it is more stiff 2)a material with a big youngs modulus may be or may not be very elastic (elasticity...
  16. E

    Vertical Vibrations and Lissajous curve

    Homework Statement a)x=cos2ωt, y=sin2ωt b)x=cos2ωt, y=cos(2ωt-∏/4) c)x=cos2ωt, y=cosωt draw the graphs of Simple Harmonic Motions.Homework Equations parametric and complex harmonic motion equation is needed.The Attempt at a Solution No attempt to solution.I can't do anything,because I can't...
  17. T

    Solving for Frictional Force in Banked Curve Problem

    Homework Statement A certain curve on a freeway has a radius of 200m and is banked at an angle of 25°. A 200-kg car moves around the curve at constant speed. 1. If the speed of the car is 35m/s, what friction force is needed to keep the car moving in a circle? 2. If the speed of the car...
  18. N

    Inversion Curve for a gas obeying Dieterici's equation of state

    Homework Statement For a gas obeying Dieterici's equation of state: P(V-b) = RTexp(-a/RTV) for one mole, prove that the equation of the inversion curve is P = ((2a/b^2) - (RT/b)) * exp((1/2) - (a/(RTb))) and hence find the maximum inversion temperature.Homework Equations N/A The Attempt...
  19. J

    Calculating the length of a curve

    Let's say I have a parabola that I know the equation of. I then asked myself the question "how do I calculate the length of the curve between two values of x, for example. After thinking about it, I realized I could use pythagoras: √δx + δy will give me a length, and then I could find the...
  20. Q

    MATLAB Fit data with a curve in MATLAB

    Hi friends. I want to fit my x datas and y datas with a function in the most exact way.my data is: x=[0.3 0.5 0.7 0.9 1 1.3 1.4 1.5 2 3 5 7 9 10 30 50 70 90 100]; y=[13.4347 8.3372 6.3107 5.27 4.93 4.28 4.14 4.0199 3.6349 3.3178 3.1282 3.0691 3.0432 3.0354 3.0043 3.0016 3.0008 3.0005...
  21. E

    Dragon Curve Fractal Using Golden Ratio

    I've been fooling around in MS Excel trying to reconstruct this fractal: I haven't had any issues here making it. I totally understand the algorithm for generating the left turn/right turn ordering. What I really want to know is how this version is generated: Original image...
  22. marellasunny

    MATLAB How can I extrude a 2D curve in MATLAB along the Z direction?

    Hi all! I would like to extrude a 2d curve in the X-Y plane along the Z direction,thereby wanting to obtain a plane of Z units long.How should I proceed? Is there a tool especially for this in MATLAB?
  23. A

    Finding the Best fitting curve for a graph?

    Hi, I m unsure whether or not this should perhaps be in the homework forum, however one might say this is more of a technical issue. I recently conducted an experiment for a report I intend to write up promptly, and I wanted to find the best fitting curve for these values. I tested how different...
  24. G

    True Stress-Strain Curve on the Log Scale: Which portion is the plastic region?

    My math is a little weak, so I'm having a hard time finding the elastic and plastic regions on this curve. Any further help will also be appreciated!
  25. C

    [Differential Geometry of Curves] Regular Closed Curve function

    Let m : [0,L] → ℝ2 be a positively oriented C1 regular Jordan curve parametrized with arc length. Consider the function F : [a,b] x [a,b] → ℝ defined by F(u,v) = (1/2) ||m(u) - m(v)||2 Define a local diameter of m as the line segment between two points p = m(u) and q = m(v) such that: The...
  26. C

    [Differential Geometry of Curves] Regular Closed Curve function

    Homework Statement Let m : [0,L] --> ℝ2 be a C2 regular closed curve parametrized with arc length, and define, for an integer n > 0 and scalar ε > 2 μ(u) = m(u) + εsin(2nπu/L)Nm(u) where Nm is the unit normal to m (1) Determine a maximum ε0 such that μ is a closed regular curve for...
  27. alane1994

    MHB How Do You Calculate the Length of a Curve Using Integrals?

    I had a question on a quiz that I missed... I am unsure how they got this answer. If someone could explain it would be great! Write the integral that gives the length of the curve. y=f(x)=\int_{0}^{4.5x} \sin{t} dt It was multiple-choice(multiple-guess;)). \text{Choice A }...
  28. R

    Finding the tangent equations of the curve

    Homework Statement Find the tangent equations to the curve y^2= x-1/x+1 at the points with x=2 Homework Equations y=mx+b dy/dx The Attempt at a Solution I tried to solve in order to y: y=sqrt((x-1)/(x+1)) Then I derived to obtain the slope, but this is the part that I don't know if it is...
  29. R

    Should I Include the Blank in My Calibration Curve Graph?

    Homework Statement Hi. I have a table where there is an absorbance for a 0.00 concentration (blank). I now that I have to substract that value to to each of the other absorbances that I have, but my question is... Do I have to graph (concentration 0.00, absorbance 0.00) in the calibration...
  30. J

    How can I find the area under a polar curve with the equation r^2 = 4cos(2θ)?

    r2 = 4cos(2θ) First I graph it. Then I set up the integral. _____π (1 / 2)∫ 4cos(2θ) dθ _____0 ________π = [sin(2θ)] ________0 I thought the limits ought to be π and 0, but that comes out to zero. I pick other limits and they come out to 0. My graph matches the one in the back of the book. I...
  31. U

    Integrate curve f ds Line Integrals

    Homework Statement Compute ∫f ds for f(x,y)= √(1+9xy), y=x^3 for 0≤x≤1 Homework Equations ∫f ds= ∫f(c(t))||c'(t)|| ||c'(t)|| is the magnitude of ∇c'(t) The Attempt at a Solution So, with this equation y=x^3 ... I got the that c(t)= <t,t^3> c'(t)=<1,3t^2> I know that from the equation...
  32. M

    Length of Curve: Finding r(t) Derivative

    Homework Statement Find the length of the curve: r(t) = e-2t i + e-2t*sin(t) j + e-2t*cos(t) k, 0 ≤ t ≤ 2π Homework Equations L = ∫\stackrel{b}{a} |r'(t)| The Attempt at a Solution I tried factoring out e-2t and got: e-2t (i + sin(t) j + cos(t) k) This is where I got...
  33. F

    Bike/motorcycle leaning in a curve. What happens to the normal force N?

    Hello Forum, If a bike is negotiation a curve, the static friction force will supply the centripetal force F_c. If we go too fast and the friction is not enough we skid along the tangent. There are two forces: the weight W=mg pointing down, the normal (contact) force N pointing up and...
  34. L

    Finding the area under a curve

    I'm trying to find the area of some shape with a straight line for the bottom, two curves on the sides and a straight top. Let's say I can only use calculus-like math. I can turn it on it's side and put it on a graph, but now there's now formula for the curved lines. What do I do? Do I split it...
  35. C

    Extra Credit- Taco Cannon Projectile (how do I find the function of the curve?)

    Homework Statement In Austin, Texas there is a Taco Cannon (modified T-shirt cannon) that will be spreading the joy of taco'ey goodness to festival goers. The only information I have is that it can fire 200 feet (60.96 meters). My physics professor allows us extra credit for applying...
  36. S

    Integrating $$\frac{ds}{(2y^2+1)^{3/2}}$$ on a Parabolic Curve

    Homework Statement Find $$\int_{C} \frac{ds}{(2y^2+1)^{3/2}}$$ where $$C$$ is the parabola $$ z^2=x^2+y^2 , x+z=1$$ Homework Equations The Attempt at a Solution I tried to parametrize the C , s.t$$ x=t, z=1-t, y=\sqrt{2t-1}$$ , but it seems to become a mess, and I don't know the...
  37. DeusAbscondus

    MHB Find volume when curve rotated about y-axis

    Hi folks, could someone please take a look at this for me: Here are the givens: $$\text{ Find the volume when this curve is rotated about the y-axis }$$ $$y=-4lnx\ \text{ where } 0\le y \le 2$$ I have set my working out in a geogebra file, taken a screenshot and attached same below. Would...
  38. B

    Calculating Area Under a Curve with Multiple Springs

    I attached the graph, provided in the problem, as a document. This is the wording of the problem: A 5000-kg freight car rolls along rails with negligible friction. The car is brought to rest by a combination of two coiled springs as illustrated in the figure below. Both springs are described by...
  39. T

    Find y' if (x-y)/(x+y)=(x/y)+1 and show that there are no points on that curve

    Homework Statement Use implicit differentiation to find y' if (x-y)/(x+y)=(x/y)+1. Now show that there are, in fact, no points on that curve, so the derivative you calculated is meaningless. Homework Equations The Attempt at a Solution I managed to get it into the form: dy/dx =...
  40. DeusAbscondus

    MHB Integration of area between curve and y-axis: transposition question/problem

    Find the area under curve $y=243x^5$ and between y=1 and y=32 Here is my working out: 1. transpose to make x the subject $$x=\frac{y^{1/5}}{3}$$ 2. integrate in y $$\int^{32}_1 \frac{y^{1/5}}{3}\cdot dy=(\frac{5\cdot 32^{6/5}}{18})-(\frac{5}{18})=17.5$$ Which is discrepant with given...
  41. S

    Mathematica Curve Fitting With Uncertainties

    I have a set of data points \{\{x_1, y_1\}, \{x_2, y_2\} ... \} each with an uncertainty \{\{dx_1, dy_1,\}, \{dx_2, dy_2\} ...\}. Is there any way of fitting a nonlinear model to the data that incorporates the uncertainties on both x and y? I know that you can use the Weights command to...
  42. N

    Automotive Ideal torque vs engine speed curve

    Just a simple question of what the ideal torque/engine speed graph would like? Also, could anyone direct me to a good website or book that would help me understand the following things better: tyre slip/traction control gear ratios and how they affect acceleration
  43. B

    Measure of the Sharpness of a curve

    Measure of the "Sharpness of a curve" I have a set of curves that belong to the family of curves y=\frac{c}{x^m}, where m and c are parameters. The attached picture (save.png) shows three such curves for different values of m and c. Now these curves have different 'sharpenss' of curvature (to...
  44. N

    Centripetal Force - Banked Curve

    Homework Statement A 1000kg car travels around a frictionless banked curve having a radius of 80m. If the banking is 20 degrees to the horizontal, at which specific speed must the car travel to maintain a constant radius? Homework Equations Fc = mv^2/r Fg(perp.) = mgcos(20) The...
  45. D

    Von Karman curve fitting to field measured spectrum

    Hello everybody, So for this wind monitoring project I'm getting data from a couple of 3d sonic anemometers, specifically 2 R.M.Young 81000. The data output is made digitally with a sampling frequency of 10Hz for periods of 10min. After all the pre-processing (coordinate rotation, trend...
  46. K

    What kind of curve would you feel more centrifugal force?

    Would you feel more centrifugal force going around a small radius curve or a large radius curve going at constant speed? Why?
  47. D

    Mathematica Fitting Curve to Data Points using Mathematica

    I am trying to use Mathematica to fit a curve to these data points ListPlot[{{2*Pi/(1 - 0^2/16), 5 (3 - Log[2])}, {2*Pi/(1 - .05^2/16), 10 (3 - Log[2])}, {2*Pi/(1 - .1^2/16), 15 (3 - Log[2])}, {2*Pi/(1 - .15^2/16), 20 (3 - Log[2])}, {2*Pi/(1 - .2^2/16), 25 (3 - Log[2])}...
  48. K

    Help understanding/evaluating line integral over a curve

    Evaluate the line integral, where C is the given curve. \int_{c} xy\:ds, when C: x=t^{2}, \ y=2t\ , \ 0\leq t\leq4 To solve this I should use the formula \int^{b}_{a} f(x(t),y(t))\sqrt{(\frac{dx}{dt})^{2}+(\frac{dy}{dt})^{2}}dt This gives me \int^{4}_{0}...
  49. G

    Centripetal acceleration question: car moving around banked curve

    If the wheels and tires of a car are rolling without slipping or sliding when turning, the bottom of the tire is rest against the road at each instant, so the force of friction is the static friction. Essentially if you are moving around a banked curve and the car is not skidding, then friction...
  50. 9

    Economics question: indifference curve?

    I know if you have two bads the slope is negative and is curved, and the closer to 0 the better. But what if you have two bads where one is worse? I.e. good a and b are both bads, but you want 5 of good b for every 1 of good a?
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