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. JoeMarsh2017

    Why Is Calculating Reluctance with an Air Gap Confusing?

    Homework Statement Homework Equations Reluctance = small "L"/mu*A The Attempt at a Solution I went the route of using B/H=Mu ...since we know that B=1.2Tesla's and Mu=4pi*10^-7 we arrive at our "magnetic field intensity "H" as 954,929.7 H" BUT if I am trying to find Reluctance...
  2. J

    IQ Distribution Curve: Is Advanced Intelligence Limited By Drift?

    What does the distribution curve of IQ in the world population look like? If the average IQ for all countries is 90 (Richard Lynn and Tatu Vanhanen “IQ and the Wealth of Nations”), with an average IQ for sub-Saharan Africans of 70, I suppose that the distribution curve is higher on the downside...
  3. D

    Proof that the line intersects the curve 3 times exactly

    Homework Statement p(x) = 0.2*(x-1)^5, q(x) = 4x-6 The Attempt at a Solution I took the diffrence h(x) = p(x) - q(x) h'(x) = ((x-1)^4) - 4 got two solutions for h'(x)=0.
  4. Cocoleia

    Find the level curve through the point on the gradient

    Homework Statement Homework EquationsThe Attempt at a Solution The answer is F. I don't how to get this. I know that it is perpendicular and must have a horizontal tangent. How do I come to this answer?[/B]
  5. R

    Is this a valid parametrisation of a curve?

    Hi, so I am learning how to parametrise curves. For the curve y^2=16x, I have said let x=t^2. Then, we can say y^2=16t^2, so that we can take the root of this and get y=4t. What I wanted to ask was do we have to say "plus or minus" in front of the 4t, or do we just leave it as positive to get...
  6. P

    B Curve where both x and y approach infinity

    What is the equation for a curve where x approaches infinity as y approaches infinity?
  7. B

    A Curve Fitting Data: Gamma Spectroscopy Lab Results

    Hi I have some data from a gamma spectroscopy lab and using a series of known radioactive sources I obtain a calibration curve. The equipment is a scintillation crystal coupled to a photo-multiplier tube connected to a multi-channel analyser to obtain an energy spectrum. Using Excel I add a...
  8. H

    How to Estimate Iron Loss at Higher Frequencies in PMSMs?

    Hi all, I am recently calculating the iron loss in a permanent magnet synchronous machine with a speed of 24000 rpm (electrical frequency 800 Hz). However, the iron loss data from the data sheet is not sufficient since only frequency of 50 Hz, 100 Hz, 200 Hz, 400 Hz, 1000 Hz and 2500 Hz are...
  9. A

    Work problem: force vs distance curve for compound bow

    Homework Statement A compound bow in archery allows the user to hold the bowstring at full draw with considerably less force than the maximum force exerted by the string. The draw force as a function of the string position x for a particular compound bow is shown in (Figure 1) . Part A How...
  10. T

    MHB What is the inequality form for a decreasing curve's slope?

    They want the curve in inequality form which i am not sure if i got it right
  11. A

    Find the equations of 2 tangents to a curve with a POI

    Homework Statement http://prntscr.com/czocek Homework EquationsThe Attempt at a Solution Okay, so the things I know before hand is that f'(x) = 4x^3 however, my question is since the problem states that the line that passes through (-1.25, -8) is tangent to the curve, wouldn't it have to...
  12. T

    MHB Find the Slope of a Curve y=f(x) at (a,f(a)) - Determine a f

    The limit below represents the slope of a curve y=​f(x) at the point​ (a,f(a)). Determine a​ f After finding the f(x) and a, I did this:8(3^2+3h(3)+3h(3)+h^2)-72 dividing by h getting h(8h^2+144) dividing by h; canceling the h's and the plugging in the limit h --> 0 getting 144. But I am...
  13. D

    Finding the curvature of a space curve

    Homework Statement Find the curvature of the car's path, K(t) Car's Path: r(t) = \Big< 40cos( \frac {2 \pi}{16}t ) , 40sin( \frac {2 \pi}{16}t ), \frac{20}{16}t \Big> Homework Equations K(t) = \frac { |r'(t)\:X \:r''(t)|}{|r'(t)|^3 } The Attempt at a Solution This is part of a massive 6...
  14. S

    Stress-strain (area under curve)

    Homework Statement Deducing what the area under the stress-strain curve shows. There are four option in the attached image. I can discount work done by considering the units. The remaining ones seem plausible, but only one is true. Homework Equations stress = force / area; strain =...
  15. toforfiltum

    Describe curve to reduce field intensity in fastest time

    Homework Statement Igor, the inchworm, is crawling along graph paper in a magnetic field. The intensity of the field at the point ##(x,y)## is given by ##M(x,y)=3x^2+y^2+5000##. If Igor is at the point ##(8,6)##, describe the curve along which he should travel if he wishes to reduce the field...
  16. D

    Given the planar curve, find the equation of the plane

    Homework Statement r(t) = < 2e^t - 5 , e^t +3t^2 , 4t^2 +1> Is a curve that lies within a plane. Find the equation of this plane. The Attempt at a Solution I am not sure if my approach is correct. These are my results: x=2e^t - 5 y = e^t +3t^2 z = 4t^2 + 1 z = 4t^2 + 1...
  17. Drakkith

    Slope of a Line Tangent to a Level Curve at a Point

    Homework Statement The equation ## f(x,y) = f(a,b) ## defines a level curve through a point ## (a,b) ## where ## \nabla f(a,b) \neq \vec 0##. Use implicit differentiation and the chain rule to show that the slope of the line tangent to this curve at the point ##(a,b)## is ##-f_x(a,b)/f_y(a,b)##...
  18. D

    What is the parametric equation for a helix on a vertical, circular cylinder?

    Homework Statement Match the parametric equations with the graphs. In this case, I am stuck on this equation: x = cos t y = sin t z = 1/(1+t^2) Homework EquationsThe Attempt at a Solution So far I have: x^2 + y^2 = cos ^2 t + sin ^2 t = 1 I know this is a circle in the xy-plane, and thus...
  19. Buckethead

    B Graph of rotation curve of a cluster (not a galaxy)

    I'm familiar with galactic rotational curves and there are plenty of graphs depicting such curves, but I'm interested in the rotation curves of entire galactic clusters at the moment and I'm not too good with sifting through what shows up in the search engines. (not a physicist, just a...
  20. I

    B Best fit curve associated with the combination formula

    Sorry if this is more of a HW question (if so then moderator please move my question. Thanks!) Hi, I'm trying to get an expression for a best fit curve of the combination formula (##_nC_r##). As far as I can tell, the curve is a simple parabolic curve, and its shape doesn't change. It's just...
  21. M

    MHB Finding Area under x^2-16 Curve: ∫_2^5

    Find the area bound by the curve y=x^2-16 , the x-axis and the lines x=2 and x=5. i am trying to use the definite integral way ∫_2^5. but i am not getting the right answer. the right answer is 53/3
  22. Turbodog66

    Length of Curve Between Points: Exploring r(t)

    Homework Statement Consider the path r(t) = <10t,5t2,5ln(t) defined for t >0. Find the length of the curve between (10,5,0) and (20,20,5ln(2)) Homework Equations L= ∫ab |r'(t)|dt The Attempt at a Solution r'(t) = <10, 10t, 5/t> t values are 1 and 2 based on the x values for the points...
  23. A

    Calculating Continuous Torque Curve for Electric Motors?

    Hi there, I have a motor peak torque, peak power, peak speed, rated torque, rated power, rated speed, torque at max speed. I was able to produce the peak torque curve. But, i don't understand how to create this continuous torque curve here. I have attach a link here, and somehow the continuous...
  24. Mohamed_Wael

    Ansys non-linear stress strain curve

    If I am using a new material in ansys, let's say I assigned it as solid linear elastic and I assigned density, young's modulus and poisson's ration and then I applied large deformation for this material so that it should be experience non-linearity ...I wonder how Ansys can give the results...
  25. J

    MHB Slope of Tangent line to Polar curve

    I am trying to find the slope of the tangent line of this polar equation: r = 4 + sin theta, (4,0) I put the equation into wolfram alpha and it gives me a 3D plot. If someone could help me find the slope of the tangent line, I would really appreciate it. Thank you.
  26. G

    B Does a tangent to a curve touch at 2 identical points?

    The graph of y = x - 1 CUTS the x-axis at x = 1 while the graph of y = x2- 1 TOUCHES the x-axis at x = 1. The point at which the tangent touches the curve is shown mathematically by having two solutions of x, i.e. x = 1 (twice). Is there some deeper meaning to these two identical solutions for x?
  27. C

    Line integral of scalar field ( piecewise curve)

    Homework Statement for the line segment c2 , why did the author want to differentiate dx with respect to dy ? and he gt dx = 0 ? I'm curious why did he didnt do so for C3 , where dy= 0 ..Why didnt he also differentiate dy with dx ? dy/dx = 0 ? Homework EquationsThe Attempt at a Solution is...
  28. B

    A How can conformal mapping be used to convert curves between different maps?

    I know the concepts of conformal mapping and complex mapping but I didn’t see none explanation about how apply this ideia and formula for convert a curve, or a function, between different maps. Look this illustration… In the Cartesian map, I basically drew a liner function f(x) = ax+b...
  29. H

    I Can an envelope curve cut its member curve twice?

    Can the envelope curve (of a family of curves) intersect a member curve of the family at more than one point? It seems possible. Consider the following. Each blue line is a member curve and the red line is the envelope curve. If we modify each blue line such that it has a protrusion like a...
  30. M

    B Experimental evidence that pure energy can curve spacetime?

    Is there any experimental evidence that pure energy (massless) can curve spacetime?
  31. Jianphys17

    Courses Gd of curve and surfaces or functional analysis before?

    Hello everyone, i just finished a course of analysis(2)\vector calculus.Now iI'm interested in doing Gd of curves and surfaces(Do Carmo), and functional analysis(Rudin'sbook), but do not know what may have precedence between the two, on which i should start before you think?
  32. KishoreAM

    Temperature Profile -- temperature-length curve of a metal bar

    Guys... If there is a bar, half of it is Copper and another half is Steel (Length wise) and both of its ends are at 1000K and I want to know how to find the temperature- length curve. this was an interview Question by the way.
  33. Robsta

    Breit Wigner Curve: Finding Reaction Rate from FWHM

    Homework Statement I've got a given Breit Wigner curve of the number of decays at given energies. I've been told by several sources that the width (FWHM) of the curve gives the rate of the reaction. I can't see however how an energy here actually translates into a rate. Homework Equations I...
  34. Evangeline101

    Drawing a heat curve of ice melting

    Homework Statement Homework Equations equation with each part. The Attempt at a Solution I did part a) b) and c) on my own: Part d: I am having trouble at this part. I understand how to draw the heat curve but I am confused on what numbers I should use for the x-axis of the graph. I...
  35. J

    I Velocity of Timelike Curve in Special Relativity

    In special relativity we can view spacetime as ##\mathbb{R}^4## with its standard smooth structure, and a metric ##\eta_{ab} = \sum\limits_{\mu, \nu = 0}^3 \eta_{\mu, \nu} (dx^\mu)_a (dx^\nu)_b## where ##\nu_{\mu \nu} = \mathrm{diag}(-1, 1, 1, )##. Given a curve ##\gamma: I \rightarrow...
  36. 5

    Calculating the Tangent to a Parametric Curve at a Given Point

    Homework Statement Consider the parametric curve given by: x=6cos(2t), y=t5/2. Calculate the equation of the tangent to this curve at the point given by t=π/4, in the form y=mx+c. The tangent is given by y= Homework Equations The Attempt at a Solution [/B] the answer that I got was...
  37. T

    I Finding the volume under the curve of a rotated function

    why can't it be like this?
  38. Kyrie

    I Finding the equation of a bitangent line to a curve

    The curve y=x^4-2x^3-2x^2-2x has a bitangent. I need to find the equation of this line. First, I started off by computing the slope. Since it touches two points on the curve, their slopes should be the same. So, I have the equation 4x^3_1-6x^2_1-4x_1-2=4x^3_2-6x^2_2-4x_2-2 I got up to the point...
  39. kq6up

    A Potential Energy Curve for Ammonia Inversion?

    For my Quantum II class I am working on a paper about masers. I am using a naive model (a coupled pair of infinite potential wells), and I would like to find out where I can find a graph of the inversion potential energy curve. This would be a simple one dimensional curve of the potential...
  40. karush

    MHB When Should Parametric Equations Be Used to Calculate Curve Length?

    Find the arc length of $$f (x)=(1/3)(x^2 +2)^{3/2}$$ On the interval [0, a] The parametric I got $$y=t$$ $$x=\sqrt{(3t)^{2/3}-2}$$ I proceeded but didnt get the answer of $$a+\frac{a^3}{3}$$
  41. Alexandra Fabiello

    I Is there a trick to tell which is the higher curve?

    The title space is too short for what I want to ask, so here it is. When determining the area bounded between two functions/curves, you have to know which is the upper curve and which is the lower curve to solve the problem. Currently I only know how to do it by graphing, but I was wondering...
  42. Dietrichw

    Curve fitting for Gravity/Conservation of Energy Lab

    Homework Statement [/B] The problem has to do with sig figs going down to 1. I've checked them multiple times by hand and with sigfig calculators but it is all the same. With 1 sig fig my standard deviation ends up being 0 which I am not sure that is acceptable. It makes sense as the points are...
  43. kenyanchemist

    Probability distribution curve for an electron in 2s and 2p

    hi, so my lecturer decides to give me manic depression by sending me on a wild goose chase. what is the general form of a plot of Ψ, Ψr2 and r2Ψ versus r for both Ψ2s and Ψ2p orbital... am not even sure i said it right So far I have only gotten the Ψ2r2 versus r
  44. K

    MHB Understanding Curve Sketching for a Challenging Function

    Hello everyone I'm having some difficulty wit curve sketching this specific function. If some one could walk through the steps and solution so that i can hopefully grab a handle on the concept it would be much appreciated! I can handle the beginner functions, but this one is giving me a hard...
  45. P

    I Calculate top of curve, and the time it takes to get there

    Hi, I'm doing a lot of temperature measurement, and I would like to add a function that could estimate how high the temperature would rise and how long it will take to get there. The calculations would be based on the current temperature relative to the past x measurements. Anyone have any...
  46. S

    Angle of average acceleration while turning a curve

    Homework Statement A cyclist is initially heading exactly due North. He then initiates a turn to the West, the turn being a quarter circle with radius 12-m. He travels at a constant speed of 5.5-m/s during the turn. What is the direction of this average acceleration, measured anti-clockwise...
  47. Shailesh Pincha

    I Rotation curve of galaxy Keplerian method

    There are 2 unknowns in the formula. The time period of rotation and the mass enclosed by orbit is Star. So how could we calculate the expected time period of rotation of stars in a galaxy and thus velocity of stars.
  48. E

    Can someone explain how a car turns on a curve?

    I am having a hard time understand why the frictional force of the tires of the car point to the center of the circle (centripetral force). http://physics.stackexchange.com/questions/138871/why-does-friction-play-the-role-of-centripetal-force-during-the-turning-of-a-car . Here is the...
  49. O

    Calculating errors after curve fitting (linearizing) graphs

    Homework Statement I did a lab this week measuring periods of swings of a simple pendulum. We need to curve fit the equation T=k*l^n and we got ln(T)=n*ln(l)+ln(k), and we need to plot the data we collected into the linear graph, meaning out y axes is ln(T) and our x axes is ln(l). So far all...
  50. chwala

    Finding the equation of a curve

    Homework Statement [/B] The normal to a curve at a point P cuts the y-axis at T and N is the foot of the perpendiculor from P to the y axis. If for all P, T is always 1 unit below N, find the equation of the curve.Homework EquationsThe Attempt at a Solution Point T has co ordinates ## (0,y)##...
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