Curves Definition and 778 Threads

A few topics we are covering in class are: Gauss map, Gauss curvature, normal curvature, shape operator, principal curvature. I am having difficulty understanding the concepts of curves on surfaces. For example, this problem: Define the map ##\pi : (\mathbb{R}^3-\{(0,0,0)\})\to S^2## by...

Homework Statement Let ##\alpha(s)## and ##\beta(s)## be two unit speed curves and assume that ##\kappa_{\alpha}(s)=\kappa_{\beta}(s)## and ##\tau_{\alpha}(s)=\tau_{\beta}(s)##, where ##\kappa## and ##\tau## are respectively the curvature and torsion. Let ##J(s) = T_{\alpha}(s)\dot\...

Homework Statement Question 2 from http://math.berkeley.edu/~mcivor/math53su11/solutions/hw6solution.pdf here. I do not understand b) and e). How do I think of the slope with respect to y? Homework Equations The Attempt at a Solution I do know that the partial derivatives are...

Homework Statement f(x,y,z) = 4x^2 + y^2 + 9z^2 another one is xy+z^2 how do u draw level curves and graphs for these? Homework Equations The Attempt at a Solution Just need somewhere to start Thanks

Homework Statement Describe the level surfaces and a section of the graph of each function/ or sketch f R3-R (x,y,z) -> 4x^2+y^2+9z^2 describe the graph of each function by computing some level sets and sections (x,y,z) -> xy+z^2 Homework Equations The Attempt at a Solution...

Homework Statement Calculate the line integral: F = <xz, (xy2 + 2z), (xy + z)> along the curve given by: 1) x = 0, y2 + z2 = 1, z > 0, y: -1 → 1 2) z = 0, x + y = 1, y: 1→0 3) z = 0, x-y = 1, y: 0 → -1 Homework Equations The Attempt at a Solution I don't think the...

So I've just been a bit confused over the concept of rounding curves and accelerating. Obviously its commonly understood that you need an acceleration to maintain a constant speed when rounding a curve, such as turning a vehicle right, left, or making a u turn. Can anyone help explain why...

Homework Statement Consider the region enclosed by the curves x=2-y^2 and y=-x Write a single integral that can be used to evaluate the area of the region. Find this area. Your answer should be a fraction reduced to its lowest terms. Homework Equations NA The Attempt at a Solution First...

Here is the question: I have posted a link there to this topic so the OP can see my work.

Homework Statement Suppose ##\sigma:I\subseteq\mathbb{R}\to\mathbb{R}^2## is a smooth plane curve parametrized by a parameter ##t\in I##. Prove that if ##\|\sigma(t_1)-\sigma(t_0)\|## depends entirely on ##|t_1-t_0|##, then the image of ##I## under ##\sigma## is a subset of either ##S^1## or a...

Homework Statement Find the area inside r = 9sinθ but outside r = 2 Homework Equations Area = 1/2(Integral of (f(θ)^2 - g(θ)^2)dθ The Attempt at a Solution f(θ)^2 = 81sin^2θ = 81((1-cos(2θ))/2) g(θ)^2 = 4 f(θ)^2 - g(θ)^2 = 36.5 - cos(2θ)/2 integral of (36.5 -...

Homework Statement Find the area inside the circle r = 3sinθ and outside the carotid r = 1 + sinθ The Attempt at a Solution Alright so I graphed it and found that they intersect at ∏/6 and 5∏/6. I can't think of a good way to approach the problem. The carotid has some of it's area...

The definition of an elliptic curve is an equation in the form: $$y^2 = x^3 + ax + b $$ Moreover, the curve must be non-singular, i.e. its graph has no cusps or self-intersections. This seems like an awfully specific definition for a family of functions. Can someone shed some light on why...

Homework Statement The question is "Use double integration to find the volume of the solid bounded by the cylinder x2+y2=9 and the planes z=1 and x+z=5" Homework Equations The Attempt at a Solution I tried to draw the curves and the solid that i formed is a cylinder with a...

Greetings, I'm simulating the principle of least action for simple object motion and reading from Feynman Vol. 2, Chpt 19 -- The Principle of Least Action. He states (with my paraphrasing) that the true path of a trajectory is the one for which the integral over all points of kinetic energy...

Homework Statement r=3+2cosθ Homework Equations The Attempt at a Solution The text shows that it's from 0 to 2pi. How did it come to those limits without graphing? I set r=o. What do I do from there?

Attached you will find the torque vs engine rpm and power vs engine rpm curves for an 'ideal engine' and also for a 'normal SI engine'. 1.(for the 1st ideal engine curve) Is it 'ideal' that the torque curve should decrease as the engine rpm increases? Why? Does this mean that 'ideally' in a...

Homework Statement For a curve in Cartesian form, show that \tan \phi = \frac{xy'-y}{x+yy'} Homework Equations The Attempt at a Solution According to the book notation, ##\phi## is the angle between the radius vector and tangent at any point of the curve. I know that ##\tan...

Homework Statement Sketch the region R bounded by the curves y = x, x = 2 - y^2 and y = 0. This is the initial part of an integral problem and I'm just curious about the method here. Homework Equations The Attempt at a Solution So, would it be proper to take the x = 2 - y^2...

I understand that GR allows for a method of time travel using closed time like curves (CTC)s. anyway i have a few question about this, first of is there some sort of (relativaly short) equation that discribes this. So my second question is based of a something i read in this thesis paper...

I don't really understand the point in Curvature and Torsion, I am wondering if someone could explain them to me. Thank you for your kindness: Why do mathematicians need Curvature and Torsion? What are their main uses??

Inverse curves related to JFET characteristics, help! Hi, This is an example given in a lesson which is closely related to a coursework question I'm trying to complete, the problem is I can't understand how they have got the results they have. Here is the statement giving the relationship...

Hello MHB, I got stuck on an old exam determine the area of the finite region bounded by the curves y^2=1-x and y=x+1 the integration becomes more easy if we change it to x so let's do it x=1-y^2 and x=y-1 to calculate the limits we equal them y-1=1-y^2 <=> x_1=-2 \ x_2=1 so we take the right...

Hello! I'm currently working on a solenoid and trying to simulate the response of a ferrous rod in the coil. I've been using FEMM 4.2 for my magnetostatic simulations, and Pspice Schematics 9.1 for the circuit part. I was thinking of combining the two by using scripts to let FEMM run through...

Homework Statement Consider curves ##C_1: (y-x)=(x+y-\sqrt{2})^2## and ##C_2: (x+y-\sqrt{2})=(y-x)^2##, then the area between ##C_1## and ##C_2## is A)1/2 B)1/3 C)1/4 D)None Homework Equations The Attempt at a Solution Finding out the points of intersection would be a lot...

1. Homework Statement [/b] A spaceship is traveling on a curved path, f(t) = (t, t2) (We'll assume that the path isn't affected by gravity, this is a math problem, not physics :-) ) It has to release a pod to intersect a space station that has an orbit described by the following: g(t) =...

One more problem that was giving me issues. Here's how I worked it out: Comments? Corrections?PS. Sorry for flooding the board with my problems. This is the last one for a bit! I have an exam coming up, and was shaky on some concepts. I get so anxious before exams. Can't hurt to ask...

Hi! Let the function be f(x,y)=x^2-2*y^2,which graph is S:z=f(x,y).Which are the contour curves?Are these hyperbolas? :confused: Thanks in advance!:)

Sketch the 2 polar curves r = -6cos(theta), r = 2 - 2cos(theta). a. Find the area of the bounded region that is common to both curves. b. Find the length around the intersection of both curves. I got a, but I don't know what to do for b because in my calculus book it only shows how to find the...

Homework Statement Consider the region bounded by the curves y= lnx and y=( x-3)^2 Find the volume of the solid obtained by rotating the region about the y-axis Homework Equations The Attempt at a Solution For this I solve for the x so i got x= e^y and x= (y)^(1/2) +3...

1. Homework Statement Find the area bounded by the following curves: --------- y = x y = -x + 6 y = (x / 2) --------- 2. Homework Equations N/A 3. The Attempt at a Solution Here is a link to what the graph looks like...

I have read about CTCs from some books and find the explanations confusing. -Are they simply natural trajectories in the given spacetimes? -How is energy-momentum conserved if a particle can simply travel back in time? i.e. Observers will observe particles traveling in a CTC to simply...

Homework Statement Find the area inside both the circles r=2sinθ; r=sinθ+cosθ. Express your answer as an integral, do not evaluate.Homework Equations \int_{\alpha}^{\beta}\frac{1}{2}(r_{1}^{2}-r_{2}^{2})d\thetaThe Attempt at a Solution So I set 2sinθ=sinθ+cosθ and solved for theta = ∏/4 and...

When i heat up an object, the kinetic energy increases. But since kinetic energy can be converted into positive or negative potential energy when it vibrates, during the heating process how can we say that the KE increases and thus temperature increases since at different points of time, the KE...

Homework Statement In photoelectric effect, why the saturated current starts at the y-axis (0,y)? and what is the reason for that gradient at the negative x-axis (Shown as a red line)? Homework Equations - The Attempt at a Solution -

Hi, i have a curve g:[0,t]->IR² with g(t)=(x(t),y(t)) in a homogenous gravitational field and i want to look at a ball rolling down this curve. therefore i want to derive some equations in order to calculate the normal force and the centripetal force at each point of this curve in order to see...

1. The function f(x,y) = x + y 2. The area A is formed by the lines : x = 0 and x = pi/4 and by the graphs : x + cos(x) and x + sin(x) 3. I have to parametrize A 4. 'Formula' : r(u,v) = (u,v*f(u)+(1-v)*g(u)) Could this be a parameterization of A : assuming f(u) = u+ cos(u)...

Homework Statement Define the vector field F = sec(x) i + k (a) Express the flow lines of F in equations form. (b) Express in equations form the particular flow line through the point (0, 3, 2). My next question is a bonus question. I'm just reading up on this now but if someone could...

I want to define (and find the sound velocity i iron). can i read it through the dispersion curves of iron? I am trying to see it through the group propagation velocity, vg = dw/dk, i. e the slope of the dispersion relation w(k). am i on the right track? thanks

Homework Statement At what point do the curves r1(t) = (t, 4-t, 63+t^2) and r2(s)= (9-s, s-5, s^2) intersect? Answer in the form: (x,y,z) = ____ Find the angle of intersection theta to the nearest degree. Homework Equations The Attempt at a Solution i: t=9-s j: 4-t=s-5...

Author: Rick Miranda Title: Algebraic Curves and Riemann Surfaces Amazon Link: https://www.amazon.com/dp/0821802682/?tag=pfamazon01-20 Prerequisities: Complex Analysis, Differential Geometry, Abstract Algebra Level: Grad Table of Contents: Preface Riemann Surfaces: Basic Definitions...

Does anyone know the mathematical form of Type Ia supernovae light curves? I am trying to analyze supernovae data. I need to fit a function to the magnitude vs time data. So I require the mathematical form for magnitude as a function of time. If anyone has any idea about that, or can suggest a...

The part of the tire on the ground of a moving car is relatively stationary compared to the ground, right? The way the wheels work is the friction parallel to the direction of movement right? So when we calculate centripetal acceleration for cars going in a circle and the friction forces, why do...

Homework Statement 2 curves f(x) and g(x) don't intersect, find the range of values of m can be. Homework Equations $$f(x)=3x^2 - 2$$ $$g(x) = mx-5$$ The Attempt at a Solution Could I work out the 2 values for m that mean g(x) is a tangent to f(x) at some point and then the range will be...

\begin{tikzpicture}[scale = 1.5] \draw (0,0) circle (1cm); \draw[-] (1,0) -- (-1,0); \draw[->] (1.2,0) -- (.5,0); \draw[->] (-1.2,0) -- (-.5,0); \draw[-] (.907107,.907107) -- (-.907107,-.907107); \draw[<-] (.907107,.907107) -- (.307107,.307107); \draw[<-] (-.907107,-.907107) --...

consider y mapping R to R^3 and such that y(0)=(1,0,0) and y'(0)=(0,1,0). suppose $y''(s)=y(s) * y'(s)$. where * is the cross product 1) show y is a unit speed curve. 2)show that $\frac{d^2}{ds^2}(|y(s)|)=2$ 3) deduce $y(s).y'(s)=s$ and further $|y(s)|=(s^2+1)^{0.5}$

Homework Statement f(x) = (x^3) + (x^2) - (x) g(x) = 20*sin(x^2) Homework Equations The Attempt at a Solution I found the zeroes of the two functions at 4 intersections, and then the zeroes of each function respectively (there's 3 for f(x) and 4 for g(x) between -3 and 3), for certain...

Homework Statement Suppose f:R^2 - {0} → R is a differentiable function whose gradient is nowhere 0 and that satisfies -y(df/dx) + x(df/dy) = 0 everywhere. a) find the level curves of f b) Show that there is a differentiable function F defined on the set of positive real numbers so that...

All, Based on pointers that I have received from a number of people on the forum, I have been reading about the support that the time dilation of supernovae (type 1a) light curves provides to the standard model– thanks to all and a special thanks to GerogeJones. The material seems to fall...

Homework Statement Find the region bounded by the two functions from y=0 to y=2 equations given: x=(y-1)2 -1 x=(y-1)2 +1 express x as a function of y and integrate it with respect to y Homework Equations equations given: x=(y-1)2 -1 x=(y-1)2 +1 The Attempt at a Solution...