Tangent vector Definition and 55 Threads

I would ask for a clarification about the following definition of tangent vector from J. Lee - Introduction to Smooth Manifold. It applies to Euclidean space ##R^n## with associated tangent space ##R_a^n## at each point ##a \in R^n##. $$D_v\left. \right|_a (f)=D_vf(a)=\left. \frac {df(a + tv)}...

I need a justification that ##|\dfrac {dr}{dt}|##=##\dfrac {ds}{dt}## cheers guys... all the other steps are easy and clear to me...

This week, I've been assigned a problem about a 3-sphere. I am confused how to approach this problem and any comments would be greatly appreciated. (a) - would I be correct to assume the metric G is simply the dot product of two vector fields with dx^2 dy^2 du^2 and dv^2 next to their...

I am trying to understand the following derivation in my lecture notes. Given an n-dimensional manifold ##M## and a parametrized curve ##\gamma : (-\epsilon, \epsilon) \rightarrow M : t \mapsto \gamma(t)##, with ##\gamma(0) = \mathbf{P} \in M##. Also define an arbitrary (dummy) scalar field...

I hope I'm asking this in the right place! I'm making my way through the tensors chapter of the Riley et al Math Methods book, and am being tripped up on their discussion of geodesics at the very end of the chapter. In deriving the equation for a geodesic, they basically look at the absolute...

For the simple case of a 2-D curve in polar coordinated (r,θ) parametrised by λ (length along the curve). At any λ the tangent vector components are V1=dr(λ)/dλ along ##\hat r## and V2=dθ(λ)/dλ along ##\hat θ##. The non-zero christoffel symbol are Γ122 and Γ212. From covariant derivative...

In 2-D Cartesian coordinate system let's there exist a scaler field Φ(x1,x2) ,now we want to find how Φ changes with a curve which is described by the parameter(arc length) s dΦ/ds=(∂Φ/∂xi)dxi/ds Can we say for Cartesian coordinate system that along the curve at any s dxi always points in the...

I am learning the basics of differential geometry and I came across tangent vectors. Let's say we have a manifold M and we consider a point p in M. A tangent vector ##X## at p is an element of ##T_pM## and if ##\frac{\partial}{\partial x^ \mu}## is a basis of ##T_pM##, then we can write $$X =...

Homework Statement The surfaces ##x^2+y^2 = 2## and ##y=z## intersect in a curve ##C##. Find a unit tangent vector to the curve ##C## at the point ##(1,1,1)##. Homework EquationsThe Attempt at a Solution So I'm thinking that we can parametrize the surfaces to get a vector for the curve ##C##...

Homework Statement My problem is: For the logarithmic spiral R(t) = (e^t cost, e^t sint), show that the angle between R(t) and the tangent vector at R(t) is independent of t. Homework Equations N/A The Attempt at a Solution The tangent vector is just the vector that you get when you take the...

Homework Statement Find the unit tangent vector T(t) for vector valued function r(t) = (e^t)(cos t ) i + (e^t)(sin t ) j + (e^t) k Homework EquationsThe Attempt at a Solution i gt stucked here ... , the ans is [1/ sqrt (3) ] [ (cos t -sin t ) i + (sin t + cos t ) j +k) [/B]

Hi all, I have long had this unsolved question about arclength parameterization in my head and I just can't bend my head around it. I seem not to be able to understand why velocity with arclength as the parameter is automatically a unit tangent vector. My professor proved in class that s(s) =...

Homework Statement http://mathwiki.ucdavis.edu/Core/Calculus/Vector_Calculus/Vector-Valued_Functions_and_Motion_in_Space/The_Unit_Tangent_and_the_Unit_Normal_Vectors In the link, I can't understand that why the Principal Unit Normal Vector is defined by N(t) = T'(t) / | T'(t) | ,can someone...

In the following book, please look at equation 3.16. Why are the components of the tangent vector given by ui = dxi/dt? I understand the velocity components would be dxi/dt and the velocity vector would be a tangent vector. Is that the same reasoning the author uses? The book is normally crystal...

Simple and basic question(maybe not). How are rotations performed in differential geometry ? What does the rotation matrix look like in differential geometry? Let us assume we have orthogonal set of basis vectors initially. I am looking to calculate the angle between two geodesics. Can this...

Homework Statement Problem statement uploaded as image. Homework Equations Arc-length function The Attempt at a Solution Tangent vector: r=-sinh(t), cosh(t), 3 Now, I just need to reparameterize it using arclength and verify my work is unit-speed. Will someone give me a hint? Should I use...

Why are tangent spaces on a general manifold associated to single points on the manifold? I've heard that it has to do with not being able to subtract/ add one point from/to another on a manifold (ignoring the concept of a connection at the moment), but I'm not sure I fully understand this - is...

I am having issues figuring out how to do the "in the direction of the vector" part of my problem I have found the equation of the tangent line but i do not know how to the the next part. My question asks: Find the equation of the tangent line to the surface defined by the function f(x,y) =...

Hello, I am attempting to calculate unit normal and tangent vectors for a scalar field I have, Φ(x,y). For my unit normal, I simply used: \hat{n}=\frac{\nabla \phi}{|\nabla \phi|} However, I'm struggling with using this approach to calculate the unit tangent. I need to express it in terms of the...

Homework Statement A determinant a is defined in the following manner ar * Ak = Σns=1 ars Aks = δkr a , where a=det(aij), ar , Ak , are rows of the coefficient matrix and cofactor matrix respectively. The second term in the equation is the expansion over the columns of both matrices, δkr is...

I have a question concerning the tangent space. Consider a manifold Mn and take Mn to be ℝn with the Euclidean metric for the purposes of this question. The directional derivative of a function in the direction of a vector v is (a) vf = ∑ vi(∂f/∂xi) where the sum runs from 1 to n. The...

Homework Statement Find the tangent vector and unit tangent vector for the curve: r=sint, theta=t/3 for 0<=t<=6pi. Homework Equations If the tangent vector is r'(t)e(hat)r + r*theta(t)e(hat)theta, how does the restriction on t affect the answer? The same for the unit tangent vector, they don't...

Given a curve ##\gamma: I \to M## where ##I\subset \mathbb{R}## and ##M## is a manifold, the tangent vector to the curve at ##\gamma(0) = p \in M## is defined in some modern differential geomtery texts to be the differential operator $$V_{\gamma(0)}= \gamma_* \left(\frac{d}{dt}\right)_{t=0}.$$...

Homework Statement Homework Equations I know the equations. See question below. The Attempt at a Solution I am just wondering with this problem, how is it that they go from that derivative to the magnitude at the bottom of that image? I know the formula, but what I mean is...

For the curve defined by r(t) = 3*t*i + 2*t^2*j − 3*t^4*k Find the tangent vector r′(t0) at the point P(4,8,−16), given that the position vector of P is r(t0). and Find the vector equation of the tangent line to the trajectory through P. Im unsure as to how to go about solving this. I've...

So let ℝ^{n}_{a}={(a,v) : a \in ℝ^{n}, v \in ℝ^{n}} so any geometric tangent vector, which is an element of ℝ^{n}_{a} yields a map Dv|af = Dvf(a) = \frac{d}{dt}|_{t=0}f(a+tv) this operation is linear over ℝ and satisfies the product rule Dv|a(fg) = f(a)Dvg + g(a)Dvf if v|a =...

"find a unit tangent vector and the equation of the tangent line to the curve r(t) = (t, t^2, cost), t>=0 at the point r(pi/2)." NOW, what I don't get is, how is that a curve? This is not like the example I have studied and I don't really get the question. So I don't know where to start. Once I...

Homework Statement Show that: \frac{dx^\nu}{d \lambda} \partial_\nu \frac{dx^\mu}{d \lambda} = \frac{d^2 x^\mu}{d \lambda^2} The Attempt at a Solution Well, I could simply cancel the dx^nu and get the desired result; that I do understand. But what about actually looking at...

This is confusing me more than it should. A curve in space is given by x^i(t) and is parameterized by t. What is the tangent vector along the curve at a point t= t_0 on the curve?

Homework Statement r(t)=cos(t)i+sin(t)j+sin(2t)k Find the curvature κ, the unit tangent vector T, the principal normal vector N and the binormal vector B at t=0. Find the tangential and normal components of the acceleration at t=∏/4 Homework Equations T(t)=r'(t)/|r'(t)| N(t)=T'(t)/|T't|...

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)||}...

Homework Statement Let c(t) be a path and T the unit tangent vector. What is \int_c \mathbf{T} \cdot d\mathbf{s} Homework Equations The unit tangent vector of c(t) is c'(t) over the magnitude of c'(t) : \mathbf{T} = \frac{c'(t)}{||c'(t)||} The length of c(t) can be represented by ...

Homework Statement Find the unit tangent vector at the indicated point of the vector function r(t) = e(19t)costi + e(19t)sintj + e(19t) kT(pi/2) = <___i+___j+___k>Homework Equations r'(t) / |r'(t)| The Attempt at a SolutionAnswers: 19e(19*∏/2)(cos(∏/2)-sin(∏/2)) /...

Homework Statement Let c(s) = \left( \begin{array}{ccc} \cos(s) & -\sin(s) & 0 \\ \sin(s) & \cos(s) & 0 \\ 0 & 0 & 1 \end{array} \right) be a curve in SO(3). Find the tangent vector to this curve at I_3 . Homework Equations Presumably, the definition of a tangent vector as a differential...

Homework Statement Find a tangent vector r that satisfies r(0)= (e^(1),0) given T(t) = (-e^(cos(t)sin(t)),cos(t)), where t is an element of [0,2π] Homework Equations Tangent vector T = r'(t)/(norm(r'(t)) The Attempt at a Solution I was thinking that r(t) = ∫r'(t), and that the norm of r(t)...

Homework Statement Find the curve's unit tangent vector. Also, find the length of the indicated portion of the curve. Homework Equations r(t) = (-3tcost)i + (3tsint)j + (2\sqrt{2})t(3/2)k 0 ≤ t ≤ ∏ The Attempt at a Solution So I found dr/dt (I think), which is v(t) =...

I can use the tangent vector and a point.

Homework Statement If a curve has the property that the position vector r(t) is always perpendicular to the tangent vector r'(t), show that the curve lies on a sphere with center the origin. Homework Equations -1/r'(t)= slope of position vector x^{2}+y^{2}=1 The Attempt at a...

The unit tangent vector, T(t) = r'(t) / || r'(t) || always has length 1. Alright, so how do we get a sense of the length of the actual tangent vector itself? Its direction is easy to imagine, but I can't understand how its magnitude changes along the curve (does it have something to do with...

Homework Statement The surfaces S1 : z = x2 + y2 and S2 : x2 + y2 = 2x + 2y intersect at a curve gamma . Find a tangent vector to at the point (0, 2, 4). Homework Equations i thought about finding gradients of the two functions and plug in the given point in the gradients and cross...

Homework Statement Here's a worked problem, I can't understand how they have evaluated T at the given point (in part c): [PLAIN]http://img31.imageshack.us/img31/3725/97856984.gif The Attempt at a Solution I just substituted (0,1, \pi/2) into r'(s) but \frac{1}{\sqrt{2}} cos...

hey there, i got stuck on an question here: Parameterise the following paths, in the dirction stated, and hence find a tagent vector(in the same dirction) to each point on the paths. (a)The upper part of the circled centred at (0,0) containing the points (-2,0) and (2,0) going anticlockwise...

Homework Statement Find the vectors T, N, and B at the given point. r(t) = (sin(t), cos(t), ln(cos(t))), P = (0,1,0) Homework Equations T(t) = r'(t) / | r'(t) | N(t) = T'(t) / | T'(t) | B(t) = T(t) x N(t) The Attempt at a Solution I am stuck on how to solve for t. I am not...

Homework Statement Find the tangent vector at the point (1, 1, 2) to the curve of intersection of the surfaces z = x2 + y2 and z = x + y. Homework Equations The Attempt at a Solution I haven't started the problem, because I'm not sure what the first thing to do is. Do I have to parametrize...

OK, this is really confusing me. Mostly because i suck at spatial stuff. If the gradient vector at a given point points in the direction in which a function is increasing, then how can it be perpendicular to the tangent plane at that point? If it's perpendicular to the tangent plane...

Homework Statement Find the unit tangent vector T(t) to the curve r(t) at the point with the given value of the parameter, t. r(t)=<e^(2t), t^(-2), 1/(3t)> t=1 Homework Equations none The Attempt at a Solution So first I took the derevative to get r'(t) which I got to be...

In Euclidean vector spaces the derivative of the position vector of a running point of a curve is the tangent vector of the curve. In thehttp://www.matematik.lu.se/matematiklu/personal/sigma/Riemann.pdf" , on page 78 appears a vector which can be regarded as position vector in a Riemann space...

My teacher wrote an alternative equation on the board for curvature, and I am wondering how it is true: k = | dT/dt / |dR/dt| | where T is the unit tangent vector. I know k = |R' x R''| / |R'|^3 = |dT/ds| but I am not sure about the formula in question. How is it true/derived?

Homework Statement r(t) = costi + 2 sint j Find the tangent vector r'(t) and the corresponding unit tangent vector u(t) at point P:(.5, 3.5,0) Homework Equations r'(t) = r(t)dt u(t) = r'(t) / |r'(t)| The Attempt at a Solution r'(t) = -sinti + 2costj |r'(t)| = [sin2t +...