Tangent Definition and 1000 Threads

Is it correct to say that: the cotangent is given by the gradients (*) to all the curves passing through a point and it actually spans the same tangent space to a point of a sphere? If you visualize them as geometric planes (**), the cotangent and the tangent spaces are more than isomorphic...

$\tiny{3.2.15}$ Find the number c that satisfies the conclusion of the Mean Value Theorem on the given interval. Graph the function the secant line through the endpoints, and the tangent line at $(c,f(c))$. $f(x)=\sqrt{x} \quad [0,4]$ Are the secant line and the tangent line parallel...

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...

For a lab, I needed to calculate the uncertainty of a refractive index that was found using Snell's law. I found an equation online for propagation of error for any general function, which was I thought that since my equation was I could just get rid of the variable y, and have After...

Hello, I am reading some material related to jet spaces, which at first glance seem to be a generalization of the concept of tangent space. I am confused about what is the correct definition of a jet space. In particular, given a map ##f: X \rightarrow Y## between two manifolds, what is the...

First I find the normal vector given any position: $$w(x, y, z) = x^2 + y^2 + z^2$$ $$∇w(x, y, z) = (2x, 2y, 2z)$$ Normal vector of plane: $$w_2 = x - 2y + 3z$$ $$∇w_2 = (1, -2, 3)$$ ##∇w = ∇w2## => point where planes are parallel = (1/2, -1, 3/2) This is completely off, but I can't find any...

Here I have some problems. I get confused when it says"with respect to the path", is it different from "with respect to the earth"? Because the path is on the Earth. Then, the vehicle is not accelerated in the vertical direction because it moves along the path, is it?

I am a masters student studying motion analysis in human running. I need to find the angle of the parable tangent derived from the theoretical arc traced by a foot during a step and the ground (see attached). The arc comprises of a persons step height and step length and I need to find the...

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...

Find the slope of the tangent line to the graph of $$f(x)=-x^2+4\sqrt{x}$$ at $x=4$ (A) $8-$ (B) $-10$ (C) $-9$ (D) $-5$ (E) $-7$

Dear all, Attached is a picture of a circle. The lower tangent line is y=0.5x. The center of the circle is M(4,7) while the point A is (3,6). I found the equation of the circle, it is: $(x-4)^{2}+(y-7)^{2}=20$ and I wish to find the dotted tangent line. I know that it is parallel to the...

So, I was able to run some numbers and get a magnitude with what looks to be the distance formula. I was able to do that by first adding the (i) and(j) components from the two vectors and then taking the sums and running them through that distance formula. So far so good, but now I have to...

I am reading Andrew McInerney's book: First Steps in Differential Geometry: Riemannian, Contact, Symplectic ... and I am focused on Chapter 3: Advanced Calculus ... and in particular on Section 3.3: Geometric Sets and Subspaces of ##T_p ( \mathbb{R}^n )## ... ... I need help with...

I am reading Andrew McInerney's book: First Steps in Differential Geometry: Riemannian, Contact, Symplectic ... and I am focused on Chapter 3: Advanced Calculus ... and in particular on Section 3.3: Geometric Sets and Subspaces of ##T_p ( \mathbb{R}^n )## ... ... I need help with an aspect...

Hi all, I am a self learner (graduated very long ago and rusty at math) working through the Riley, Hobson and Bence text, chapter 1. 1. Homework Statement Use the fact that ##sin(\pi/6) = 1/2## to prove that ##tan(π/12) = 2 − \sqrt{3}.## Homework Equations ##tan(2x) = \frac { 2 tan(x)} {1...

Hi all, this is my first thread! I am having problems trying to find the way of drawing a line which is tangent to a circle and intersects another circle making a 30º intersection. Let´s say I have circle A with coordinates 479183.87, 4365099.87 (x1,y1) and a radius of 27780m. I have a second...

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...

https://www.google.com/url?sa=t&source=web&rct=j&url=http://www.damtp.cam.ac.uk/user/hsr1000/part3_gr_lectures_2017.pdf&ved=2ahUKEwi468HjtNbgAhWEeisKHRj9DNEQFjAEegQIARAB&usg=AOvVaw3UvOQyTwkcG7c7yKkYbjSp&cshid=1551081845109 Here in page 55 it is written that geodesic is a curve whose tangent...

Hi guys! I had a question on the bolded sections below and wondered how to solve this in a step by step solution. I already have the answers, but don't know how to solve it. I will show the whole problem so that any content needed is available for sections e and d which I am stuck on. Thanks! A...

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...

Find the slope of the curve at the given point} $2y^8 + 7x^5 = 3y +6x \quad (1,1)$ Separate the variables $2y^8-3y=-7x^5+6x$ d/dx $16y^7y'-3y'=-35x^4+6$ isolate y' $\displaystyle y'=\frac{-35x^4+6}{16y^7-3}$ plug in (1,1)...

How do you know whether two points ##p## and ##q## of a manifold have the same tangent space? If the two tangent spaces are equal, then the vectors in the two tangent spaces are exactly the same. I suspect that it's equivalent to picking a vector at ##p## and dragging it to ##q## and the vector...

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 =...

How do you know if two given points on a manifold have the same tangent space? Checking if a vector does not change when transported from one point to the other is enough?

I am reading Loring W.Tu's book: "An Introduction to Manifolds" (Second Edition) ... I need help in order to fully understand Tu's section on tangent vectors in $$\mathbb{R}^n$$ as derivations... In his section on tangent vectors in \mathbb{R}^n as derivations, Tu writes the following: In the...

I'm given a simple table (time(s),position) to make into a graph: 0,0 2,0.6 4.,2.4 6,5.4 8,9.6 10,15 I'm asked to draw tangent lines at 4, 6, and 8 seconds. There is no indication of how they should be drawn, are tangent lines just eyeballed? I see equation involving "lim" online and equations...

Homework Statement Write the equations of tangent lines to the curve of the implicit function x2+2x+2y2-4y=5 that are normal to the line y=x+122. The attempt at a solution I know that the slope has to be m=-1 I found the derivative using implicit differentiation: dy/dx=(-2x-2)/(4y-4) Now I am...

Hi, I'm stuck on a homework problem in my Calculus III class. I solved 3a really easily, but 3b is giving me a lot of trouble. I know that to find the tangent line, I first have to find the slope, which is represented by the vector: <3cos^2(t)(-sin(t)), 3sin^2(t)(cos(t))>. I know the formula...

Homework Statement Back for more questions. This section has been pretty tricky. The graph of a folium with equation ##2x^3+2y^3-9xy=0## is given. Find the equation for the tangent line at the pont ##(2,1)## Find the equation of the normal line to the tangent line in the last question at the...

Homework Statement Find the equation of the tangent line to the graph of the given equation at the indicated point. ##xy^2+sin(πy)-2x^2=10## at point ##(2,-3)## Homework EquationsThe Attempt at a Solution Please see attached image so you can see my thought process. I think it would make more...

Homework Statement For ##y=f(x)##, find the slope of the tangent line to its inverse function ##f^{-1}## at the indicated point P. ##f(x) = -x^3-x+2## , ##P(-8,2)## Homework Equations The Inverse Function Theorem: ##(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}## The Attempt at a Solution So...

Homework Statement "Find the center and radius of the circle that passes through A(1,1) and is tangent to the line y=2x-3 at the point B(3,3). (Picture of the graph: https://imgur.com/a/0wAnqcU) Homework Equations Here's a link: https://imgur.com/a/y71Z9GY The Attempt at a Solution Soo, I've...

Homework Statement Homework Equations [/B]The Attempt at a Solution I got -1, but the answer says "6". Could you help me, please?

How do you show that there can be only one tangent space at a given point of a manifold? Geometrically it's pretty obvious in 3 dimensions, as one notices that there can be only one tangent plane at a point. But how could we show that using equations?

Homework Statement in title Homework Equations x=2cotθ y=2sin2θ dy/dx = 2sin3θcosθ y-y1=m(x-x1) point = (-2/√3,(3/2) The Attempt at a Solution Have been stuck for hours I solved for the dy/dx above, now I need to figure out how to get rid of the θ to get my equation in terms of x so I was...

I got 138.17, but that isn't correct. I don't know how to do it, since the only way I thought, gave me the wrong answer. Can anyone help?

Ok, i have a question. First, i am an engineering student and have done all math requirements up to linear alg. HOWEVER, my geometry is terrible, oh so terrible and i need some spoon feeding right now because i am stuck on a problem. Ok, i saw a really cool tool the other day called a radius...

While studying Relativity I decided to take over a concrete case. So I thought of (what I think is) the simplest case which is the Sphere ##S^2##. So I want to construct the tangent space to the sphere. I think for this I need to embbed it in ##R^3##. I worked out similar problems in the early...

Hi, Given a smooth distribution of lines in R2, could we assert that there is a unique distribution of curves such that: - the family of curves "fill in" R2 completely - every curve is tangent at every point to one of the smooth distribution of lines

Hey guys, I've got this problem I can't seem to get past. I need to find the tangent line to a parametric curve at t=\frac{\pi}{4} I thought I solved the equation, but my answer doesn't seem to be registered as correct. I'm guessing that means I stuffed up the equation, but I can't see where...

[FONT=Times New Roman]If the line $x=k$ is tangent to the curve $$\large y\:=\:x+\sqrt2\,e^{\frac{x+y}{\sqrt2}}$$ what is the value of $k$?

If the line x + my = 1 is a tangent of the circle $$x^2+y^2-4x+6y+8=0$$, the value of m is ... A. -2 B. $$\frac{1}{4}$$ C. $$\frac{1}{4}$$ D. 3 E. 4 Looking at the circle's equation, the center is (2, -3) and the radius is $$\sqrt5$$. If I know the coordinate where the line meet the circle I...

One of the tangent line equation of the circle $$x^2+y^2+6x-8y+12=0$$ at the point whose absis is -1 is ... A. 2x - 3y - 7 = 0 B. 2x - 3y + 7 = 0 C. 2x + 3y - 5 = 0 D. 2x - 3y - 5 = 0 E. 2x - 3y + 5 = 0 By substituting x = -1, I got: $$(-1)^2+y^2+6(-1)+8y+12=0$$ $$1+y^2-6+8y+12=0$$...

Homework Statement Determine the equations of the tangent lines to the graph of f(x)=3x(5x^2+1) that are parallel to the line y=8x+9Homework Equations y=m(x-x_1 )+y_1 The Attempt at a Solution 11.f(x)=3x(5x^2+1) The slope of the tangent line y=8x+9 f^' (x)=3(5x^2+1)+3x(10x) f^'...

Homework Statement Graph ##y=tan\left(x-\frac {π}{4}\right)## Homework Equations N/A The Attempt at a Solution To graph a tangent function, I first find the vertical asymptotes to set the boundaries for the graph: To do so, set what's inside the parentheses equal to ##\frac π 2## and ##-\frac...

Mod note: Moved from a technical forum section, so missing the homework template. @fab13 -- please post homework problems in the appropriate section under Homework & Coursework. I have the following exercise to solve : I have to find all the points on the surface ##x^2+y^2+z^2=36## (so a sphere...

I am trying to figure how one arrives at the following: dxμ∂ν = ∂xμ/∂xν = δμν Where, dxμ is the gradient of the coordinate functions = basis of cotangent space ∂ν = basis of tangent space I know that dual vectors 'eat' vectors to produce scalars. Is this demonstrated by absorbing d into ∂...

Hi :) The question is in dutch so i'l translate it. on an ellipse E with vertex P and P' on the major axis and vertex Q and Q' on the minor axis. chose R(x1,y1), the projection of R on the major axis is R' and on the minor axis is R''. Define the perpendicular projection of the intrersection...

Homework Statement Use gradients to find an equation of the tangent plane to the ellipsoid ##\frac {x^2}{4} + \frac {y^2}{9} + \frac {z^2}{25} = 3## at ##P = (2, -3, -5)##. Homework Equations ##\triangledown f## is a normal vector of f. The Attempt at a Solution Let ##w = \frac {x^2}{4} +...

So, I can't wrap around my head of why the Equation of the Tangent Line is: y = f(a) + f'(a)(x - a) I get it that it's the equation of a line, and so it should be something like y = mx + b. I also understand why f(a) = b (since it's a point in that line) and why f'(a) = m (since it's the slope)...