Tangent Definition and 1000 Threads

Hello. I was trying to prove that the tangent bundle TM is a smooth manifold with a differentiable structure and I wanted to do it in a different way than the one used by my professor. I used that TM=M x TpM. So, the question is: Can the tangent bundle TM be considered as the product manifold...

I need urgent help. I have this question: Use implicit differentiation to find an equation of the tangent line to the curve at the given point. \begin{equation} {x}^{2/3}+{y}^{2/3}=4 \\ \left(-3\sqrt{3}, 1\right)\end{equation} (astroid) x^{\frac{2}{3}}+y^{\frac{2}{3}}=4 My answer is...

I am reading "An introduction to manifolds" by Tu. He starts off in Chapter 1 by introducing some definitions on ##\mathbb{R}^n## that will carry across to general manifolds. In Chapter 1, 2.2, he defines germs of functions as a certain equivalence class of smooth functions ##C^\infty_p##. I...

Start with a circle of radius r and center c. Inside of that circle is an arbitrary point p. Given an arbitrary normalized direction vector d, I need to find the radius and center the circle that (1) intersects p, (2) is tangent with the circle centered at c, and (3) has its center lying on the...

Hey! :o A differentiable function $f(x,y,z)$ has $\nabla f (x_0, y_0, z_0) \neq (0,0,0)$ and zero instant rate of change from $(x_0, y_0, z_0)$ in the direction $\left( \frac{2}{3},-\frac{1}{3},-\frac{2}{3}\right)$. Which could be the cartesian equation of the tangent plane of the level surface...

Homework Statement To calculate the resultant pedal force from the variables given: Crank (degrees) measured clockwise from vertical, spindle (degrees) measured anti-clockwise from horizontal, tangent force (N) applied to the pedal surface, normal force (N) applied to the pedal surface and...

hello everyone, I'm trying to model a system in simscape using the simMechanics blocks. The system I am trying to model is: a ball bearing connected eccentrically to a motor axis. On that ball bearing lays a beam. The edge of the beam is connected to an axis. so, the system looks like this: I...

Homework Statement the line goes through (0, 3/2) and is orthogonal to a tangent line to the part of parabola y = x^2, x > 0 Homework EquationsThe Attempt at a Solution I have problems regarding finding the equation of tangent line to the part of parabola because the question not specifically...

Hey! :o Let $K$ be a circle with center $M=(x_0 \mid y_0)$ and radius $r$ and let $P_1=(x_1\mid y_1)$ be a point of the circle. I have done the following tofind the equation of the tangent that passes through $P_1$: The tangent passes through $P_1$ and is perpendicular to $MP$, then let...

it is possible to have two different tangents at the same point of a function?

I decided to review a little trigonometry. Why does tan(x + pi/2) = -cotx? I cannot use the tangent of a sum formula because tan(pi/2) does not exist. How about tan(x + pi/2) = [sin(x + pi/2)]/[cos(x + pi/2)] and then apply the addition rules for sine and cosine?

Homework Statement The curve ##C## has equation $$y=x^2+0.2sin(x+y)$$ Show that ##C## has no tangent(no point where ##dy/dx=∞##), that is parallel to the y axis. Attempt $$1=2x\frac{dx}{dy}+0.2cos(x+y)(1+\frac{dx}{dy})$$ For a tangent to be parallel to y-axis, $$\frac{dx}{dy}=0$$...

The tangent passes through just one point Now i can draw it such that the angle of incidence is always 0 and incident angle is always 90 Then no image will be formed. Why does this not happen??

Homework Statement Find the line tangent to the curve f(x)=0.5x2+3x-1 which is parallel to the line g(x)=x/2+0.5 Homework Equations f'(x)=x+3 The Attempt at a Solution I know it involves taking the derivative of f(x) and using it somehow, but I don't know where to go from there.

The tangent and the normal to the conic $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ at a point $$(a\cos\left({\theta}\right), b\sin\left({\theta}\right))$$ meet the major axis in the points $$P$$ and $$P'$$, where $$PP'=a$$ Show that $$e^2cos^2\theta + cos\theta -1 = 0$$, where $$e$$ is the...

Prove that the paraboloids: $$\frac{x^2}{a_1^2}+\frac{y^2}{b_1^2}=\frac{2z}{c_1}$$; $$\frac{x^2}{a_2^2}+\frac{y^2}{b_2^2}=\frac{2z}{c_2}$$; $$\frac{x^2}{a_3^2}+\frac{y^2}{b_3^2}=\frac{2z}{c_3}$$ Have a common tangent plane if: $$\begin{bmatrix}a_1^2 & a_2^2 & a_3^2\\ b_1^2 & b_2^2 & b_3^2\\...

Working through Schutz "First course in general relativity" + Carroll, Hartle and Collier, with some help from Wikipedia and older posts on this forum. I am confused about the gradient one-form and whether or not it is normal to a surface. In the words of Wikipedia (gradient): If f is...

Homework Statement http://i.imgur.com/4FPnTNS.jpg Homework Equations (Written in above photo) The Attempt at a Solution (Written in above photo) I have tried hard in figuring out what's wong I have done done, but what I finally got is still option d instead of the model answer e. Are there...

Homework Statement Equation: x^2+y^2-6x-2y+8=0 Find the center and the radius. (Help) : Find the equation of the tangent to the circle above that passes through the beginning of axis O (0,0)The Attempt at a Solution I found the center and radius and i believe the values are : C (3,1) and R...

Homework Statement Find the equation of the tangent line to the curve ##\ xy^2 + \frac 2 y = 4## at the point (2,1). Answer says ##\ y-1 = -\frac 1 2(x-2)## And with implicit differentiation I should have gotten ##\frac {dy} {dx}= -\frac {y^2} {2xy-\frac {2} {y^2}}## Homework Equations ##\...

stevendaryl submitted a new PF Insights post Solve Integrals Involving Tangent and Secant with This One Weird Trick Continue reading the Original PF Insights Post.

Find the equation of the straight line(s) which pass through the point (1, −2) and is (are) tangent to the parabola with equation y = x2 No calculus is to be used.I can substitute the point into the equation for the straight line giving -2=m+c And into the parabola (-2)2 = m+c Not sure if...

Good Morning All: I am now understanding a bit -- just a bit: still struggling - about the tangent bundle. But I have no idea WHY this is important. As I understand, at every point on a manifold (or, more appropriately: at the coordinates placed on a manifold by a mapping), we study the union...

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 Provide a complete proof that a regular plane curve γ : I → R2 can near each point γ(t0) be written as a graph over the tangent line: more precisely, there exists a smooth real valued map x → f(x) for small x with f(0) = 0 so that x → xT(t0) + f(x)JT(t0) parametrizes γ near...

I'm have trouble understanding a fundamental question of a derivative. So a derivate gives me a tangent line at any given point on a function. this makes sense for me for a function y=x^2 because the derivative is y'=2x which is a straight line function. But what about y=x^3 where the...

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...

Hi All, This question is about vector calculus, gradient, directional derivative and normal line. If the gradient is the direction of the steepest ascent: >> gradient(x, y) = [ derivative_f_x(x, y), derivative_f_y(x, y) ] Then it really confuse me as when calculating the normal line...

In an AC circuit with only a capacitor this diagram represents the relation between the current and the voltage in it (the current leads the voltage by 90 degrees). and because: (I= dQ/dt) and ( Q=C*V) where: Q is the amount of charge, C is the capacitance and V is the potential difference...

Homework Statement The problem is from D'Inverno's book on GR, problem 5.6. We're using the Jacobian/transformation matrix to convert the tangent to a circle centered at the origin of radius A from Cartesian to polar coordinates. I can do the problem and get the book answer, that's okay...

Homework Statement A cylindrical vessel of height ##H## and radius ##R## contains liquid of density ##\rho##. Determine the circumferential tension at a height ##h##; also determine its maximum and minimum values. This is the scan of the original question(solved), I couldn't understand what...

My book had these two solved examples that gave contradictory replies. The power output of an alternator is 100 kW. Now if the tangent of pf angle is 0.8 lagging, the KVAR rating must be -80KVAR. I drew phasor and got this. I took it that since tangent is negative. Now tangent is opposite...

1. Let p be an arbitrary point on the unit sphere S2n+1 of Cn+1=R2n+2. Determine the tangent space TpS2n+1 and show that it contains an n-dimensional complex subspace of Cn+1Homework Equations3. It is easy to find tangent space of S1; it is only tangent vector field of S1. But what must do for...

Homework Statement The following point (x0,y0), is on the curve sqrtx +sqrty = 1Show that line equation of the tangent line in the point. (x0,y0) Is x/sqrtx0 + y/sqrty0 = 1 I've found the slope which is -sqrty/sqrtx. So slope of the point is -sqrty0/sqrtx0 Homework EquationsThe...

Hallo, I have a question which elements are responsible for the increase of tangent delta in a X Capacitor and the reduction of its insulation resistance?

I need to prove that: $ \arctan{\dfrac{1}{x}}=\dfrac{\pi}{2}- \arctan{x}, \forall x>0$. Now, I assumed $\arctan{\dfrac{1}{x}}=\arccot{x}$. So, I've tried to do this: $\cot{y}=x \implies y=arccot{x} \\ \tan{y}=\dfrac{1}{\cot{y}}=\dfrac{1}{x} \implies y=\arctan{\dfrac{1}{x}} \\ \implies...

$\textsf{got ? on the Greek notation. if Period = T}$ \begin{align} \displaystyle Y_{tan}&=A\tan\left[\omega\left(x-\frac{\phi}{\omega} \right) \right]+B \implies A\tan\left(\omega x-\phi \right)+B \\ T&=\left(\frac{\phi}{\omega}\right) \\ PS&=\phi \end{align} $\textsf{so on:}$ \begin{align}...

I'm sure that I am not the first one to notice this, but I found that for angles between 0 and 90 degrees, tan(90-10^n) approximately equals 5.7296*10^(-n+1). Is that purely a coincidence?

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) =...

stuck on how to simplify this?

My professor shown us in class how to find the slope but I still don't understand

When I press these buttons on my calculator to find the third side or an angle in a triangle, what calculation is happening? What is the logic behind it all? I know it's a very basic question, but I am only in grade 10 and have not started Math yet this year.

Homework Statement ##\vec { \dot { r } } =(t+1)\vec { A } +(1-sint)\vec { B } \quad \vec { r(0) } =\vec { C } ## a. Find an equation of the tangent line to the curve at ##\vec { r(0) } =\vec { C } ##. b. Use a definite integral to find ##\vec { r(t) } ## c. If ##A## and ##B## are non...

Homework Statement Show that the tangent to ##x^2-y^2=1## at points ##x_1=\cosh (u)## and ##y_1=\sinh(u)## cuts the x-axis at ##{\rm sech(u)}## and the y-axis at ##{\rm -csch(u)}##. Homework Equations Hyperbolic sine: ##\sinh (u)=\frac{1}{2}(e^u-e^{-u})## Hyperbolic...

Show that the tangent to ##x^2-y^2=1## at points ##x_1=\cosh (u)## and ##y_1=\sinh(u)## cuts the x-axis at ##{\rm sech(u)}## and the y-axis at ##{\rm -csch(u)}##. $$2x-2yy'=0~\rightarrow~\frac{x}{y}=y'=\frac{\cosh (u)}{\sinh (u)}=\frac{e^u+e^{-u}}{e^u-e^{-u}}$$ The equation...

Homework Statement find the slope of tangent line to curves cut from surface z = (3x^2) +(4y^2) - 6 by planes thru the point (1,1,1) and parallel to xz planes and yz planes ... Homework EquationsThe Attempt at a Solution slope of tnagent that parallel to xz planes is dz/dy , while the slope of...

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)##...

Homework Statement Suppose that you have the following information concerning a differentiable function ##f##: ##f(2,3)=12##, ##\space## ##f(1.98,3)=12.1##, ##\space## ##f(2,3.01)=12.2## a) Give an approximate equation for the plane tangent to the graph of ##f## at ##(2,3,12)##. b) Use the...

Hello I am trying to determine the Fourier transform of the hyperbolic tangent function. I don't have a lot of experience with Fourier transforms and after searching for a bit I've come up empty handed on this specific issue. So what I want to calculate is: ##\int\limits_{-\infty}^\infty...