Coordinate Definition and 868 Threads

Is there some geometry in which a coordinate transformation of a vector of magnitude zero transforms to a vector that does not have a zero magnitude? Since the formula for the magnitude of a vector is √(x12+x22+...xn2), I can see no way for it to have magnitude zero unless every component is...

Homework Statement This isn't exactly a "problem" per se , but I need to understand it for a course I'm taking. I'm trying to understand the significance and when to use the vector conversion matrices, or just the identities. I'll use an example that I made up, using rectangular to polar...

Homework Statement Starting from the coordinate representation for the vectors, show the result in Equation 1.16 of Griffith's book. (1.16)A \cdot (B \times C) = \left[ \begin{array}{ccc} A_x & A_y & A_z \\ B_x & B_y & B_z \\ C_x & C_y & C_z \end{array} \right] Note: Here, I use * to...

Homework Statement Homework Equations none The Attempt at a Solution Two possible locations on the coordinate axis for the terminal arm of angle A: Two possible values for the measure of angle A and the related acute angle: Can someone please tell me if I did this correctly?

In my introduction to manifolds the following is stated: Polar coordinates (r, phi) cover the coordinate neighborhood (r > 0, 0 < phi < 2pi); one needs at least two such coordinate neighborhoods to cover R2. I do not understand why two are needed. Any point in R2 can be described by polar...

This question really pertains to motivating why vectors have components whereas scalar functions do not, and why the components of a given vector transform under a coordinate transformation/ change of basis, while scalar functions transform trivially (i.e. ##\phi'(x')=\phi(x)##). In my more...

Is the "gradient" vector a concept that that is coordinate independent ? For example, the concept of a vector representing a force is independent of what coordinate system is used to represent the vector. So is a "gradient vector" such a physical vector ? The web page...

As I understand it, the notion of a distance between points on a manifold ##M## requires that the manifold be endowed with a metric ##g##. In the case of ordinary Euclidean space this is simply the trivial identity matrix, i.e. ##g_{\mu\nu}=\delta_{\mu\nu}##. In Euclidean space we also have that...

Hello! I understand the the polar coordinate system without vectors. But when it is related to vector, it is confusing. Do the unit vectors r and phi keep changing? How do I interpret it as they changes? For example, F = 2 r + 3 phi. Based on the vector addition and scale multiplication, it...

Homework Statement why x is p(cosθ)(sinφ) ? and y=p(sinθ)(cosφ)? z=p(cosφ) As we can see, φ is not the angle between p and z ... Homework EquationsThe Attempt at a Solution

I'm fairly new to differential geometry (learning with a view to understanding general relativity at a deeper level) and hoping I can clear up some questions I have about coordinate charts on manifolds. Is the reason why one can't construct global coordinate charts on manifolds in general...

hi, I always see that jacobian matrix is derived for just 2 dimension ( ıt means 2x2 jacobian matrix) in books while ensuring the coordinate transformation. After that kind of derivation, books say that you can use same principle for higher dimensions. But, I really wonder if there is a proof...

Homework Statement Here is my question Homework Equations I don't know with what formula does the book find Fourier series? The Attempt at a Solution

Homework Statement [/B] In a set of axes where the z axis is the axis of rotation of a finite rotation, the rotation matrix is given by ## \left[ \begin{array}{lcr} &\cos\phi \ \ \ &\sin\phi \ \ \ &0 \\ & -\sin\phi \ \ \ &\cos\phi \ \ \ &0 \\ &0 \ \ \ &0 \ \ \ &1 \end{array} \right]##...

Without regard to a coordinate system (I only wish to consider special relativity) the stress-energy-momentum tensor defines a linear transformation from a 4-vector to a 4-vector. Let T be the linear transformation then b = T(a), a and b are 4-vectors. What is the physical meaning of a and b...

(Note that the title of this thread might be incorrect - I'm just drawing on the vocabulary people use when discussing Lagrangian Mechanics...) Hi, I'm trying to set up a coordinate system to represent points in space where one of the coordinates is the distance along a parametric curve, one is...

Let's say we have r=R( theta, phi, t) on the surface of the particle and need to find the normal vector in Spherical Coordinate system. We know that, the unit vector =grad(r-R( theta, phi, t)) / |grad((r-R( theta, phi, t))| where grad is Spherical gradient operator in term of e_r, e_\theta...

While investigating about the curl I have found this interesting perspective: http://mathoverflow.net/a/21908/69479 I lack the knowledge to do the derivation on my own so I would like to ask for your help. I am an undergraduate. I do not understand what a "first order differential operator"...

MENTOR note: moved from General Math hence no template What would be the Y-Axis if: X-Axis: theta=266.4 phi=-28.94 Z-Axis: theta=192.85 phi=27.13 where: theta=atan(Y/X) phi=asin(Z/R) My thinking, theta is +90 from X-Axis and phi is -90 from the Z-Axis. Is the Y-Axis theta=356.4 phi=-62.87?

We know, that the infinitesimal area element in Cartesian coordinate system is ##dy~dx## and in Polar coordinate system, it is ##r~dr~d\theta##. This inifinitesimal area element is calculated by measuring the area of the region bounded by the lines ##x,~x+dx, ~y,~y+dy## (for polar coordinate...

Homework Statement In Euclidean three-space, let ##p## be the point with coordinates ##(x,y,z)=(1,0,-1)##. Consider the following curves that pass through ##p##: ##x^{i}(\lambda)=(\lambda , (\lambda -1)^{2}, -\lambda)## ##x^{i}(\mu)=(\text{cos}\ \mu , \text{sin}\ \mu , \mu - 1)##...

where ##□=\nabla^{\mu}\nabla_{\mu}## is the covariant D'Alembertian. ##□x^{\mu}=0## ##g^{\rho\sigma}\partial_{\rho}\partial_{\sigma}x^{\mu}-g^{\rho\sigma}T^{\lambda}_{\rho\sigma}\partial_{\lambda}x^{\mu}=0## So this line is fine by subbing in the covariant derivative definition and lowering...

I was reading through some questions online and one asked the reader to calculate the distance between a star and Zenith given Sidereal Time 17hrs, RA*=16hr30mins, DEC*=50degrees. Could someone explain to me how you would do this please? There were no examples and so far I haven't managed to...

Studying the acceleration expressed in polar coordinates I came up with this doubt: is this frame to be considered inertial or non inertial? (\ddot r - r\dot{\varphi}^2)\hat{\mathbf r} + (2\dot r \dot\varphi+r\ddot{\varphi}) \hat{\boldsymbol{\varphi}} (1) I do not understand what is the...

I'm trying to understand the nature of coordinate transformations in physics. In classical mechanics, we can transform to a different coordinate frame by means of a Galilean transformation. In special relativity, this is replaced by a Lorentz transformation. I am now wondering whether there...

Homework Statement I have an a-b coordinate system which is skewed with an angle = 60 deg. I also have a particle position defined by vector V1 (a1, b1, 0) which follows the coordinate system. The problem I have is that I need to get V2 (a2, b1, 0) which is perpendicular to V1. Homework...

Homework Statement [/B]Hello, I am seeking help solving the following problem: find the transformation matrix that rotates a rectangular coordinate system through an angle of 120° about an axis making equal angles with the original three coordinate axes.Homework Equations none, we need to find...

My question is mostly about notation. I know the general definitions for divergence and curl, which can be derived from the divergence and Stokes' theorems respectively, are: \mathrm{div } \vec{E} \bigg| _P = \lim_{\Delta V \to 0} \frac{1}{\Delta V} \iint_{S} \vec{E} \cdot \mathrm{d} \vec{S}...

Hi, I have an interesting problem. I have three GPS coordinates, creating two lines across the surface of a sphere (assuming the Earth is spherical). I want to be able to create a new line (across the surface of a sphere) with a gradient that is in between the gradient of the two existing...

I am trying to make sure that I have a proper understanding of contravariant transformations between coordinate systems. The contravariant transformation formula is: Vj = (∂yj/∂xi) * Vi where Vj is in the y- frame of reference and Vi is in the x-frame of reference. Einstein summation...

Homework Statement Given the line element ##ds^2## in some space, find the transformation relating the coordinates ##x,y ## and ##\bar x, \bar y##. Homework Equations ##ds^2 = (1 - \frac{y^2}{3}) dx^2 + (1 - \frac{x^2}{3}) dy^2 + \frac{2}{3}xy dxdy## ##ds^2 = (1 + (a\bar x + c\bar y)^2) d\bar...

Hi.. I am new to this forum and not sure whether this is the right place to ask for a help. I have to transform angular velocity vector of a satellite from Earth Centered Inertial (ECI) coordinate system to Local Vertical Local Horizontal(LVLH) system. How can I do that..? Any help appreciated..

Iam having trouble understanding how one arrives at the transformation matrix for spherical to rectangular coordinates. I understand till getting the (x,y,z) from (r,th ie., z = rcos@ y = rsin@sin# x = rsin@cos# Note: @ - theta (vertical angle) # - phi (horizontal angle) Please show me how...

Homework Statement Evening all, I'm trying to solve the 2-D diffusion equation in a region bounded by y = m x + b, and y = -m x -b. The boundary condition makes it complicated to work with numerically, and I recall a trick that involves a coordinate transformation so that y = m x + b, and y =...

Evening all, I'm trying to solve the 2-D diffusion equation in a region bounded by y = m x + b, and y = -m x -b. The boundary condition makes it complicated to work with numerically, and I recall a trick that involves a coordinate transformation so that y = m x + b, and y = -m x -b are mapped...

The problem gives you a function describing the position of a particle moving along an x axis: x(t) = 12t2 - 2t3 With this function, one must determine the maximum positive coordinate reached by a particle and the maximum positive velocity. The first step to the problem is to take the...

Given the metric c^2 d\tau^2 = c^2 B(r) dt^2 - A(r) dr^2 - C(r) r^2 d\phi^2 and solving only for a static, spherically symmetric vacuum spacetime, I want to reduce the number of coordinate functions A, B, and C from three to only one using the EFE's. We can then make a coordinate choice for...

Hey all,just coded this SOM based on the explination on the ai-junkie website, it seems to place all the input vectors in one single common coordinate, what am I doing wrong? I suspect its the training loop as that was the one part that the site was unclear about, Any help appreciated. static...

Homework Statement I'm trying to find the direction and magnitude of Earth's gravity on some projectile. The question states that I can ignore z, and that the origins of the x and y axes should be on the surface of the planet. I should then use Newton's law of Gravity to find the direction and...

At time 1:11:20, Lenny introduces the metric for ordinary flat space in the hyperbolic version of polar coordinates? Is that what he is doing here? d(tau)^2 = ρ^2 dω^2 - dρ^2. He then goes on to say that this metric is the hyperbolic version of the same formula for Cartesian space, i. e...

I am wondering how can I find the infinitesimal displacement in any coordinate system. For example, in spherical coordinates we have the folow relations: x = \, \rho sin\theta cos\phi y = \, \rho sin\theta sin\phi z = \, \rho cos\theta And we have that d\vec l = dr\hat r +rd\theta\hat \theta...

Tried to post this in the resources forum but could not start a new thread. I can not find graphing software to plot data in polar coordinates. Anyone got links??

Homework Statement If i want to show which direction is "positiv" I can do like this right? (Or is it wrong) 2. But if the figure would look like this, could i draw a coordinate system rather? Is this way to show which way i say as positive? or should i rather draw like this? Or Is...

https://www.particleincell.com/2012/curvilinear-coordinates/ http://www.jfoadi.me.uk/documents/lecture_mathphys2_05.pdf Hi, I have a question about the curvilineare coordinate system. I wonder why is normal to the isosurfaces?isnt ei a tangent vector to the surface ui since "With these...

Assume that you are given a coordinate change on a line which changes the coordinate x to a new coordinate z given by the formula z=U⋅x+P where U,P are real numbers with U non zero. If the new coordinate of the point -13 is 12 and the new coordinate of the point -7 is 6 then we must have U= ...

Hi, I have a reference device that outputs euler angles, which are angles that relate the sensor body frame to the north east down frame. These angles are called pitch roll and yaw. The sensor is an accelerometer. I know how to get the rotation matrix that will put accelerations from the...

Homework Statement A model for the suspension of a vehicle is shown where the spring has stiffness k = 178 N.mm and an unstretched length of 347 mm. Here is the picture: http://i.imgur.com/1dTVs12.jpg Part a asked to determine the value of P and the force supported by member AB so that the...

This system of coordinates: can be "translated" in terms of x and y, so: x = \sqrt{\frac{\sqrt{u^2+v^2}+u}{2}} y = \sqrt{\frac{\sqrt{u^2+v^2}-u}{2}} Exist another form more simplified of write x and y in terms of u and v? I tried rewrite these expressions using the fórmulas of half angle but...

Homework Statement Sketch the solid whose spherical coordinates (ρ, φ, θ): 0≤ρ≤1, 0≤φ≤(pi/2) Homework EquationsThe Attempt at a Solution I was thinking that since ρ represented the distance from the point of the origin and φ represented the angle between the positive z-axis and the ray through...

Homework Statement Say how coordinate lines of the z plane transform when applied the following transformation ##e^z=\frac{a-w}{a+w}## Homework EquationsThe Attempt at a Solution This is exactly the way the problem is stated. It is a pretty weird transformation in my opinion and I'm guessing...