Cartesian Definition and 561 Threads

Ok, so randomizing three random variables, X, Y and Z, each from a standard normal distribution, then plotting these in an ordinary cartesian coordinate system gets me a spherically symmetric cloud of points. Now I want to create this cloud having the same probability distribution but by using...

Hi, I have to resample images taken from camera, whose target is a spherical object, onto a regular grid of 2 spherical coordinates: the polar and azimutal angles (θ, Φ). For best accuracy, I need to be aware of, and visualise, the "footprints" of the small angle differences onto the original...

Hi, I'm currently making a three-body simulator and I'm trying to add the eccentricity to Kepler's Law to turn the circular orbits to more of a elliptical orbit? I'm using Newton's Law of Gravitational to plot the new positions. How would I add in the eccentricity to this equation? I'm...

Hello! Good morning to all forum members! I am studying general relativity through the wonderful book: "General Relativity: An Introduction for Physicists" by M.P. Hobson (Cambridge University Press) (2006). My question is about Riemannian manifolds and local cartesian coordinates (Chapter 02 -...

Could someone please help with me with part $(b)$? By "the other set of vectors" they mean $R$, and the linear combination is $(1,2,9) = -3(1,0,-1)+2(2,1,3)$. https://i.imgsafe.org/b80212f.png

Homework Statement Provide an expression in Cartesian coordinates for a plane wave of amplitude 1 [V/m] and wavelength 700 nm propagating in u = cosθx + sinθy direction, where x and y are unit vectors along the x and y-axis and θ is the measured angle from the x axis. Homework Equations...

What is the physical interpretation of Cartesian coordinates in GR? Say, e.g., a system centered at the center of a spherical mass. What are x,y, and z physically, i.e., how are they measured?

Homework Statement Change the Cartesian integral into an equivalent polar integral and then evaluate. Homework Equations x=rcosθ y=rsinθ I have: ∫∫r2cosθ dr dθ The bounds for theta would be from π/4 to π/2, but what would the bounds for r be? I only need help figuring out the bounds, not...

Homework Statement I am currently trying to calculate the moment and products of inertia of a ring rotating about the x-axis at the moment the ring lies in the xy plane. The problem is that the notations I have from textbook are denoted for Cartesian coordinates. i.e. Ixx=∑i mi(yi2+zi2), and...

This is a problem that has been bugging me all day. While working with the well-known dydx = rdrdθ, where r is a function of θ I divided both sides of the equation by dxdθ to get dy/dθ = r(dr/dx) For the left side, I use y = rsinθ and derive with respect to θ to get dy/dθ = sinθdr/dθ + rcosθ...

Homework Statement if the alpha is larger than 90 degree ( which means the resultant F is lean towards -x axis , then the angle between the Ax and the line from Ax to A will be less than 90 , am i right ? Homework EquationsThe Attempt at a Solution

Homework Statement Consider a unit sphere centered at the origin. In terms of the Cartesian unit vectors i, j and k, find the unit normal vector on the surface Homework Equations A dot B = AB cos(theta) A cross B = AB (normal vector) sin(theta) Unit sphere radius = 1 The Attempt at a...

Hello! (Wave) I want to find the cartesian equation of the following parametrized curve: $$r(t)=(\cos^2 t, \sin^2 t)$$ I have tried the following: Since $\cos^2 t+ \sin^2 t=1, \forall t$, the coordinates $x= \cos^2 t, y= \sin^2 t$ of $r(t)$ satisfy $x+y=1$. Is the above sufficient or is a...

Homework Statement A highway has an exit ramp that beings at the origin of a coordinate system and follows the curve ##y=\frac{1}{32}x^{\frac{5}{2}}## to the point (4,1). Then it take on a circular path whose curvature is that given bt the curve ##y=\frac{1}{32}x^{\frac{5}{2}}## at the point...

Homework Statement Translate the following equations from the given coordinate system into equations in each of the other two systems. Also, identify the surfaces so described by providing appropriate sketches. Homework EquationsThe Attempt at a Solution For my solutions, I obtained z=2r^2 for...

Homework Statement In this question, I didn't see why the given 90 degree is 90 degree becoz it doesn't look like 90 degree. Can someone draw me a better diagram? It's hard to visualize it's 90 degree Homework EquationsThe Attempt at a Solution

...system, I mean as in the Cartesian Vector/Tensor definition. I get that if you have two mutually orthogonal basises which are theta degrees apart and the transformation from one basis to the other follows the same as a rotation by theta degrees i.e: V'i = Rij Vj then it is a Cartesian...

How can I find the parametric vector form of a cartesian equation under a specific condition? Cartestian equation: $$-2x-y+z=6$$ I know to find the parametric vector form we can find any 3 points P, Q and R which satisfy the cartesian equation. $$ \begin{pmatrix} x_1\\ y_1\\ z_1...

If we expres cartesian cordinates in polar coordinates we get: x=r*cos(theta) y=r*sin(theta) let's differentiate those 2 eqs: dx= dr cos(theta) -r* d(theta) * sin(theta) dy=dr sin(theta) + r* d(theta) * cos(theta) why isn't dx*dy= r* dr* d(theta) ( like when taking the jacobian , or when...

Suppose we have the sets $A=\left\{2,3\right\}$ and $B=\left\{5\right\}$, then $A$ X $B$ is defined as $\left\{(x,y)|x \in A, y\in B\right\}=\left\{(2,5), (3,5)\right\}$. But what happens when $A$ contains elements that are not in $\Bbb{R}$? Example: $A=\left\{(2,3),(3,4)\right\}\subset...

I was reading the Wikipedia page on Dynamism in order to get an idea of the motivation and thinking behind Liebniz's physics. In it there is this paragraph: In the opening paragraph of Specimen dynamicum (1692), Leibniz begins by clarifying his intention to supersede the Cartesian account of...

I have an original function ##z_{xy}## that I converted into cylindrical coordinates, now denoted ##z_{rθ}##. I have shown the steps I took to get here in the image file posted named "Work." Now, I have taken that work and converted it into code to plot in Python. I plotted it in another piece...

Hi, I was wondering if anyone could help with a vector question that I have. If I have a unit vector defined in cartesian co-ordinates as p= (0,1,0) how would I go about converting this vector to a cylindrical geometry. I understand that I will probably need to use p_r=sqrt(px^2+py^2) and...

Homework Statement Homework EquationsThe Attempt at a Solution a) The schrödinger equation $$i \hbar \frac {\partial \Psi}{\partial t} = - \frac {\hbar^{2}}{2m} \nabla^{2} \psi + V \psi $$ For the case ##0 \le x,y,z \le a##, ##V = 0## $$i \hbar \frac {\partial \Psi}{\partial t} = - \frac...

Homework Statement Solve the following equation: v is the dependent variable, x is the independent variable Homework Equations \frac{d^2v/dx^2}{(1+\frac{dv}{dx}^2)^{3/2}}=1 The Attempt at a Solution Hi, I am trying to solve the following equation...

I want to convert this system of corrdinates (see image beloow) to cartesian system. How make this? https://www.physicsforums.com/attachments/c2-png.82342/?temp_hash=1cfcfdb56cb59e415f556c06ffbe270a Tip: x = a exp(+u) cosh(v) y = b exp(-u) sinh(v)

Assume that G is some group with two normal subgroups H_1 and H_2. Assuming that the group is additive, we also assume that H_1\cap H_2=\{0\}, H_1=G/H_2 and H_2=G/H_1 hold. The question is that is G=H_1\times H_2 the only possibility (up to an isomorphism) now?

I've been trying to solve this question all day. If somebody could point me in the right direction I would really appreciate it! (ii) A particle’s motion is described by the following position vector r(t) = 4txˆ + (10t − t)ˆy Determine the polar coordinate unit vectors ˆr and ˆθ for r. [4]...

Homework Statement Working in Cartesian coordinates (x,y,z) and given that the function g is independent of x, find the functions f and g such that: v=coszi+f(x,y,z)j+g(y,z)k is a Beltrami field. Homework Equations From wolfram alpha a Beltrami field is defined as v x (curl v)=0 The Attempt...

The usual change of variables in this case (mentioned in the title of this topic) is this: ##x = rcos(t)## ##y = rsin(t)## When I rewrite (say my integral) in polar coordinates I have to change ##dxdy## to ##rdrdt## My question is why can't I just compute dx and dy the usual way (the already...

Homework Statement r = 4sec(θ) Homework Equations x2 + y2 = r2 y = rsin(θ) x = rcos(θ) The Attempt at a Solution Given that r = 4sec(θ), I replaced sec(θ) with 1/cos(θ) and got x = 4. The problem is that I'm not sure if that's the final answer because I have been unable to find r, y or θ.

(This is actually a calculus problem, not a physics one, but physics is based on calculus, so I hope it's fine) 1. Homework Statement Eliminate the parameter to find the Cartesian equation of x = (1/2)cos(θ) y = 2sin(θ) Homework Equations x^2 + y^2 = 1 (eq of circle) The Attempt...

Hello PF, I have a problem to solve in the following form: Given a vector with Cartesian components, V={Vx,Vy,Vz}, find its components in circular cylindrical coordinate. Given the actual vector components, it'd be very easy to convert. But I have no idea where to start on this. Any guide to...

Hey! (Mmm) Proposition The Cartesian product of two at most countable sets is countable. Proof Let $A,B$ sets both of which are at most countable. That means that there are functions: $f : \omega \overset{\text{surjective}}{\rightarrow} A, \ g : \omega \overset{\text{surjective}}{\rightarrow}...

Hello! (Wave) The set $n \times m$ is equinumerous with the natural number $n \cdot m$ and thus $n \times m \sim n \cdot m$, i.e. $Card(n \times m)=n \cdot m$. Which bijective function could we pick in order to show the above? (Thinking)

Homework Statement Why is it necessarily true that for a hyperbola, the focus length, ##f ## has got to be greater than the semi-major axis , ## a## - ## f >a ## ? Homework Equations - The Attempt at a Solution I needed to derive the cartesian equation of a hyperbola with centre at ##...

Hello I have this problem - From a generator, I get a compton scattering with the electrons theta and phi angles. where I having the following equations for a particle px = E_particle * sin (theta) * cos (phi); py = E_particle * sin (theta) * sin (phi); pz = E_particle * cos (theta)...

Homework Statement The problem and its solution are attached as TheProblemAndTheSolution.jpg. If you don't want to view the attached image, the cartesian-coordinate version that the problem wants me to convert to a polar-coordinate version is the following (let "int" = "integral").: int int (1...

Hello! (Wave) Sentence: If $A,B$ are sets, there is the (unique) set, of which the elements are exactly the following: $\langle a,b\rangle: a \in A \wedge b \in B$. Proof: Remark: $\langle a,b\rangle:=\{ \{a\},\{a,b\}\}$ If $a \in A$, then $\{ a \} \subset A \rightarrow \{ a \} \in...

In T. S. Blyth's book on Module Theory, the author uses a large 'times' symbol (similar to a capital X) for the Cartesian Product as seen in the text below (taken from Blyth page 58) Can someone help me with the Latex code for such a symbol?Peter

Hi everyone! I've got a vector index notation proof that I'm struggling with. (sorry ignore the c, that's the question number) I've simplified it u * (del X del) and from there I've sort of assumed del X del = 0. Is that right and if so could somebody please explain it? Else any help on...

I'm trying to find the azimuthal angle unit vector \vec{\phi} in the cartesian basis by taking the cross product of the radial and \vec{z} unit vectors. \vec{z} \times \vec{r} = <0, 0, 1> \times <sin(\theta)cos(\phi), sin(\theta)sin(\phi), cos(\theta)> = <-sin(\theta)sin(\phi)...

Homework Statement Q.[/B] A body dropped from a height H above the ground strikes an inclined plane at a height h above the ground. As a result of the impact, the velocity of the body becomes horizontal. The body will take the maximum time to reach the ground if : (a) ##h=\frac H4## (b)...

Homework Statement Convert 2cis(-pi/3)cis(pi/6) into cartesian form. Show all working to obtain full marks Homework Equations I know that the equation for it is 2((cos(theta) +isin(theta))+(cos(theta)+isin(theta))) The Attempt at a Solution Okay so cos of (-p/3) = 1/2 Sin of (-p/3) =...

Hello! (Wave) There is the following sentence in my notes: Let $A$ be a set. We define the set $I_A=\{ <a,a>, a \in A \}$. $$A \times A=\{ <a_1,a_2>: a_1 \in A \wedge a_2 \in A \}$$ Then $I_A$ is a relation, but does not come from a cartesian product of sets. Could you explain me the last...

Hey! (Nerd) If $A,B$ are sets, the unique set $\{ <a,b>: a \in A \wedge b \in B \}$ is called cartesian product of $A,B$ and is symbolized as $A \times B$. I want to find the cartesian product $\mathbb{Z} \times \{ 1, 2 \}$. I thought, that it is equal to $\{ <x,1>,<x,2>: x \in \mathbb{Z}\}$...

Mod note: This post with template not used and no effort shown received a warning. Okay I am totally confused in this. This is not a homework question but rather one I saw online and was wondering for example how to solve it The question was -3-i/-8+6i to be expressed into Cartesian form...

I've learned that a vector in coordinate system can be expressed as follows: A = axAx+ayAy+azAz. ai, i = x, y, z, are the base vectors. The transformation matrix from cylindrical coordinates to cartesian coordiantes is: Ax cosΦ -sinΦ 0 Ar Ay = sinΦ cosΦ...

Hi! (Wave) If $A,B$ are sets, the set $\{ <a,b>=\{ a \in A \wedge b \in B \}$ is called Cartesian product of $A,B$ and is symbolized $A \times B$. If $A,B,C$ sets, then we define the Cartesian product of $A,B,C$ as: $$A \times B \times C:=(A \times B) \times C$$ But.. is it: $(A \times B)...

Homework Statement I have this equation and i need to find the cartesian equation, so i apreciate your help Homework Equations X=cost ' y=2sin2t The Attempt at a Solution I am usign this [/B] Sin2t=2costsint So x+y/2=cost+2costsint But i don't know what to do after, I also try to solve that...