Cartesian Definition and 561 Threads

Let A={1, 2} and B={∅}. First, I find the power set of A and the power set of B: P(A)= { ∅, {1}, {2}, {1, 2} } P(B)= { ∅, {∅} } I believe the power sets are correct. I'm still new to the concept of power sets. Anyway, my main question is regarding cartesian product of power sets. I'm...

I'm reading through a multivariable calculus book and it starts off with some linear algebra. It defines vector addition as V \times V \rightarrow V. My text describes V as a set and describes the above process as a mapping. I believe the \times may represent a Cartesian product. Could someone...

Homework Statement Eliminate the parameter to find the Cartesian equation of the curve. Problem #1: x = sin t, y = csc t, 0 < t < ∏ / 2 Problem #2: x = sin θ, y = cos 2θ Homework Equations What's shown above is what's listed in the book. However, the authors felt compelled to...

(This is NOT homework) just my personal interpretation, because these are the formulas as you already know: r = √(x^2 + y^2 + z^2) φ = arctan(y/x) θ = arccos(z/r) using (x,y,z) = (∞,∞,∞) I come across a bit of a sinister problem: r = √(∞^+∞^+∞^) = √(3∞^2) which is right because if we just...

I googled it, and it says: \dot{x}=\dot{r}sinθcos∅ + (rcosθcos∅)\dot{θ} - (rsinθsin∅)\dot{∅} . . and so on for \dot{y} & \dot{z} And then they wrote "We will also need the inverse transformation obtained by solving the equations above w.r.t \dot{r}, \dot{θ}, and \dot{∅} for example...

Homework Statement Do you see how y gets converted to csc? I don't get that. I would y would be converted to sin in polar coordinates.

Homework Statement An airplane flies at an air speed of 300 miles per hour, in the direction toward southwest. There is a head wind of 75 mi/hr in the direction toward due east. (A) Determine the ground speed. (B) Determine the direction of motion of the plane, expressed as an angle...

Homework Statement Find cartesian equations of the line of intersection of the planes x+3y-6z =2 and 2x+7y-3z=7 The Attempt at a Solution What I did first was I cross product the 2 equation and then I got 33i-9j+k Then I took both of the equation and let y = 0. After that my answer seems...

Homework Statement For each part, find the cartesian equation of the plane through the given points. (1,0,3), (2,-4,3),(4,-1,2) The Attempt at a Solution No attempt. Dunno how to do :(

Homework Statement Use polar coordinates to evaluate: ∫sqrt(2)0 ∫sqrt(4-y2)y 1/(1+x2+y2) dxdy Homework Equations The Attempt at a Solution I graphed it and I see r is the part of the elipse sqrt(4-y2) and goes from 0 to ∏/4. I'm not sure how to make the bounds for r or how to...

Can someone help with this multi-part question. i did the first three but it doesn't seem right! --------------------------------------… (a) Find the equation of the line l through P(1, 1, 2) and Q(1, 0, 4) in vector, parametric and Cartesian forms. (b) Find the vector form of the line k...

Homework Statement Express z=-1+4i in polar for then find z^4 converting to Cartesian form Homework Equations r = sqrt(x^2+y^2) theta = y/x z= r cos (theta) + i r sin (theta) The Attempt at a Solution r= sqrt(-1^2+4^2) = sqrt(17) theta = tan a = 4/1 a = tan^-1...

Homework Statement ((-1+i)/(√2))^1002 find polar and cartesian form Homework Equations The Attempt at a Solution So I started by finding |z|=1 and Arg(z)= arctan (-1) = 5pi/6 so 1^(1002)*e^(i*5pi/6)*1002 =1*e^(i*835pi) but that's as far as I got because the answer...

Homework Statement Evaluate the matrix elements x_{nn'} = \left<n\left|x\right|n'\right> and p_{nn'} = \left<n\left|p\right|n'\right> and map the energy eigenstates \left|n\right> to Cartesian unit vectors. Homework Equations x = \sqrt{\frac{\hbar}{2m...

Homework Statement This exercice is in a Chapter named Introduction to Cartesian tensors. The following is the original question of the exercise: Homework Equations Compute the vector: (x1^2 + 2x1*x2^2 + 3x2^2*x3), i The Attempt at a Solution Plz help me, i don't understand what...

All necessary information is attached except the answer in Cartesian coordinates, which is -ix-jy+2kz and my work converting back from cylindrical to Cartesian, which I used WolframAlpha for, as the trig is a mess (that is, if the way I am doing this is correct)...

All necessary information is attached except the answer in Cartesian coordinates, which is -ix-jy+2kz and my work converting back from cylindrical to Cartesian, which I used WolframAlpha for, as the trig is a mess (that is, if the way I am doing this is correct)...

Homework Statement Given the tensor F_{\mu \nu }= \left[ \begin{array}{cccc} 0 & -E_{x} & -E_{y} & -E_{z} \\ E_{x} & 0 & B_{z} &-B_{y} \\E_{y} & -B_{z} & 0 & B_{x} \\E_{z} & B_{y} & -B_{x} & 0 \end{array} \right] F^{\mu \nu }F_{\mu \nu }=2(B^2-\frac{E^2}{c^2}) and metric tensor n_{\mu \nu...

EDIT: Figured it out. Stupid me. I should have solved in terms of x, giving me x=1-(y+3)^2 as my answer. Homework Statement x= 1−t^{2}, y= t−3, −2 ≤ t ≤ 2 Eliminate the parameter to find a Cartesian equation of the curve for −5 ≤ y ≤ −1 Homework Equations N/A The Attempt at a Solution...

Forgive me if this is in the wrong thread I'm new here. I am trying to plot an orbit in MatLab using Kepler's First law of motion. In polar form it works fine r(θ) = h^2/μ*(1/(1+e*cos(θ))) h = angular momentum μ = standard gravitational constant and e = eccentricity. The problem is I'd...

Homework Statement I have a result which is in the form (cylindrical coordinates): $$ A\boldsymbol{e_{\theta }}=kr\boldsymbol{e_{\theta }} $$ And I have to provide the answer in cartesian coordinates.Homework Equations I know that the unit vectors: $$ \boldsymbol{\hat{\theta}...

Let W be a hyperplane in R4 spanned by the column vectors v1 , v2, and v3, where Note that these are suppose to be COLUMN vectors: v1 = [3,1, -2 , -1], v2 = [0, -1, 0 , 1], v3= [1,2 ,6, -2] Find the Cartesian (i.e., linear) equation for W. I'm not quite sure where to start or how to...

Homework Statement This is not really coursework. Instead, this is some sort of curiosity and proposition formulation rush. Then the initial questions are that if this is a valid result that is worth to be proven. Let X,Y be metric spaces and X\times Y with another metric the product metric...

Ok I have a quick question. I have this problem that is doable with polar coordinates and triple integrals but I was wondering if it would be possible to do this problem in the cartesian coordinate system (odd question I know...). Homework Statement A sprinkler distributes water in a circular...

Homework Statement Homework Equations The Attempt at a Solution Do you see that 2 between A and the integral? There's no 2 in the above equation. I don't see where that 2 came from. Everything else is fine.

Homework Statement The magnetic field around a long, straight wire carrying a steady current I is given in spherical coordinates by the expression \vec{B} = \frac{\mu_{o} I }{2∏ R} \hat{\phi} , where \mu_{o} is a constant and R is the perpendicular distance from the wire to...

Homework Statement In order to advance on a problem I'm working, I need to covert this ellipsoid from cartesian to spherical coordinates. \frac{x^2}{a^2} +\frac{y^2}{b^2} +\frac{z^2}{c^2} = 1 Homework Equations x^2 +y^2+z^2= \rho ^2 x=\rho sin \phi cos \theta y= \rho sin \phi sin...

I'm working with 3D geometry, and I've been at this for days. I'm beating my head against a wall, because I'm nearly done with the project. There's only one glitch in my system. The Situation: I have a 3D cartesian coordinate system with a Spherical system overlaid over it (the "poles"...

I know the Cartesian product for an algebraic structure: A x B = {(a,b): a ∈ A, b ∈ B} Which naturally gives An = {(a1, a2, ... , an): ai ∈ A ∀ i} Some of the time, at least we can also have a non integer n. For example [A x A x A]2/3 = A x A. Is there any way of continuing the...

Homework Statement Prove that the unit vector r{hat} of two-dimensional polar coordinates is equal to r{hat}= x{hat}cosθ + y{hat}sinθ and find the corresponding expression for θ{hat}. all I need is the last part... I'm just not sure what θ{hat} is? How do I go about doing this? Nothing in my...

Hi! I have to do cartesian mesh of 3d object. I have to do this in Matlab. It will be a part of FEM programme. I've read sth about Delanuay triangulation etc, but I have to do really simple disretization sth like in http://en.wikipedia.org/wiki/Regular_grid (Example of a Cartesian grid.)...

Homework Statement The plane that passes through the point (1, 6, 4) and contains the line x = 1 + 2t; y = 2 - 3t; z = 3 - t where t is an element of R Homework Equations x = 1 + 2t; y = 2 - 3t; z = 3 - t The Attempt at a Solution Let L be the solution. L = (1,6,4) - ? t = (x -1)/ 2 =...

Homework Statement Let Tx and Ty be topologies on X and Y, respectively. Is T = { A × B : A\inTx, B\inTy } a topology on X × Y? The attempt at a solution I know that in order to prove T is a topology on X × Y I need to prove: i. (∅, ∅)\inT and (X × Y)\inT ii. T is closed under...

we all know that triple integral can be solved by either cartesian coordinates , spherical ,or cylindrical coordinates i just need like some advice in knowing when the variable used is constant and when it is not for example : r in cylindrical coordinates can it be constant or not?? because i...

If I have an integral: \int\int_{R} x^{2} + y^{2} dy dx Where the region R is the area enclosed by a circle centered on the origin of any given radius, is it possible to just convert x^2 + y^2 to r^2 and integrate from 0 to r over dr and 0 to 2 pi over d\theta? So it would become...

Solving the Laplace equation in Cartesian Coordinates leads to the 2nd order ODEs: \frac{X''}{X}=k_1, \qquad \frac{Y''}{Y}=k_2 \qquad \frac{Z''}{Z}=k_3 In each case the sign of k_i will determine if the solution (to the particular ODE) is harmonic or not. Hence, if two people solve the...

Homework Statement Find volume bounded by: x - z = 0; x + z = 3; y + z = 1; z = y + 1; z = 0 Homework Equations Vol = \int\int\int_{V}dV The Attempt at a Solution I really don't know how to begin this problem because I have trouble visualizing what all those intersection of...

Okay, so I have just broken into the polar coordinate system, and I like to derive things on my own to strengthen my intuition. I decided to try and derive the equation of an ellipse swiftly on my own, and had the high ambitions of eventually deriving the area of an ellipse with polar...

Hi, I have to solve a numerical problem namely how air is flowing first through a porous medium followed by streaming of the air coming out of the porous medium in a very narrow channel flowing to the ambient (journal porous air bearing). This problem can be described with the use of two...

Transformation from Cartesian to spherical polar coordinates In dimensions: x=r sinθ cos \varphi and y= r sin θ sin \varphi z=r cos θ Show one example of: ∂z\alpha/ ∂xμ . ∂xμ/ ∂z\alpha = δ\alpha\beta Now here is my answer: δyx=(∂y/∂r . ∂r/∂x) + (∂y/∂θ . ∂θ/∂x) + (∂y/∂\varphi...

Homework Statement Two long metal plates of length L>>H (their height) intersect each other at right angles. Their cross section is a cross with each line of length H. This configuration is held at a potential V=Vo and the total charge up to a distance d (d<<H) from the center of the cross is...

Hi I'm getting confused solving the Laplace eqn in Cartesian coordinates. The equation can be solved by solving each of \frac{X''(x)}{X(x)}=-k_x^2, \qquad\qquad \frac{Y''(y)}{Y(y)}=-k_y^2, \qquad\qquad \frac{Z''(z)}{Z(z)}=k_z^2 and then substituting into the equation...

can i get some help in how i can convert from cartesian to cylindrical and spherical unit vectors and vice versa ? thank you

Cartesian Diver Help!? Homework Statement So for a class assignment we have to choose from the following laws/principles that potentially apply to the function of a Cartesian Diver: (Our teacher stated that there are only two that do not apply from this list - Please state which two do not...

Homework Statement Alright, here's the question, A stream function for a plane, irrotational, polar-coordinate flow is ψ=9r^2sin^θ. Find out the velocity potential in Cartesian Co-ordinate! Homework Equations The Attempt at a Solution Well, I can easily find out the velocity...

Do tuples exist which aren't elements of a cartesian product of sets? Can you just write an ordered list of elements which does not necessarily have to be defined in sets? (or does every tuple need to be defined through sets in order for it to rigourously exist in mathematics?)

we know that for any (x, y) in Cartesian system, there is such (r, θ) in Polar system, so x = r * cosθ and y = r * sinθ how you can calculate what corresponds to (Δx, Δy) in polar system? how come Δx * Δy = r * Δr * Δθ? Maybe this is very stupid question and has obvious answer...

1. The position vector of a particle at time t ≥ 0 is given by r = sin(t)*i + cos(2t)*j. Find the cartesian equation for the path of the particle. 2. I was told that the answer is: y = 1 - 2x^2 But I don't know how to obtain that solution. 3. r = sin(t)*i + cos(2t)*j At first I...

Homework Statement So I am just trying to understand the concepts here. My main question is what exactly is the cartesian product of an omega tuple? Homework Equations Given a set X, we define an ω-tuple of elements of X to be a function x:N\rightarrowX We denote x as Let...

How can I prove this: Let A, B, and C be non empty sets. If A is bijective to B, then A x C is bijective to B x C. also if A and B are bijective Power set of A is bijective to Power set of B and finally Fun(A,C) is bijective to Fun(B,C)