Cartesian Definition and 561 Threads

In mathematics, specifically set theory, the Cartesian product of two sets A and B, denoted A × B, is the set of all ordered pairs (a, b) where a is in A and b is in B. In terms of set-builder notation, that is




A
×
B
=
{
(
a
,
b
)

a

A



and



b

B
}
.


{\displaystyle A\times B=\{(a,b)\mid a\in A\ {\mbox{ and }}\ b\in B\}.}
A table can be created by taking the Cartesian product of a set of rows and a set of columns. If the Cartesian product rows × columns is taken, the cells of the table contain ordered pairs of the form (row value, column value).One can similarly define the Cartesian product of n sets, also known as an n-fold Cartesian product, which can be represented by an n-dimensional array, where each element is an n-tuple. An ordered pair is a 2-tuple or couple. More generally still, one can define the Cartesian product of an indexed family of sets.
The Cartesian product is named after René Descartes, whose formulation of analytic geometry gave rise to the concept, which is further generalized in terms of direct product.

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  1. N

    B Cartesian Space vs. Euclidean Space

    For a while I've been trying to get a better understanding of how Descartes' invention of Cartesian space revolutionized math. It seems like an invention on par with the invention of agriculture in how it led to human progress. I am still having trouble, though, pinpointing examples of what can...
  2. kirito

    The div in cartesian coordinates

    I am currently studying a section from \textit{Electricity and Magnetism} by Purcell, pages 81 and 82, and need some clarification on the following concept. Here’s what I understand so far: 1. The integral of a function $ \mathbf{F} $ over a surface \( S \) is equal to the sum of the integrals...
  3. M

    Is My Solution to the Driven Spring Problem Correct?

    For this problem, For part(a), I am not sure if I am solving it correctly. I define the usual cartesian x-y coordinate system at the base of the wall. This gives ##x = l_0 + q(t) + x_w(t) = l_0 + q(t) + d\sin(\gamma t)## which implies that ##\dot x = \dot q + d \gamma \cos (\gamma t)##...
  4. E

    I Schwarzschild in Cartesian: Tricks for Transformation

    According to Schutz, the line element for large r in Schwarzschild is $$ ds^2 \approx - ( 1 - \frac {2M} {r}) dt^2 + (1 + \frac {2M} {r}) dr^2 + r^2 d\Omega^2 $$ and one can find coordinates (x, y, z) such that this becomes $$ ds^2 \approx - ( 1 - \frac {2M} {R}) dt^2 + (1 + \frac {2M} {R})...
  5. Trysse

    Geometry Looking for books (or papers) on the Cartesian coordinate system

    I am looking for more books like this one: https://archive.org/details/MethodOfCoordinateslittleMathematicsLibrary Method of Coordinaes (Little Mathematics Library) by A. S. Smogorzhevsky I am also interested in papers if you can suggest any. I am interested in texts, that explore the idea of...
  6. bob012345

    I Helmholtz Equation in Cartesian Coordinates

    So given the Helmholtz equation $$\nabla^2 u(x,y,z) + k^2u(x,y,z)=0$$ we do the separation of variables $$u=u_x(x)u_y(y)u_z(z)= u_xu_yu_z$$ and ##k^2 = k_x^2 + k_y^2 +k_z^2## giving three separate equations; $$\nabla^2_x u_x+ k_x^2 u_x=0$$ $$\nabla^2_y u_y+ k_y^2 u_y=0$$ $$\nabla^2_z u_z+ k_z^2...
  7. chwala

    Find the Cartesian equation given the parametric equations

    hmmmmm nice one...boggled me a bit; was trying to figure out which trig identity and then alas it clicked :wink: My take; ##x=(\cos t)^3 ## and ##y=(\sin t)^3## ##\sqrt[3] x=\cos t## and ##\sqrt[3] y=\sin t## we know that ##\cos^2 t + \sin^2t=1## therefore we shall have...
  8. chwala

    Find the Cartesian equation of the curve

    Find ms solution; My approach; ##xt=t^2+2## and ##yt=t^2-2## ##xt-2=t^2## and ##yt+2=t^2## ##⇒xt-2=yt+2## ##xt-yt=4## ##t(x-y)=4## ##t=\dfrac{4}{x-y}## We know that; ##x+y=2t## ##x+y=2⋅\dfrac{4}{x-y}##...
  9. chwala

    Comparing Hyperbolic and Cartesian Trig Properties

    I came across this question; i noted that the hyperbolic trigonometry properties are somewhat similar to what i may call cartesian trigonometry properties... My approach on this; ##\tanh x = \sinh y## ...just follows from ##y=\sin^{-1}(\tan x)## ##\tan x = \sin y## Therefore...
  10. Father_Ing

    Cartesian and polar coordinate in Simple pendulum, Euler-Lagrange

    $$L = \frac {mv^2}{2} - mgy$$ It is clear that ##\dot{x}=\dot{\theta}L## and ##y=-Lcos \theta##. After substituting these two equations to Lagrange equation, we will get the answer by simply using this equation: $$\frac {d} {dt} \frac {∂L}{∂\dot{\theta}} - \frac {∂L}{∂\theta }= 0$$ But, What if...
  11. nuclearsneke

    I Convert cylindrical coordinates to Cartesian

    Good day! I am currently struggling with a very trivial question. During my studies, I operated with a parameter called "geometrical buckling" for neutrons and determined it in cylindrical coordinates. But thing is that we usually do not consider buckling's dependence on angle so its angular...
  12. A

    Difference between a closed solid and a Cartesian surface

    Greetings All! I have hard time to make the difference between the equation of a closed solid and a cartesian surface. For example in the exercice n of the exam I thought that the equation was describing a closed solid " a paraboloid locked by an inclined plane (so I thought I could use...
  13. K

    I A little clarification on Cartesian tensor notation

    Goldstein pg 192, 2 edIn a Cartesian three-dimensional space, a tensor ##\mathrm{T}## of the ##N## th rank may be defined for our purposes as a quantity having ##3^{N}## components ##T_{i j k}##.. (with ##N## indices) that transform under an orthogonal transformation of coordinates...
  14. M

    I Transform from polar to cartesian

    Probability distribution - uniform on unit circle. In polar coordinates ##dg(r,a)=\frac{1}{2\pi}\delta(r-1)rdrda##. This transforms in ##df(x,y)=\frac{1}{2\pi}\delta(\sqrt{x^2+y^2}-1)dxdy##. The problem I ran into was the second integral was 1/2 instead of 1.
  15. M

    B Exploring Holonomic Basis in Cartesian Coordinates

    Are cartesian coordinates the only coordinates with a holonomic basis that's orthonormal everywhere?
  16. chwala

    Find the Cartesian equation of a curve given the parametric equation

    My interest on this question is solely on ##10.iii## only... i shared the whole question so as to give some background information. the solution to ##10.iii## here, now my question is, what if one would approach the question like this, ##\frac {dy}{dx}=\frac{t^2+2}{t^2-2}## we know that...
  17. R

    Transformation of Reynolds Equation from Cartesian to cylindrical

    ∂/∂x ((ρh^3)/12μ ∂p/∂x) + ∂/∂z ((ρh^3)/12μ ∂p/∂z) = ∂/∂x (ρh (U_1+U_2)/2) + ∂/∂z (ρh (W_1+W_2)/2) + (∂(ρh))/∂t (1) 1/r ∂/∂r (r (ρh^3)/12μ ∂p/∂r) + 1/r ∂/∂θ ((ρh^3)/12μ ∂p/r∂θ) = rω/2 ∂(ρh)/r∂θ + (∂(ρh))/∂t (2)
  18. L

    A Tensor product in Cartesian coordinates

    I am confused. Why sometimes perturbation ##V'=\alpha xy## we can write as ##V'=\alpha x \otimes y##. I am confused because ##\otimes## is a tensor product and ##x## and ##y## are not matrices in coordinate representation. Can someone explain this?
  19. Pipsqueakalchemist

    Engineering Using Cartesian vs. Normal/Tangential Coordinates for Centripetal Motion

    So for this problem the solution used Cartesian coordinates but I was wondering wouldn’t it be easier to use Normal and tangential coordinate because the bar is undergoing centripetal motion? Also on the right diagram shouldn’t the acceleration be down and not up. The reason I think that is...
  20. Eclair_de_XII

    Cartesian sum of subspace and quotient space isomorphic to whole space

    Let ##n=\dim X## and ##m=\dim Y##. Define a basis for ##X: y_1,...,y_m,z_{m+1},...,z_n##. The first ##m## terms are a basis for ##Y##. The remaining ##n-m## terms are a basis for its complement w.r.t ##X##. Let's call it ##Z##. ##X## is the direct sum of ##Y## and ##Z##; denote it as ##X=Y+Z##...
  21. srfriggen

    I Cartesian to Polar form.... Is it just a transformation of the plane?

    Hello, Today I started to think about why graphs, of the same equation, look different on the Cartesian plane vs. the polar grid. I have this visualization where every point on the cartesian plane gets mapped to a point on the polar grid through a transformation of the grids themselves...
  22. Athenian

    I Transforming Cartesian Coordinates in terms of Spherical Harmonics

    As the subject title states, I am wondering how would one go about transforming Cartesian coordinates in terms of spherical harmonics. To my understanding, cartesian coordinates can be transformed into spherical coordinates as shown below. $$x=\rho \sin \phi \cos \theta$$ $$y= \rho \sin \phi...
  23. JorgeM

    A How do I express an equation in Polar coordinates as a Cartesian one.

    I got a polar function. $$ \psi = P(\theta )R(r) $$ When I calculate the Laplacian: $$ \ \vec \nabla^2 \psi = P(\theta)R^{\prime\prime}(r) + \frac{P(\theta)R^{\prime}(r)}{r} + \frac{R(r)P^{\prime\prime}(\theta)}{r^{2}} $$ Now I need to convert this one into cartesian coordinates and then...
  24. T

    I Dot product in Euclidean Space

    Hello As you know, the geometric definition of the dot product of two vectors is the product of their norms, and the cosine of the angle between them. (The algebraic one makes it the sum of the product of the components in Cartesian coordinates.) I have often read that this holds for Euclidean...
  25. E

    I Velocity Vector Transformation from Cartesian to Spherical Coordinates

    Hi all, I can't find a single thing online that translates a cartesian velocity vector directly to spherical vector coordinate system. If I am given a cartesian point in space with a cartesian vector velocity and I want to convert it straight to spherical coordinates without the extra steps of...
  26. BadgerBadger92

    B Time in Cartesian Coordinate Systems: Math Q&A

    I am teaching myself math and have a question about cartesian coordinate systems. How is time illustrated in such a graph? [Moderator's note: Moved from a math forum after post #13.]
  27. M

    MHB Sets so that the cartesian product is commutative

    Hey! :o Let $A,B$ be sets, such that $A\times B=B\times A$. I want to show that one of the following statements hold: $A=B$ $\emptyset \in \{A,B\}$ I have done the following: Let $A$ and $B$ be non-empty set. Let $a\in A$. For each $x\in B$ we have that $(a,x)\in A\times B$. Since...
  28. Z

    I Area Differential in Cartesian and Polar Coordinates

    The area differential ##dA## in Cartesian coordinates is ##dxdy##. The area differential ##dA## in polar coordinates is ##r dr d\theta##. How do we get from one to the other and prove that ##dxdy## is indeed equal to ##r dr d\theta##? ##dxdy=r dr d\theta## The trigonometric functions are used...
  29. Arman777

    I Radial Vector in Cartesian form

    If I wanted to write ##\hat{r}##in terms of ##\hat{x}##and ##\hat{y}##, is it ##\frac{\hat{x} + \hat{y}}{\sqrt{2}}## ?
  30. n3pix

    Converting Velocity Formula: Polar to Cartesian

    I have a little question about converting Velocity formula that is derived as, ##\vec{V}=\frac{d\vec{r}}{dt}=\frac{dx}{dt}\hat{x}+\frac{dy}{dt}\hat{y}+\frac{dz}{dt}\hat{z}## in Cartesian Coordinate Systems ##(x, y, z)##. I want to convert this into Polar Coordinate System ##(r, \theta)##...
  31. K

    I Locally Cartesian Coordinates on the Sphere

    I was trying to construct locally Euclidean metrics. Consider the sphere with the usual coordinate system induced from spherical coordinates in ##\mathbb R^3##. Consider a point ##p## in the Equator having coordinates ##(\theta_0, \phi_0) = (\pi/2, 0)##. If you make the coordinate change ##\xi^1...
  32. Y

    MHB Does AxA Equal BxB Imply A Equals B?

    Dear all, I am trying to prove a simple thing, that if AxA = BxB then A=B. The intuition is clear to me. If a pair (x,y) belongs to AxA it means that x is in A and y is in A. If a pair (x,y) belongs to BxB it means that x is in B and y is in B. If the sets of all pairs are equal, it means...
  33. A

    Converting a Cartesian Integral to a Polar Integral

    the graph of x= √4-y^2 is a semicircle or radius 2 encompassing the right half of the xy plane (containing points (0,2); (2,0); (0-2)) the graph of x=y is a straight line of slope 1 The intersection of these two graphs is (√2,√2) y ranges from √2 to 2. Therefore, the area over which we...
  34. M

    Convert cylindrical coordinate displacement to Cartesian

    Summary: I can't figure out how the solver carries out the conversions from cartesian to cylindrical coordinates and vice-versa. I have a set of points of a finite element mesh which when inputted into a solver (ansys) gives the displacement of each node. I can get the displacement values of...
  35. colemc20

    Hollow Sphere Inertia in Cartesian Coordinates

    Problem Statement: How do you calculate the rotational inertia of a hollow sphere in cartesian (x,y) coordinates? Relevant Equations: I=Mr^2 My physics teacher said its his goal to figure this out before he dies. He has personally solved all objects inertias in cartesian coordinates but can't...
  36. Z

    I Derivation of Divergence in Cartesian Coordinates

    In section 1-5 of the third edition of Foundations of Electromagnetic Theory by Reitz, Milford and Christy, the authors give a coordinate-system-independent definition of the divergence of a vector field: $$\nabla\cdot\mathbf{F} = \lim_{V\rightarrow 0}\frac{1}{V}\int_S\mathbf{F\cdot n}da$$...
  37. K

    I Definition of Cartesian Coordinate System

    I was asking myself what is the definition of a Cartesian Coordinate System. Can we say that it's a coordinate system such that - the basis vectors are the same ##\forall x \in R^n## - the basis vectors are orthonormal at each ##x \in R^n## So for instance, normalized polar coordinates do not...
  38. cromata

    I Integration over a part of a spherical shell in Cartesian coordinates

    I am modeling some dynamical system and I came across integral that I don't know how to solve. I need to integrate vector function f=-xj+yi (i and j are unit vectors of Cartesian coordinate system). I need to integrate this function over a part of spherical shell of radius R. This part is...
  39. A

    A Representing harmonic oscillator potential operator in. Cartesian basis

    My question is given an orthonormal basis having the basis elements Ψ's ,matrix representation of an operator A will be [ΨiIAIΨj] where i denotes the corresponding row and j the corresponding coloumn. Similarly if given two dimensional harmonic oscillator potential operator .5kx2+.5ky2 where x...
  40. Carrie233

    Why Can't All Subsets of A×B Be Expressed as Cartesian Products?

    Homework Statement Prove: If A and B each have at least two elements, then not every element of P(A×B) has the form A1 ×B1 for some A1 ∈ P(A)and B1 ∈ P(B). Homework EquationsThe Attempt at a Solution Suppose A = {1, 2}, B = {3, 4}. AXB = {(1,3), (1,4), (2,3), (2,4)} P(A) = {{1}, {2}, {1,2}...
  41. T

    Converting Cartesian to Polar (Double Integral)

    Homework Statement Integrate from 0 to 1 (outside) and y to sqrt(2-y^2) for the function 8(x+y) dx dy. I am having difficulty finding the bounds for theta and r. Homework Equations I understand that somewhere here, I should be changing to x = r cost y = r sin t I understand that I can solve...
  42. S

    Cartesian to Cylindrical coordinates?

    Homework Statement I want to convert R = xi + yj + zk into cylindrical coordinates and get the acceleration in cylindrical coordinates. Homework Equations z The Attempt at a Solution I input the equations listed into R giving me: R = i + j + z k Apply chain rule twice: The...
  43. Calculuser

    I How Are Infinite Cartesian Products Interpreted in Set Theory?

    I was studying Group Theory on my own from a mathematics journal and got confused at some point where it defines Cartesian products, from binary one, say (A × B), to n-tuples one, say (A_1 × A_2 × ... × A_n). What confuses me when I tried to read it is that the definition made for infinite...
  44. M

    Solve Cartesian Product Without Symbol: Find Answers Here

    <Moderator's note: Moved from a technical forum and thus no template.> Here is a problem and solution given in my book. this is using a symbol in cartesian product . I checked other books for cartesian product examples but there was no such symbol being used. Here is my solution without...
  45. F

    Convert the polar equation to the Cartesian equation

    Homework Statement Replace the polar equation with an equivalent Cartesian equation. ##r^2 = 26r cos θ - 6r sin θ - 9## a)##(x - 13)^2 + (y + 3)^2 = 9## b)##(x + 26)^2 + (y - 6)^2 = 9## c)##26x - 6y = 9## d)##(x - 13)^2 + (y + 3)^2 = 169## Homework Equations ##x= r cos \theta## ##y= r sin...
  46. F

    Convert the Cartesian equation to the polar equation

    Homework Statement Replace the Cartesian equation with an equivalent polar equation. ##x^2 + (y - 18)^2 = 324## a)##r = 36 sin θ## b)##r^2 = 36 cos θ## c)##r = 18 sin θ## d)##r = 36 cos θ## Homework Equations ##x= r cos \theta ## ##y= r sin \theta ## ##x^2 + y^2 = r^2 ## The Attempt at a...
  47. F

    Graph the Cartesian equation: x=4t^2, y=2t

    Homework Statement Parametric equations and a parameter interval for the motion of a particle in the xy-plane are given. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. Indicate the portion of the graph traced by the particle and the direction...
  48. F

    Graph the Cartesian equation: x = 2 sin t, y = 4 cos t

    Homework Statement Parametric equations and a parameter interval for the motion of a particle in the xy-plane are given. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. Indicate the portion of the graph traced by the particle and the direction...
  49. F

    Finding the Particle's Path: Graphing a Cartesian Equation

    Homework Statement Identify the particle's path by finding a Cartesian equation for it . Graph the Cartesian equation . indicate the portion of the graph traced by the particle . ##x=2sinh t## , ##y=2cosh t## , ##-\infty < t < \infty## Homework Equations ##cosh^2 t - sinh^2 t = 1## The...
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