# What is Cartesian: Definition and 558 Discussions

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
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B
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{
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a

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{\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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4. ### 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...
5. ### 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}##...
6. ### 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...
7. ### 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...
8. ### 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...
9. ### 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...
10. ### 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...
11. ### 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.
12. ### B Exploring Holonomic Basis in Cartesian Coordinates

Are cartesian coordinates the only coordinates with a holonomic basis that's orthonormal everywhere?
13. ### 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...
14. ### 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)
15. ### 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?
16. ### 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...
17. ### 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##...
18. ### 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...

35. ### 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...
36. ### 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...
37. ### 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...
38. ### Cartesian Product of Sets

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}...
39. ### 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...
40. ### 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...
41. ### I Cartesian Product Definitions

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...
42. ### 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...
43. ### 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...
44. ### 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...
45. ### 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...
46. ### 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...
47. ### 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...
48. ### Express exp(3+Pi*i) in Cartesian Form

The problem statement Express exp(3+π*i) in Cartesian Form. The attempt at a solution Equating e^(3+πi) = e^(x)e^(iy) = e^(x)(cos(y) + isin(y)) then e^(x)cos(y) = 3 e^(x)sin(y)=π now |e^(3+πi)| = e^(x) so x = sqrt(9+π^2) then cos(y) = 3/sqrt(9+π^2) sin(y) = π/sqrt(9+π^2) at this point i don't...
49. ### I Cartesian and polar terminology

I have a scalar quantity ##V## (let's call it a voltage for concreteness) that is a function of angle ##\theta##. There are two obvious ways to plot it, as a Cartesian plot (see A above) or as a polar plot (see B). I can also express the polar plot in terms of Cartesian coordinates ##V_x = V \...
50. ### I From Geographical coordinates to Cartesian coordinates

I have 2 points expressed in (latitude,longitude) and I want to calculate the angle with respect to the north pole. Since the two points are very near (like hundred of meters), is it possible to consider the two points in the carthesian system simply as: x=longitude y=latitude Then...