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.
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
$$ ds^2 \approx - ( 1 - \frac {2M} {R}) dt^2 + (1 + \frac {2M} {R})...
I am looking for more books like this one: https://archive.org/details/MethodOfCoordinateslittleMathematicsLibrary
Method of Coordinaes (Little Mathematics Library) by A. S. Smogorzhevsky
I am also interested in papers if you can suggest any. I am interested in texts, that explore the idea of...
So given the Helmholtz equation $$\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...
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...
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}##...
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...
$$L = \frac {mv^2}{2} - mgy$$
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...
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...
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...
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...
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.
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...
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?
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...
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##...
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...
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.
$$x=\rho \sin \phi \cos \theta$$
$$y= \rho \sin \phi...
I got a polar function.
$$ \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...
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...
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...
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.]
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...
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...
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)##...
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...
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...
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...
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...
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...
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$$...
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...
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...
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...
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}...
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...
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...
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...
<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...
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...
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...
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...
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...
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...
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 \...
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...