Cartesian Definition and 549 Threads

Hi, I have a problem understanding the difference between Cartesian product of vector spaces and tensor product. Let V1 and V2 be vector spaces. V1 x V2 is Cartesian product and V1 xc V2 is tensor product (xc for x circled). How many dimensions are in V1 x V2 vs V1 xc V2? Thanks, Monte

Homework Statement A field is given in spherical coordinates as F=[cos(θ)/r2]∙ar+[sin(θ)/r]∙aθ. Express F in terms of x, y, z, ax, ay, azHomework Equations ar∙ax=sin(θ)cos(∅) ar∙ay=sin(θ)sin(∅) ar∙az=cos(θ) aθ∙ax=cos(θ)cos(∅) aθ∙ay=cos(θ)sin(∅) aθ∙az=-sin(θ) x=r*sin(θ)*cos(∅)...

Homework Statement Picture is attached Homework Equations The Attempt at a Solution I feel like I am right and the computer is wrong, but I just wanted to check here first. I attached a picture of the problem and my answer. This is how I solved the problem: Fz=750sin(45)=530.33N...

I am having trouble solving this problem, I also have it attached: http://i160.photobucket.com/albums/t193/John123321_bucket/Capture.png Here is a short version of how I got my answers, but it says it’s wrong: X=500cos(30)=433N i= 433sin(45)=306.2 j=-433cos(45)=-306.0...

Hey Guys, I've been passed this from a friend to help them out, but I don't know much about it. So this is a bit of annoying first post... SORRY! Homework Statement if z = √2 + i, use de Moivre's theorem to find z^5 in Cartesian form. AND. Convert z = -8 + 8 √3 i to polar...

Hi all! Is it possible to derive x-y-z vectors of c in 3d cartesian space? Is there any way we could then observe the photon (or measure its speed) in parallel with x-axis for example? Thanks.

Dear readers, Let X be the product space of a countable family \{X_n:n\in\mathbb{N}\} of separable metric spaces. If X is endowed with the product topology, we know that it is again separable. Are there other topologies for X such that is separable? Is there a natural metric on X such that X...

I'm going through the set theory material in the appendix of Knapp's Basic Algebra. I want to make sure that I understand what he says is the set theoretic notion of the indexed cartesian product, \prod_{x\in S}A_{x}. He says that this can be thought of as the set of all functions...

Homework Statement Consider a velocity field where the radial and tangenetial components of velocity are V_r=0 and V_theta=cr, respectively, where c is a constant. Obtain the equations of the streamlines. Homework Equations x=rcos(theta) y=rsin(theta) The Attempt at a Solution I...

URGENT HELP! cartesian form vector help! Homework Statement given the following vectors, calculate Adotb and BxA. Express your answers in cartesian Form A= +6X-3Y+4Z B= -3X+1Y-5Z Homework Equations The Attempt at a Solution My teacher went over this in about five minutes making...

Homework Statement The Carteian coordinates of a point on a circle with its center at the origin are [0.40, 0.30]. What is the arc length measured counterclockwise on the circle from the positive x-axis to this point? Homework Equations The Attempt at a Solution Wouldn't they be...

Hello all, I'm new, so please go easy on me if this is a silly question. If I have a source set of characters, say: "abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ0123456789!\"#$%&'()*+,-./:;<=>?@[\\]^_`{|}~ " And I compute the Cartesian product of this source set (sort of). This gives...

Hello, I was wondering if there were alternative definitions to a "function" ( alternative to the standard f is a subset of A X B if f : A -> B ). I was introduced to the "general" definition of a cartesian product ( with respect to an indexing set H ) , it is weird to me because the general...

Homework Statement A pilot wishes to fly form city A to city B, a distance of 720 km on a bearing of 070°. The speed of the plane is 700 km/h. An 60 km/h wind is blowing on a bearing of 110°. What heading should the pilot take to reach his or her destination? How long will the trip take? Use...

Homework Statement Transform to polar coordinates and evaluate... \int^{a/\sqrt{2}}_{0} dx\int^{\sqrt{a^2-x^2}}_{x}\sqrt{x^2 + y^2}dy Homework Equations x^2 + y^2 = r^2 x = r cos \theta y = r sin \theta I've been struggling to make sense of this problem, it should be easy, I'm...

Why is \sqrt{9+36} = 3\sqrt{5} and not 6.708 ? I wasnt interested in maths at school but now I'm trying to self teach, so pardon my ignorance. Edit: Ok i feel foolish now, no need to correct me as I've just worked out i WAS correct. I'm still unsure as to why it would be shown...

I'm interested how to solve the following problem: if we have a triangle, where a,b,c are sides of that triangle and we know that (a+b+c)x(a+b-c)=3ab, we need to find the angle opposite to side c. How to do this?

Homework Statement integrate 1/((1+x^2+y^2)^2) dx dy Both x and y going from 0 to infinity Homework Equations x^2+y^2 =r The Attempt at a Solution After that I get 1/(1+r^2) ^2 Cannot visualize the function, do not know what the limits are. If I could have any help it...

I'm writing a function for Matlab and I'm trying to figure out how to apply a torque matrix in cartesian coordinates to an object in spherical coordinates. The short story is this: For interest's sake, a friend and I have written a function with creates a tree which random branch...

Homework Statement Given the point P(-2,0,6) and the line "x-4/3= y+2/-5=z-1/2" determine the cartesian equation of the plane. Homework Equations the lines vector equation is (4,-2,1)+T(3,-5,2)The Attempt at a Solution using A(x,y,z) i attempted to find a directional vector for P...

Homework Statement Determine the Cartesian equation for the plane with the following: 1) through the points (1,2,1) and (2,1,4) 2) parallel to x-axis Homework Equations Parametric Equations Vector Eqs Plane Eqs The Attempt at a Solution I basically understand how perpendicular...

Homework Statement evaluate the following triple integral in spherical coordinates:: INT(=B) = (x^2+y^2+z^2)^2 dz dy dx where the limits are: z = 0 to z = sqrt(1-x^2-y^2) y = 0 to z = sqrt(1-x^2) x = 0 to x = 1 Homework Equations The only thing I know for sure is how to set...

Homework Statement x=2acos(theta- pi/3)cos(theta) y=2acos(theta- pi/3)sin(theta) Write everything in terms of x and y Homework Equations cos(a-b)=cos(a)cos(b)+sin(a)sin(b) sina/cosa=tana etc... The Attempt at a Solution I've tried a lot of rearranging, can't figure it out.

Homework Statement Question 3 (a)A long metal cylinder of radius a has the z-axis as its axis of symmetry.The cylinder carries a steady current of uniform current density J = Jzez. Derive an expression for the magnetic field at distance r from the axis,where r<a. By resolving the...

Homework Statement Prove for all sets A, B, and C that: Ax(B^C) = (AxB)^(AxC) Homework Equations AxB is the Cartesian Product of A and B. That is, the set of all ordered pairs (x,y) such that x is an element of A and y is an element of B. ^ denotes intersection (and) The Attempt...

"finite cartesian product of connected space is connected" hi am not able understand the theorem that.. "finite cartesian product of connected space is connected".. what is a base point? how it is related to homeomorphism? can anyone explian?

Hi, Everyone: I just read recently a comment to the effect that Descartes never intented his use of coordinates in his layout of analytic geometry to be used globally; there was also a follow-up comment about " no one really using any coordinates in a global way. Does anyone...

I have a series of maps from various parts of the world. Each map is 512 pixels by 512 pixels. I also have approximate left, top, right and bottom latitude and longitude values for each map. I'm wanting to plot variable lat/long value on the maps which need to be converted to x,y values -...

Homework Statement Andrew and David kick a football at the same time. Andrew kicks it with a force 155N and David kicks it with a force 210N. Angle between the two forces is 30 degrees. Calculate the magnitude and direction of the resultant force. Homework Equations Cosine law...

Find all units in Z12 X Z6 and their inverses. What i did was find the units of Z12 which are 1,11,5,7 then the ones of Z6 which are 1,5 and take the cartesian product of those two sets?

Homework Statement Let a surface be define by z = x^(a=3) = f[x^(a=1,2)] Show that the Christoffell sybols of the 2nd kind are: [Christoffell symbol]^abc = { fafbc }/ { f^\alpha f_sub_\alpha } where indices on f indicates partial derivatives Homework Equations (d^2...

Hi there, a few days ago I derived the probably well-know expression for the magnetic field of a current loop including elliptic integrals of the first and second kind (it can be seen here http://plasmalab.pbworks.com/f/bfield.pdf" ). As I'd like to rotate and shift the position of the...

Homework Statement I'm given 2 unit vectors a_x and a_theta. I need to find the dot product between the two. Homework Equations Conversion from Cylindrical to Cartesian x = r * sin(theta) y = r * cos(theta) z = z Conversion from Cartesian to Cylindrical r = sqrt(x^2 +...

Dear Frnds, my problem is very simple. please help me. I have a dielectric tensor like: epsilon_T= |1 0 0| |0 1 0| %% note that it is in cartesian co-ordinate system |0 0 a| now the potential...

I do not understand when we are given a vector at point P(x,y,z) or in different forms cylindrical and spherical. What does it mean at point?? I mean aren't vectors supposed to start at origin, even if they don't how will that make a difference in their magnitude or angle between them. For...

Homework Statement Not really a homework question, but more of a concept question which I'm unfamiliar with. So as we know, equations can be in any coordinate, but how do you convert them from one to another? For example, a few equations from fluid mechanics. the first equation is the vector...

[SOLVED] Electric field in cartesian coordinates Homework Statement Suppose the electric potential is V(r) = C1 /r + C2 cosθ /r^2 where (r, θ, φ) are the spherical polar coordinates for points in three dimensions. [Data: C1 = 4.3 Vm ; C2 = 1.6 Vm^2 ] (A) Determine the electric...

Hi. First off I don't know if this is the right topic area for this question so I'm sorry if it isn't. So my current situation is that I can find the jacobian matrix for a transformation from spherical to cartesian coordinates and then take the inverse of that matrix to get the mapping from...

Hello. For a physics course, I need to often make use of the binomial series and it's corollary, the expansion of: \sqrt{1-x^2} This probably sounds rather stupid, but for some reason, when I do a MacClaurin expansion of this series, I cannot seem to generate the correct series, which I...

Homework Statement Specifically, this question is about computing the expectation value of the Hamiltonian of a variational calculation of a hydrogen atom *neglecting the potential term. I'm assuming the trial wavefunction e^{-\alpha r}. The question, however, is purely math based, not...

Homework Statement Write the following polar equation in Cartesian coordinates: r= 2/(3cos(theta)-9sin(theta)) Homework Equations r = (x^2 + y^2) ^.5 x=rcos theta y=rsin theta sin^2(x)+cos^2(x) = 1 The Attempt at a Solution I'm stuck on how to do this. Any push in the right...

Homework Statement Find a bijective map g : X^{m} \times X^{n} \rightarrow X^{m + n} Homework Equations The Attempt at a Solution I don't even know where to begin. How would I map X^{m} \times X^{n} in the first place? How could I map X^{m + n}?

Is the cartesian product (A \times B) the set of ALL POSSIBLE ordered pairs (a, b) such that a is an element of A and b is an element of b, or is it simply the set of "all ordered pairs?"

Homework Statement Find a Cartesian equation for the curve and identify it. r = 3sin(θ)Homework Equations The Attempt at a Solution I know that the answer is as follows... r^2=3(r*sin(\theta)) x^2+y^2 = 3y x^2 + (y-\frac{3}{2})^2 = (\frac{3}{2})^2 I understand everything but the last part...

Homework Statement 6...y ∫...∫ x dx dy 0 ...0 The Attempt at a Solution for reference: π=pi 6...y ∫...∫ x dx dy 0 ...0 y=6=x=6. rcosΘ=6 r=6/cosΘ goes to π/2 6secΘ ∫ ...∫... r cosΘ r dr dΘ π/4...0 π/2.....r=6secΘ ∫ ... [((r^3)/3)cosΘ] dΘ π/4......r=0 π/2...

Homework Statement Velocity vector, v: v = (yi + xj) / (x^2 + y^2 + z^2)^(3/2) "Re write "v" in spherical polar coordinates and unit vectors" Homework Equations Obviously r = (x^2 + y^2 + z^2)^(1/2) theta = tan^(-1)((x^2 + y^2)^(1/2)/z) phi = tan(^-1)(y/x) The Attempt at a Solution...

I'm trying to create a program which will do these calculations. I am given Latitudes, Longitudes, and ellipsoidal heights for ellipsoids and another set of x, y, and z coordinates. I am so confused looking at how to convert these things. I know I need to iterate in order to go from cartesian to...

Howdy everyone, I'm on a quest for something that is proving a bit elusive at the moment: a Cartesian to polar transform (along with its inverse) for \mathbb{R}^4. I'm well aware of how to derive the transform for both \mathbb{R}^2 and \mathbb{R}^3, as it is just a matter of looking at the...

Hello I am carrying out some analysis on the magnetic field generated over a 3D region by a single current loop. The present form of the equations is in cylindrical coordinates and is as follows \vec{B}={Brc,0,Bz} There is no angular component in this present from. Note: The following website...

Homework Statement Okay here's the problem: Consider the region R interior to a circle(of r =2) and exterior to a circle(r=1). 1.Using cartesian coords and double integral, calc the area of annulus. 2. repeat calculation above but using double integral with polar coords The...