Coordinate System Transformation Question

Factual Questions
1

Ok. I am going to make a feeble attempt at asking this question without the use of the a symbolic editor. Its gonna be a mess so strap yourself in for the ride.

I am in my final math class for my EE (Advanced Eng. Mathematics) and we have just begun with a “refresher” on coord system transformations. I have gotten every thing fine up until this question on the practice problems (please assume that every capital letter is a unit vector):

Transform the following vectors into spherical coordinates and then evaluate them at the indicated points:

A= Xy^2 + Yxz + Z4: In cartesian the points are (1, -1, 2)

Now I have the book which gives the 3 equations which you solve to get the 3 spherical coordinates: R, theta, and phi. They are too long for me to input into this post, but I am thinking that anyone who would be able to help will already know them or have access to them. The thing is that they are packed with sin and cos functions. I punch all three equations into my TI-89 (best calc ever made) and the thing comes back with some crazy answers that I cant even understand. I am very comfortable with this calc (I’ve used for three years straight) so I know I am punching it in correctly.

The answers are right next to the problem so we can check ourselves.

A=R2.858 -THETA2.887+PHI2.121

I am not even getting numbers for my answer on my calc, just a copy of the same equations.

This is NOT a graded assignment nor is it a quiz or anything like that. We are going to be going over it in class on Wendsday so please don’t get all mad at me for making you do my homework. This is an optional refresher. We have the answer for petes sakeI just want to get it together before class on Wendsday so I can follow along with his instruction. I just want a good starting point. I think I may be making this harder than it is.

Are there any weblinks that cover this type of problem in laymans terms? Does anyone have any tips or a good place to start?

2

I’m a bit confused by your notation. These uppercase X,Y,Z, are they indepentent parameters, unit-vectors or something else? In which case, what are the lower case x,y,z?

It’s hard to write scientific notation i plain ascii, but one way of writing vectors is to use a comma separated notation (A,B,C) to indicate the point (x=A, y=B,z=C).

I think we need more info to help you here.

3

I’m assuming A is the vector evaluated at (x,y,z) = (1,-1,2). Then

A = X*(-1)^2 + Y12 + Z*4

|A| = R = sqrt(1^2 + 2^2 + 4^2) = sqrt(21) = 4.58258

So right away there’s a problem. Maybe I’m misunderstanding the notation, or maybe the given answer is wrong.

You might want to consider doing this in two steps, from spherical (R, theta, phi) to cylindrical (Rho, Phi, Z) to Cartesian coordinates.

4

Are you sure you are in RADIAN mode, and not degrees?

for (1,-1,2) shouldn’t R be sqrt(6)? (~2.449)
(tho this has nothing to do with the equation)

if X, Y and Z are unit vecors, then the result at the point indicated is (1,2,4) (cartesian), which would have an R of sqrt (21) (~4.58)

Hmm, no idea sorry

Brian

5

Gah. I have all the required information to do this problem in a textbook at home.

Don’t forget, the vector has to be transformed from the i, j, k system into the [symbol]r, j, q[/symbol].

6

I always forget how to do this, so I went and looked it up again:

  1. You have a vector A= (y^2,xz,4) in a cartesian basis. You want to rewrite this vector in a spherical basis, and evaluate it at given values of x,y, and z.

I think much of the confusion people are having in this thread so far comes from trying to transform the point to spherical coordinates. We want to transform the entire vector to spherical coordinates, and then evaluate it at that point.

Oh, and just to write them down:
x = r sin(o) cos(p)
y = r sin(o) sin(p)
z = r cos(o)

where o=theta, p=phi.

  1. To transform from one coordinate system to another, you need what is called a Jacobian. It’s the matrix for rewriting the unit vectors i,j,k as a combination of (e.g.) spherical unit vectors r, theta, phi, based on the partial derivatives.

  2. For spherical coordinates, the differential of some vector l can be written as

dl = dr R + r do O + r*sin(o) dp P

where r=r, o=theta, p=phi and R,O,P are the corresponding spherical unit vectors.

The transformation of the unit vectors is:

X = dx/dr R + (1/r)dx/do O + (1/r*sin(o))dx/dp P
= sin(o)cos(p) R + cos(o)cos(p) O - sin(p) P

Y = dy/dr R + (1/r)dy/do O + (1/r*sin(o))dy/dp P
= sin(o)sin(p) R + cos(o)sin(p) O + cos(p) P

Z = dz/dr R + (1/r)dz/do O + (1/r*sin(o))dz/dp P
= cos(o) R - sin(o) O

So, the vector A can now be written in a spherical basis:

A = y^2X + xzY + 4Z (y^2 = 1, xz = 2)
= X + 2Y + 4Z
= [sin(o)cos(p) + 2sin(o)sin(p) + 4cos(o)]R + [cos(o)cos(p) + 2cos(o)sin(p) -4sin(o)]O + [-sin(p)+2cos(p)]*P

Calculating the value at (1,-1,2):

r = sqrt( x^2 + y^2 + z^2 ) = sqrt(6)
cos(o) = z/r = 2/sqrt(6)
sin(o) = sqrt(x^2+y^2)/r = 1/sqrt(3)
cos(p) = x/(rsin(o)) = 1/sqrt(2)
sin(p) = y/(rsin(o)) = -1/sqrt(2)

Hence,

A = [1/sqrt(6) - 2/sqrt(6) + 8/sqrt(6)]R + [2/sqrt(12) - 4/sqrt(12) - 4/sqrt(3)]O + [1/sqrt(2)+2/sqrt(2)]P
= 2.858R - 2.887O + 2.121P

I hope that was clear. (Probably not, but at least you have the formulas now…) :slight_smile:

7

Yoinks! Thanks for typing that all out for me Giraffe.

I had those same formulas and came up with the same result but thought the spherical coordinates were peculiarly long. Guess they were right. I was having a hard time evaluating it for some reason as well.

GQ to the rescue! Thanks Giraffe