How do you do a cross product in spherical coordinates?

I am doing E&M and need to do a cross product in spherical. I am pretty sure I can’t just use a straight determinant like in rectangular coordinates, but I can’t remember what changes in a spherical determinant. I can’t find the definition of it in my books or online. Appreciate the help.

It is a bit different, though the same basic concept. Let’s say you have two coordinates, (A, B, C) and (X, Y, Z). Your cross product will be

(BZ – CY, XC – ZA, AY – BX).

Give me a second to type how to remember it:

First, position the coordinates above each other visually, with the first one on top (yes, order does matter):

(A, B, C)
(X, Y, Z)

For the first term, cover up the first values:

( , B, C)
( , Y, Z)

and then multiply the B times Z, subtracting from that C times Y (BZ – CY).

For the second term, cover the middle values:

(A, , C)
(X, , Z)

And multiply and subtract again, but this time switch directions. Start with the first value in the second coordinate (XC – ZA)

For the third term, cover the last values:

(A, B, )
(X, Y, )

and then multiply and subtract, starting with the value in the first coordinate again (AY – BX).

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February 18 2010 06:18 pm | Cross

One Response to “How do you do a cross product in spherical coordinates?”

  1. Mr. Smith on 18 Feb 2010 at 11:39 pm #

    It is a bit different, though the same basic concept. Let’s say you have two coordinates, (A, B, C) and (X, Y, Z). Your cross product will be

    (BZ – CY, XC – ZA, AY – BX).

    Give me a second to type how to remember it:

    First, position the coordinates above each other visually, with the first one on top (yes, order does matter):

    (A, B, C)
    (X, Y, Z)

    For the first term, cover up the first values:

    ( , B, C)
    ( , Y, Z)

    and then multiply the B times Z, subtracting from that C times Y (BZ – CY).

    For the second term, cover the middle values:

    (A, , C)
    (X, , Z)

    And multiply and subtract again, but this time switch directions. Start with the first value in the second coordinate (XC – ZA)

    For the third term, cover the last values:

    (A, B, )
    (X, Y, )

    and then multiply and subtract, starting with the value in the first coordinate again (AY – BX).
    References :

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