A problem in Elementary Differential Geometry

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My teacher has defined [itex]U_1 = \langle1, 0, 0\rangle[/itex], [itex]U_2 = \langle0, 1, 0\rangle[/itex], and [itex]U_3 = \langle0, 0, 1\rangle[/itex].

So it seems like the function maps [itex]L(\langle1, 0, 0\rangle, \langle0, 1, 0\rangle) = a, L(\langle1, 0, 0\rangle, \langle0, 0, 1\rangle) = b,[/itex], and [itex]L(\langle0, 1, 0\rangle, \langle0, 0, 1\rangle) = c[/itex]

I'm not sure how that helps me determine what [itex]L(\langle v_1, v_2, v_3\rangle, \langle w_1, w_2, w_3\rangle)[/itex] is.

Could the function perhaps be [itex]L(u, v) = L(\langle v_1, v_2, v_3\rangle, \langle w_1, w_2, w_3\rangle) = a\cdot{}v_1 + b\cdot{}w_3 + c\cdot{}v_2[/itex]

That seems to satisfy our initial conditions.

My textbook doesn't cover this and my teacher hasn't shown an example of how these functions work, so I'm unsure what to do.
 

Answers and Replies

  • #2
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Joseph.png



My teacher has defined [itex]U_1 = \langle1, 0, 0\rangle[/itex], [itex]U_2 = \langle0, 1, 0\rangle[/itex], and [itex]U_3 = \langle0, 0, 1\rangle[/itex].

So it seems like the function maps [itex]L(\langle1, 0, 0\rangle, \langle0, 1, 0\rangle) = a, L(\langle1, 0, 0\rangle, \langle0, 0, 1\rangle) = b,[/itex], and [itex]L(\langle0, 1, 0\rangle, \langle0, 0, 1\rangle) = c[/itex]

I'm not sure how that helps me determine what [itex]L(\langle v_1, v_2, v_3\rangle, \langle w_1, w_2, w_3\rangle)[/itex] is.

Could the function perhaps be [itex]L(u, v) = L(\langle v_1, v_2, v_3\rangle, \langle w_1, w_2, w_3\rangle) = a\cdot{}v_1 + b\cdot{}w_3 + c\cdot{}v_2[/itex]

That seems to satisfy our initial conditions.

My textbook doesn't cover this and my teacher hasn't shown an example of how these functions work, so I'm unsure what to do.

I would start with the definition of "skew-symmetric multilinear function".

The next thing to do would be to note that v = <v1, v2, v3> = v1U1 + v2U2 + v3U3. You can write w in a similar fashion, as a linear combination of U1, U2, and U3.

That's a start...
 
  • #3
Dick
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Write v=v1*U1+v2*U2+v3+U3 and similar for w. Use linearity to factor out the vi's and wi's and skew-symmetry to change, for example, <U2,U1> to -<U1,U2>.
 

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