A problem in Elementary Differential Geometry

[jdinatale]

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.

 

[Mark44]

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...

 

[Dick]

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>.