A problem in Elementary Differential Geometry

[jdinatale]

My teacher has defined $U_1 = \langle1, 0, 0\rangle$, $U_2 = \langle0, 1, 0\rangle$, and $U_3 = \langle0, 0, 1\rangle$.

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

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

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

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 $U_1 = \langle1, 0, 0\rangle$, $U_2 = \langle0, 1, 0\rangle$, and $U_3 = \langle0, 0, 1\rangle$.

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

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

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

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>.