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>.