# A problem in Elementary Differential Geometry

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.

## Answers and Replies

Mark44
Mentor

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