# 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.

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.