Evaluating a $1$-form on a vector field

Question

I have a homework question that says to evaluate the $1$-form $\phi=x^2 dx - y^2 dz$ on the vector field $V= x U_1 + y U_2 + z U_3$. How do I go about solving this? I've tried reading the book but I can't make any sense of it.

Answer 1

Assuming that $\require{cancel}\{U_1,U_2,U_3\}$ is the natural frame in $\Bbb R^3$, just remember that ${\rm d}x(U_1) = 1$, ${\rm d}x(U_2) = {\rm d}x(U_3) = 0$, and similarly for ${\rm d}z$. Use linearity: $$\phi(V) = x^2{\rm d}x(V) - y^2{\rm d}z(V),$$but for example, $${\rm d}x(V) = x\,\cancelto{1}{{\rm d}x(U_1)} + y\,\cancelto{0}{{\rm d}x(U_2)} + z\,\cancelto{0}{{\rm d}x(U_3)} = x.$$Now you finish it!