My differential geometry course introduced the following formula for computing pullbacks by a $k$-form:
$f^{*}\alpha = \sum (f \circ a_J) df^{j_{1}} \wedge \cdots \wedge df^{j_{l}}$.
My instructor computed the pullback $f^{*} \alpha$, where $f(x_1, x_2, x_3) = (x_1^2 + x_2^2, \sin(x_3))$, and $\alpha = y_1^2 dy_1 + y_2^3 dy_2$. He then got as the answer before simplification:
$f^{*}\alpha = (x_1^2 + x_2^2)^2 d(x_1^2 + x_2^2) + (\sin(x_3))^3 d(\sin(x_3))$.
Could someone please explain how he obtained this result from the general formula written above? I am slightly confused on the indices.
Thanks.