Question

Compute both dip and for: . ф-2,2dz + zydy and ydz + z3dy are one forms on R2. . cosz)dx +sin(z)dy+rydz andzdx +rdy+ydz are one forms on R3

Answer 1

To compute both d psi and psi intersection psi for (i) phi = x^2 dx + x y dy and psi = y dx + x^3 dy are one forms on R^2 Given , phi = x^2 dx = xy dy psi = y dx + r^3 dy phi intersection psi = (c^2 dx + x y dy) intersection (y dx + x^3 dy) = x^2 y dx + x^5 dx dy +x y^2 dy dx + x^4 y dy dy but dx dx = dy dy = 0 and dy dx = negative dx dy so, phi intersection psi = x^5 dx dy - x y^2 dx dy phi intersection psi = (x^5 - x y^2) dx dy the product at two I - forms is a 2 - form Def: If phi = Sigma fid x is a I - form in R^3, then the exterior derivative of phi is a 2 - form d phi = sigma d fi x dx: Given phi