What is expand-(u-zu)(zdu+udz) ?

The solution to expand-(u-zu)(zdu+udz) is -uzdu-u^2dz+uz^2du+u^2zdz

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expand-(u-zu)(zdu+udz)

expand

- (u - z u) (z d u + u d z)

- u z d u - u^{2} d z + u z^{2} d u + u^{2} z d z

Solution steps

- (u - z u) (z d u + u d z)

= - (u - u z) (z d u + u d z)

Expand

(u - z u) (z d u + u d z) : u z d u + u^{2} d z - u z^{2} d u - u^{2} z d z

= - (u z d u + u^{2} d z - u z^{2} d u - u^{2} z d z)

Distribute parentheses

= - (u z d u) - (u^{2} d z) - (- u z^{2} d u) - (- u^{2} z d z)

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - u z d u - u^{2} d z + u z^{2} d u + u^{2} z d z

(2 - i)^{7}

(x y^{2} - y^{2} + x - 1) d x + (x^{2} y - 2 x y + x^{2} + 2 y - 2 x + 2) d y

(- 1 + 2 i)^{15}

12 (1 - e^{- \frac{t}{τ}})

3 (3 - 3)

What is expand-(u-zu)(zdu+udz) ?
The solution to expand-(u-zu)(zdu+udz) is -uzdu-u^2dz+uz^2du+u^2zdz