Question on baby Rudin examples 10.12

I have two questions:

$(a)$.

Fix $a > 0$ and $b > 0$ and define $\gamma(t) = (a\cos(t),b\sin(t))$ where $(0\leq t \leq 2\pi)$. So that $\gamma$ is a closed curve in $R^2$. (Its range is an ellipse.)

Then

$$\int_\gamma x dy = \int_0^{2\pi} ab \cos^2(t)dt = \pi ab \tag{1}$$

$(b)$.

Let $D$ be the $3$-call defined by $0 \leq r \leq 1$, $0 \leq \theta \leq \pi$ and $0 \leq \varphi \leq 2\pi$. Define $\phi ( r,\theta,\varphi ) = (x,y,z)$, where $x = r\sin\theta \cos\varphi$, $y = r\sin\theta \sin\varphi$, $z = r\cos\theta$

Then $$ J_\phi (r,\theta,\varphi) = \frac{\partial(x,y,z)}{\partial(r,\theta,\varphi)} = r^2\sin\theta. $$

Hence $$ \int_{\phi} dx \land dy \land dz = \int_D J_\phi\ \color{red}{dr\,d\theta\, d\varphi} = \frac{4\pi}{3}. \tag{2} $$

I don't understand how do we get $(1)$ and $(2)$ .

(in the $(2)$ I also don't understand why is $\int_D J_{\phi}$ equal of $\frac{4\pi}{3}$ ).

Any help would be appreciated.

For (a) the 1-surface $\gamma$ gives the determinants as \begin{align} \frac{\partial\gamma_1(t)}{\partial t} &= \partial(a\cos t)/\partial t = - a \sin t & \frac{\partial\gamma_2(t)}{\partial t} &= \partial(b\sin t)/\partial t = b \cos t \\ \end{align} The 1-form we have is $\omega = x dy$, i.e. \begin{align} a_1(\gamma(t)) &= 0 & a_2(\gamma(t)) &=x(t) = \gamma_1(t) = a \cos(t) \end{align} Thus $$ \omega(\gamma) = \int_\gamma \omega = \int_{[0,2\pi]}(a_1(\gamma(t))\frac{\partial\gamma_1(t)}{\partial t} + a_2(\gamma(t))\frac{\partial\gamma_2(t)}{\partial t}) dt = \dots $$

For (b) we have $\omega = dx \land dy \land dz$ and so $a(x,y,z) = 1$ $$ \int_{\phi} dx \land dy \land dz = \int_D J_\phi\ \color{red}{dr\,d\theta\, d\varphi} = \int_{r = 0}^1 \int_{\theta = 0}^\pi \int_{\varphi= 0}^{2\pi} r^2\sin\theta dr\,d\theta\, d\varphi \\ = \dots $$ For your other question look at the Appendix in this answer to learn how you get the $$\frac{\partial (x_{i_1},\dots,x_{i_k})}{\partial (u_1,\dots, u_k)}$$ from $\Phi$.