Boundary of the sphere (Differential forms)

Question

I have this problem from an example in "D.Bachman,A Geometric Approach to Differential Forms".
Suppose:
$$\phi_1(r,\theta)=(r\cos 2\pi\theta,r\sin2\pi\theta,\sqrt{1-r^2}) $$ $$\phi_2(r,\theta)=(-r\cos 2\pi\theta,r\sin2\pi\theta,-\sqrt{1-r^2})$$ Let $\sigma_1=Im(\phi_1)$, $\sigma_2=Im(\phi_2)$, then $\sigma=\sigma_1+\sigma_2$ is a unit shpere in $\mathbb{R}^3$.
Show that: $\partial\sigma=\emptyset$
My solution is:
First we have: \begin{align*} \partial\sigma_1&=(\phi_1(1,\theta)-\phi_1(0,\theta)-(\phi_1(r,1)-\phi_1(r,0) ) \\ &= (\cos2\pi\theta,\sin2\pi\theta,0)-(0,0,1)-(r,0,\sqrt{1-r^2})+(r,0,\sqrt{1-r^2})\\ &=(\cos2\pi\theta,\sin2\pi\theta,0)-(0,0,1) \end{align*} And: \begin{align*} \partial\sigma_2&=(\phi_2(1,\theta)-\phi_2(0,\theta)-(\phi_2(r,1)-\phi_2(r,0) ) \\ &= (-cos2\pi\theta,\sin 2\pi\theta,0)-(0,0,-1)-(r,0,-\sqrt{1-r^2})+(r,0,-\sqrt{1-r^2})\\ &=(-\cos2\pi\theta,\sin2\pi\theta,0)-(0,0,-1) \end{align*} Therefore: \begin{align*} \partial\sigma&=\partial \sigma_1+\partial \sigma_2\\ &=(\cos2\pi\theta,\sin 2\pi\theta,0)-(0,0,1)+(-\cos2\pi\theta,\sin 2\pi\theta,0)-(0,0,-1) \end{align*} I don't see how can $\partial \sigma=\emptyset$.
But when I use a parameterization of $\sigma$ (a unit shpere), that is: $$\phi(\varphi,\theta)=(\cos(2\pi \varphi) \cos(\pi\theta),\sin(2\pi \varphi) \cos(\pi\theta),\sin (\pi\theta))$$ It can easily lead straight to $\partial\sigma=\emptyset$
I am really confused about this.

Here are some previous definitions from the book:
Definition 1: Let $I=[0,1]$, a n-cell,$\sigma$, is the image of a differentable map, $\phi:I^n\to \mathbb{R}^m$, with a specified orientation. We denote the same cell opposite orientation as $-\sigma$. We define a $0$-cell to be a oriented point of $\mathbb{R}^m$
Definition 2: A $n$-chain is a formal linear combination of $n$-cells, and we assume the following relation holds:
$$\sigma-\sigma=\emptyset$$ $$\sigma+\tau=\tau+\sigma$$ $$(m+n)\sigma=m\sigma+n\sigma$$ Definition 3: Let a $n$-cell $\sigma$ is the image of the parameterization $\varphi:I^n\to\mathbb{R}^m$ with the induced orientation, we define the boudary of $\sigma$ (denoted as $\partial \sigma$) as:
$$\partial \sigma=\displaystyle \sum_{i=1}^n (-1)^{i+1}(\phi|_{(x_1,...,x_{i-1},1,x_{i+1},...x_n)}-\phi|_{(x_1,...,x_{i-1},0,x_{i+1},...x_n)})$$ Definition 4: Let $C=\sum n_i\sigma_i$ is a $n$-chain, we define the boundary of $C$ as: $$\partial C=\sum n_i\partial \sigma_i$$ Notes: The book intially lets both $r,\theta$ on above problem vary from $0$ to $1$.