In other words; Why can't I integrate the whole equation in one go like this?
$$\begin{align}f=\int df&=\int yz\,dx +\int xz\,dy+\int xy\,dz+\int a\,dz\\&=xyz+xyz+xyz+az+C\\&=3xyz + az +C\end{align}$$
This is strangely remarkably close to the correct answer, which is
$$f=xyz+az+C$$
I know that the differential
$$df=yz\,dx+xz\,dy+(xy+a)\,dz\tag{a}$$
can be written as
$$df=\frac{\partial f}{\partial x}\,dx+\frac{\partial f}{\partial y}\,dy+\frac{\partial f}{\partial z}\,dz\tag{b}$$
Matching equations $(\mathrm{a})$ and $(\mathrm{b})$ leads to $3$ more equations, namely:
$$\frac{\partial f}{\partial x}=yz\tag{1}$$
$$\frac{\partial f}{\partial y}=xz\tag{2}$$
$$\frac{\partial f}{\partial z}=xy+a\tag{3}$$
Now integrating $(1)$, $(2)$, and $(3)$ with respect to their partial derivatives
$$f=xyz + P\quad\text{from} \quad(1)\tag{A}$$
$$f=xyz + Q\quad\text{from} \quad(2)\tag{B}$$
$$f=xyz + az+R\quad\text{from} \quad(3)\tag{C}$$
where $\mathrm{P}$, $\mathrm{Q}$ and $\mathrm{R}$ are constants of integration.
Now 'somehow' we decide that equation $(\mathrm{C})$ best describes the parent function $f$ and is therefore the function we desire:
$$f=xyz + az+C$$
with $R$ replacing $C$, since they are both constants.
So apart from the obvious "because it gives the correct answer", my question is as follows: Why do we have to integrate each term separately (independently) instead of the method I used at the beginning of this question (integrating the whole equation in one go)?
Also; What is the precise logic behind choosing $(\mathrm{C})$ to represent $f$ instead of $(\mathrm{A})$ or $(\mathrm{B})$?
Many thanks.
Best Answer
Observe that $P,Q,R$ are not mere constants but functions themselves as well, i.e. $P=P(y,z)$ and $Q=Q(x,z)$ and $R=R(x,y)$. This is because $x,y,z$ are not at all independent but are functionally dependent by some relation. So your method of integrating all together does not work. Note that when you are integrating by your method, you are considering $2$ variables independent of the integrating variable.
And if you use the 3rd equation, it most aptly describes the function from among the other options. And it can be realized if you write $P,Q,R$ as $P=P(y,z)$ and $Q=Q(x,z)$ and $R=R(x,y)$.