Let $z=f(x,y)$ be a neat function of two variables $x$ and $y$. We consider a point $P=(a,b,c)$ with $c=f(a,b)$. We know that the vectors $$\mathbf{r}=\left(\begin{array}{c} 1 \\ 0 \\ f_x(a,b)\end{array}\right)\quad \text{and}\quad \mathbf{s}=\left(\begin{array}{c} 0 \\ 1 \\ f_y(a,b)\end{array}\right)$$ span the tangent plane. An arbitrary tangent vector $\mathbf t$ at a point $P$ on the graph of $f(x,y)$ is a linear combination of these vectors, say of the form $$\begin{aligned}\mathbf{t} &= \mathbf{r}\,\dd x+\mathbf{s}\,\dd y\\ \\ &=\left(\begin{array}{c} \dd x \\ 0 \\ f_x(a,b)\,\dd x\end{array}\right)+\left(\begin{array}{c} 0 \\ \dd y \\ f_y(a,b)\,\dd y\end{array}\right)\\ \\ &=\left(\begin{array}{c} \dd x \\ \dd y \\ f_x(a,b)\,\dd x+f_y(a,b)\,\dd y\end{array}\right)\end{aligned}$$

The first component, $\dd x$, is the increase of the $x$ coordinate along $\mathbf{t}$, the second component, $\dd y$, is the increase $y$ coordinate along $\mathbf{t}$, and the third component is the increase in the $z$ coordinate along $\mathbf{t}$. For very small $\dd x$ and $\dd y$, the tangent plane at $P$ practically coincides with the graph of $f(x,y)$ near that point, and the increase in $z$ thus virtually coincides with the increase in $f(x,y)$. This increase of $f(x,y)$, denoted as ${\vartriangle}f$, is given by $${\vartriangle}f=f(a+\dd x,b+\dd y)-f(a,b)$$ and for very small $\dd x$ and $\dd y$ almost equal to $$f_x(a,b)\,\dd x+f_y(a,b)\,\dd y$$ The last expression is called the total differential of $f(x,y)$ at the point $P=(a,b)$. The notation for the total differential is $\dd\bigl(f(a,b)\bigr)$, but as with functions of one variable, we leave in the notation out the dependency of the differential of the chosen point $P$ and we write the following:

In the figure above, the function is once more $f(x,y)=\tfrac{1}{2}\bigl(1-\sin(2x^2-y-1)\bigr)$ and we have selected the point $P=(0.3,0.5)$. For the clarity of the figure, the increase $\dd x=0.2, \dd y=0.3$ have not been chosen too small and the deviation between the tangent plane and the graph of $f$ is not small: the difference between the total differential $\dd f$ at $P$ and the function value difference ${\vartriangle}f$ is about $0.01$. But when we choose our $\dd x$ and $\dd y$ ten times smaller ( $\dd x=0.02$ and $\dd y=0.03$ ), then the total differential $\dd f$ at $P$ equals $$\dd f=f_x(0.3,0.5)\,\dd x+ f_y(0.3,0.5)\,\dd y\approx -0.14891\times 0.02 + -0.12409\times 0.03\approx 0.00074$$ while the function value difference ${\vartriangle}f$ in this case equals $${\vartriangle}f= f(0.32, 0.53)-f(0.3,0.5)\approx 0.00064$$ The difference between $df$ and ${\vartriangle}f$ is only $0.0001$. When we make $\dd x$ and $\dd y$ another ten times smaller, the difference is about $0.000001$!

The total differential can be defined in a similar way for functions of three or more variables; for instance