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I think I understand now the intuitive reasoning behind the total derivative of a multivariate function $z = z(x, y)$, which is

$$ dz = \frac{\partial{z}}{\partial{x}}dx + \frac{\partial{z}}{\partial{y}}dy $$

So let's take an example, $z = x^2 + y^2$, a paraboloid and a surface of revolution. Here $z_x = 2x$ and $z_y = 2y$, so

$$ dz = 2xdx + 2ydy. $$ How would I evaluate this numerically? For example, if I had a program that calculated the infinitesimal change $dz$ at a point $\langle x_0, y_0 \rangle$, I could plug in the values of $x_0$ and $y_0$ for $x$ and $y$ respectively, but what would I plug in for $dx$ and $dy$?

EDIT: My guess is that you actually can't evaluate it directly as a number, but instead have to use it as a combination of infinitesimals, kind of like how you can't treat problems involving the imaginary unit $i$ as all real numbers. So instead of treating $dz$ as some actual value, you use it to derive other values that are useful. That's my guess but I'm not sure if it's valid or not.

user3002473
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The differential is not a numerical value, so it's meaningless to associative a number to a differential. But it indeed has a representation using a set of basis, in which the differential is regarded as some kind of vector. Say instead we use binary tuple $(2x,2y)$ to represent differential $dz$. In practice, we also don't care about $dz$. What we concern is how it applies to a vector $w=(u,v)$. So now we can give $dz(w)$ a numerical value by $2(ux+vy)$.

Shuchang
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  • Hmm, I think I see. So my initial guess was correct? – user3002473 Aug 27 '14 at 23:56
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    @user3002473 Yes. Nice guess. – Shuchang Aug 27 '14 at 23:58
  • Awesome, thanks! One last thing, what kind of problems would you suggest I do to get a better intuitive understanding of the way differentials behave? – user3002473 Aug 27 '14 at 23:59
  • @user3002473 Get some knowledge on differential form. You will know differential forms, like tangent vectors, form a vector space and the space is dual to tangent space, in which view, differential form has an intuitive explanation, regarded as, say in $\mathbb R^3$ area pieces orthogonal to vectors. – Shuchang Aug 28 '14 at 00:07