Differentials in Multivariable Calculus: Encoding Changes and Approximations

Differential of f(x,y,z)

Let f(x, y, z) be a function of three variables. If we want to find the differential dx, dy,
or dz of f at point (x, y, z), we can input the values for x and y into the equation df =
f(x + dx, y + dy, z + dz).

df = fxdx + fydy + fzdz

You might want to review the other notation, partial f, partial x, dx plus partial f, partial
y, dy plus partial f over partial z, dz.

df =dfdxdx +dfdydy +dfdzdz

This object is called a differential. It is not a number. It is not a vector. It is not a
matrix. Differentials have their own set of rules, and we must learn how to
manipulate them.
First of all, it is important to take into account the difference between delta f (∆f) and
df when thinking about these phenomena.
df is NOT ΔF

So that thing is a number. It's going to be a number once you have a small variation
of x, a small variation of y, and a small variation of z.

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These are numbers. Delta x,
delta y, and delta z are actual numbers. And this becomes a number. In
mathematical equations, all you can do with a differential is express it in terms of
other differentials. So in fact, this dx, dy, and dz are the differentials of x, y, and z. So
in fact, you can think of these differentials as placeholders where you will put other
things. For example, they represent changes in x, y, z and f.

One way to explain it is to say that they represent infinitesimal changes. Another way
to say it is that these things are placeholders for values and tangent approximations.
So, for example, if we replace these symbols with delta x, delta y, and delta z
numbers, then we will actually get a numerical quantity. That quantity is called the
linear approximation or tangent plane approximation for f.
The first thing that differential equations do is encode how changes in x, y, and z
affect the value of f. The most general answer to what a differential equation is—a
relation between x, y, and z and f. In particular—this is a placeholder for small
variations like delta x, delta y, delta z—to get an approximation formula which is
approximately equal to delta f is equal to f sub x delta x plus fy delta y plus fz delta z.

1. Encode how change in x, y, z affect f.

2. Placeholder for small variations Δx, Δy, Δz to get approx. formula

Δf≈fxΔx + fyΔy + fzΔz

However, this one appears to be equal to that one, although the two values are
actually not exactly the same.