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 = 𝑓𝑥 𝑑𝑥 + 𝑓𝑦 𝑑𝑦 + 𝑓𝑧 𝑑𝑧 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 = 𝑑𝑓 𝑑𝑥 𝑑𝑥 + 𝑑𝑓 𝑑𝑦 𝑑𝑦 + 𝑑𝑓 𝑑𝑧 𝑑𝑧 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. 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≈𝑓𝑥 Δ𝑥 + 𝑓𝑦 Δ𝑦 + 𝑓𝑧Δ𝑧 However, this one appears to be equal to that one, although the two values are actually not exactly the same.