Multivariable Vector Calculus

Question

I am just starting my introduction to vector calculus and it is very confusing to me. I was watching a Khan Academy video and he found the vector of a multivariable function [f(x,y)]. For this example, the function was

f(x,y) = [y^3 - 9y , x^3 - 9x]

He plugged in the point (1,2) and got [-10, -8]. Where he confused me was drawing the vector line. From the point (1,2), he went 10 units to the left and 8 units down. However, he got the -10 from the y-input and the -8 from the x-input. Why did the y-input become the x-output and the x-input the y-output?

Also, if there are any other study materials for vector calculus you would recommend, those would be greatly appreciated! Thank you for the help!

Answer 1

Consider an easier example, to see better what's going on:

f(x,y) = (y,x) Plug in the point (1,2) to get (2,1). the x and y values are switched. It's just the way the function treats them. the f(x,y) function could have been (y,y) or (x,x) or any combinations of x and y.

It's sort of similar to finding the inverse function. Take f(x) = y=x² Switch the x and y's to get the inverse function x=y² or f^-1(x) = y=x^1/2