I am confused about how multivariable calc works

1. Feb 19, 2015

Calpalned

1. The problem statement, all variables and given/known data
My teacher introduced the third dimension ($R^3$) and higher dimensions to my class using vectors. Later on, my teacher introduced functions of two or more variables and now there's no mention of vectors. I am confused as to how vectors (i + j + k) and functions of two or more variables f(x, y, z) are related.

2. Relevant equations
N/A

3. The attempt at a solution
I'm not sure how to start. Thank you all!

2. Feb 19, 2015

Pythagorean

Typical convention is that i is the eigenbasis of x, j is the eigenbasis of y, and k is the eigenbasis of z.

Other than that, it depends on what you're doing. For instance, taking the gradient requires taking the partial of your function, f(x,y,z) with respect to each variable, and multiplying each of those by their respective eigenvector as below: