I am confused about how multivariable calc works

[Calpalned]

Homework Statement

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.

Homework Equations

N/A

The Attempt at a Solution

I'm not sure how to start. Thank you all!

 

[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:

http://mathworld.wolfram.com/Gradient.html

 

[HallsofIvy]

I would not use "eigen" here. The vectors i, j, and k are the unit vectors pointing in the directions of the x, y, and z axes, respectively. The gradient of a scalar valued function points in the direction of fastest increase and its length is the rate of change in that direction.