Multivariable Functions

Question 1

What’s a level curve?

Answer 1

It’s a family of curves that are gotten as the intersection between the planes z = k and the surface z = f(x, y)

Question 2

Limits along paths or trajectories, what are they?

Answer 2

They are curves that are continuous and differentiable that pass through the point of tendency

Question 3

If two trajectories have different limits, what happens?

Answer 3

Then the actual limit does not exist

Question 4

If a limit at a point exists, then what happens with any trajectory?

Answer 4

They all exist and give you the same limit

Question 5

What happens if a trajectory does not exist for a given limit?

Answer 5

Then, the limit does not exist either.

Question 6

What’s an open ball around a point xo?

Answer 6

An open ball B(xo, r) is the set of all x in R^{n} such that:

|x - xo| < r

Question 7

What’s an interior point of a set U?

Answer 7

It’s a point x in such a set such that there is an open ball around the point that is contained in the set U.

Question 8

When is a set said to be an open set?

Answer 8

When all of its points are interior points.