Multivariable Functions (Part 1) Flashcards
(30 cards)
What’s a level curve?
It’s a family of curves that are gotten as the intersection between the planes z = k and the surface z = f(x, y)
Limits along paths or trajectories, what are they?
They are curves that are continuous and differentiable that pass through the point of tendency
If two trajectories have different limits, what happens?
Then the actual limit does not exist
If a limit at a point exists, then what happens with any trajectory?
They all exist and give you the same limit
What happens if a trajectory does not exist for a given limit?
Then, the limit does not exist either.
What’s an open ball around a point xo?
An open ball B(xo, r) is the set of all x in R^{n} such that:
|x - xo| < r
What’s an interior point of a set U?
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.
When is a set said to be an open set?
When all of its points are interior points.
What’s a boundary point of a set U?
It’s a point x in R^{n} such that, for all possible values r > 0, we have that B(x,r) has points both in U and its complement.
When is a set U said to be closed?
When the boundary set of U is contained in U.
As well, a set is said to be closed when its complement is open.
What is the formal definition of the limit when x approaches a of a function f(x) = L?
For every epsilon there is a delta such that:
||x - xo|| < delta implies that ||f(x) - L|| < epsilon.
By changing from rectangular (x,y) to polar (r, theta) or to spherical a 2D or 3D limit passes to?
Lim r —> 0+ F(r, theta)
Lim rho —> 0+ F(rho, theta, phi)
Formal definition of continuity of f at a point a?
|x-a|< delta —> |f(x) -f(a)| < epsilon
Let f and g be continuous function, what operations on them satisfy continuity?
f+g, f-g, fg, f/g are all continuous in their domains
f(g(x) or g(f(x)) are continuous as long as they’re defined.
What is the partial derivative with respect to a variable?
It’s the rate of change of that variable when intersecting the surface of f(x,y) with the planes parallel to the coordinates ones.
What do partial derivatives tell you?
How a function f(x,y) changes when one of the variables remains fixed.
Does differentiability assures continuity in multivariable calculus?
Nope, a function can be differentiable and not continuous in multivariable calculus
Geometric interpretation of a partial derivative at a point P?
It will represent the slope of the tangent line of the surface at the point P and in the direction of the variable that’s being derivated.
Vector equations for the two tangent lines in the direction of the coordinate axis?
L1 = (a,b,c) + t(1,0, fx (a,b) )
L2 = (a,b,c) + t(0, 1, fy(a,b) )
What’s the cross partial derivatives theorem?
Fxy = Fyx
When is a function differentiable?
When all of its partial derivatives are continuous.
When is a function f of type C(k)?
When f and all of its partial derivatives until order k are continuous.
What happens if a function is C(1)?
That means that all of its first partial derivatives are continuous and therefore, the function is differentiable
What is the tangent plane of a surface at a point conceptually?
It’s the plane that is tangent to the surface at the point and contains all tangent lines at that point.
