Differentiation of Multivariate Functions_Chap. 4 Flashcards
What’s a function of two variables?
It’s a function z= f(x,y) where z is the height above the xy plane and (x,y) is the input from a subset D in the 2D space that gets mapped to a unique real number z.
The set D is the domain of the function. The image of f
is the set z.
What’s the level curve of a function of 2 variables?
It’s a set of points satisfying the equation f(x, y) = c when you’re given a function f(x, y) and a number c, which will be in the image of f.
They’re always graphed in the xy plane.
Ex: the contour lines of a topographical map.
What’s a vertical trace?
It’s the set of points that solves the function z=f(x,y) when you set either x or y (not both) equal to some constant.
What’s a level surface of a function with 3 variables?
It’s the set of points satisfying f(x, y, z) = c where c is in the range of f.
What’s a sphere?
A delta-ball of the point P naught in 2d space is the sets {(x, y) : square root[ (x- x0)^2 + (y- y0)^2 ] < delta }
What’s an interior point?
The point P naught is an interior point of S, which is some subset of 2d space, if there exists a radius delta > 0 such that the delta-ball of P naught is entirely within S.
What’s a boundary point?
The point P naught is a boundary point of S, which is some subset of 2d space, if for all delta-balls of P, there’s a point inside of S and outside of S.
What does it mean for S, which is a subset of 2d space, to be open?
It means every point of S is an interior point.
What does it mean for S, which is a subset of 2d space, to be closed?
It means S contains all of its boundary points.
What does it mean for S, which is a subset of 2d space, to be connected?
S is connected if it is not a union of non-intersecting open sets.
What’s a region?
A region is any open, connected, non-empty subset of 2d space.
