代做21-259: Calculus in Three Dimensions Lecture #7 Spring 2025代做Prolog

21-259: Calculus in Three Dimensions

Lecture #7

Spring 2025

Functions of Several Variables

Definition: Let D be a subset of R n for some positive integer n. A function f of the n variables x1,x2,...,xn is a rule that assigns to each n-tuple (x1,x2,...,xn) in D a value f (x1,x2,...,xn) in R. The set D is the domain of f and the set of all outputs is the range of f .

While functions can be described for inputs of size any positive integer n, the most commonly discussed multivariate functions are functions of two variables z = f (x, y), three variables w = f (x, y, z), and four variables w = f (t,x, y, z) (the latter being the case where the output value w depends on the position (x, y, z) of a point in 3-D space and time t).

Example 1. Sketch the domain and evaluate f (3,2) of the functions and f(x, y) = x ln(y2 − x).

Example 2. Let Find the domain of f and evaluate f (4,0,2). Can you describe the range of f?

Definition: The graph of a multivariate function f : R n →R is the set of all points (x1,x2,...,xn, f (x1,x2,...,xn)) in Rn+1.

Graphs of functions of two variables can be visualized as surfaces in R 3 . However, we have some trouble visualizing functions of three or more variables. One approach for plotting a function f (x, y, z) is to fill the domain with colors at every point, each point representing a different color that represents the output of the function.

Definition: The level curves of a function f of two variables are curves with the equations f (x, y) = k, where k is a constant in the range of f . The level surfaces of a function f of three variables are surfaces with the equations f (x, y, z) = k where k is in the range of f.

Example 3. Match the following functions to their graphs:

(a) f (x, y) = |x| +|y|

(b) f (x, y) = |x y|

(c) f (x, y) = 1+ x2 + y2/1

(d) f (x, y) = (x2 − y2)2

(e) f (x, y) = (x − y)2

(f) f (x, y) = sin(|x| +|y|)

Limits and Continuity

Definition: Let f be a function of two variables whose domain D includes points arbitrarily close to (a,b). Then we say that the limit of f (x, y) as (x, y) approaches (a,b) is L and we write

if for every number ε > 0 there is a corresponding δ > 0 such that if (x, y) ∈ D and then |f (x, y)−L| < ε.

This definition means that if any small interval (L −ε,L +ε) is given around L, then we can find a disk Dδ with center (a,b) and radius δ > 0 such that f maps all the points in Dδ (except possibly (a,b)) into the interval (L −ε,L +ε).

For functions of a single variable, when we let x approach a, there are only two possible directions of approach - from the left or from the right. However, for functions of two or more variables, the situation is not as simple as because we can let (x, y) approach (a,b) from an infinite number of directions in any manner whatsoever, as long as (x, y) stays in the domain of f .

This means that, in general, it is a very difficult problem to prove that a limit of a function of several variables exists, as we must consider all possible trajectories towards the point (a,b). This also means that if we approach (a,b) along multiple paths and get different limit values, then the limit itself cannot exist.

If f (x, y)→L1 as (x, y)→(a,b) along path C1 and f (x, y)→L2 as (x, y)→(a,b) along path C2 where L1 ≠ L2, then lim(x,y)→(a,b) f (x, y) does not exist.

Example 4. Show that the limit does not exist.

Example 5. Show that the limit does not exist.

Definition: A function f of two variables is called continuous at(a,b) if f (x, y) = f (a,b).

We say f is continuous on D if f is continuous at every point (a,b) ∈ D.

This means that the surface of f (x, y), where f is continuous on D, has no hole or break for any (x, y) ∈ D. We can also apply our standard limit laws to immediately show that sums, differences, products, and quotients of continuous functions are also continuous wherever they are defined.

Example 6. Find