Description
This is a course in calculus in several variables. The following is core material: Basic topology of n-dimensional Euclidian space; limits and continuity of functions; the derivative as a linear transformation; Taylor's formula with remainder; the Inverse and Implicit Function Theorems, change of coordinates; integration (including transformation of integrals under changes of coordinates); Green's Theorem.
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QUIZ 5. ADVANCED MULTIVARIABLE CALCULUS,
SPRING 2021
1. (5 pts) Let n ≥ 3, and, for every f ∈ C 2 , f : Rn → R, let
n
X
∂ 2f
∆f =
∂x2k
k=1
be the Laplace operator. Find α 6= 0 so that
∆ (kxkα ) = 0
everywhere outside of the origin in Rn . Here kxk is the Euclidean norm
in Rn .
2. (5 pts) Suppose that a function f : B1 (0) → R has continuous
bounded partial derivatives on the unit Euclidean ball B1 (0) in Rn .
Prove that f is uniformly continuous on B1 (0).
3. (5 pts) Compute partial derivatives of the first order of the function
wz
x
f (x, y, z, w) =
.
y
1
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