1 #complex/qual/work
Is the following function continuous, differentiable, continuously differentiable? f:R2→Rf(x,y)={xy√x2+y2(x,y)≠(0,0)0else.
? #complex/qual/work
Show that f(z)=z2 is uniformly continuous in any open disk |z|<R, where R>0 is fixed, but it is not uniformly continuous on C.
6 #complex/qual/work
Let F:R2→R be continuously differentiable with F(0,0)=0 and ‖∇F(0,0)‖<1.
Prove that there is some real number r>0 such that |F(x,y)|<r whenever ‖(x,y)‖<r.
2 Multivariable derivatives #complex/qual/work
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Complete this definition: “f:Rn→Rm is real-differentiable a point p∈Rn iff there exists a linear transformation…”
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#complex/qual/work Give an example of a function f:R2→R whose first-order partial derivatives exist everywhere but f is not differentiable at (0,0).
- #complex/qual/work Give an example of a function f:R2→R which is real-differentiable everywhere but nowhere complex-differentiable.
Implicit/Inverse Function Theorems
3 #complex/qual/work
Let f:R2→R.
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#complex/qual/work Define in terms of linear transformations what it means for f to be differentiable at a point (a,b)∈R2.
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#complex/qual/work State a version of the inverse function theorem in this setting.
- #complex/qual/work Identify R2 with C and give a necessary and sufficient condition for a real-differentiable function at (a,b) to be complex differentiable at the point a+ib.
5 #complex/qual/work
Let P=(1,3)∈R2 and define f(s,t):=ps3−6st+t2.
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State the conclusion of the implicit function theorem concerning f(s,t)=0 when f is considered a function R2→R.
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State the above conclusion when f is considered a function C2→C.
- Use the implicit function theorem for a function R×R2→R2 to prove (b).
There are various approaches: using the definition of the complex derivative, the Cauchy-Riemann equations, considering total derivatives, etc.
7 #complex/qual/work
State the most general version of the implicit function theorem for real functions and outline how it can be proved using the inverse function theorem.
Complex Differentiability
4 #complex/qual/work
Let f=u+iv be complex-differentiable with continuous partial derivatives at a point z=reiθ with r≠0. Show that ∂u∂r=1r∂v∂θ∂v∂r=−1r∂u∂θ.
Tie’s Extra Questions: Fall 2016
Let u(x,y) be harmonic and have continuous partial derivatives of order three in an open disc of radius R>0.
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Let two points (a,b),(x,y) in this disk be given. Show that the following integral is independent of the path in this disk joining these points: v(x,y)=∫x,ya,b(−∂u∂ydx+∂u∂xdy).
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In parts:
- Prove that u(x,y)+iv(x,y) is an analytic function in this disc.
- Prove that v(x,y) is harmonic in this disc.
Tie’s Questions, Spring 2014: Polar Cauchy-Riemann #complex/qual/work
Let f=u+iv be differentiable (i.e. f′(z) exists) with continuous partial derivatives at a point z=reiθ, r≠0. Show that ∂u∂r=1r∂v∂θ,∂v∂r=−1r∂u∂θ.
? #complex/qual/work
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Show that the function u=u(x,y) given by u(x,y)=eny−e−ny2n2sinnxfor n∈N is the solution on D={(x,y) |x2+y2<1} of the Cauchy problem for the Laplace equation ∂2u∂x2+∂2u∂y2=0,u(x,0)=0,∂u∂y(x,0)=sinnxn.
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Show that there exist points (x,y)∈D such that lim sup.