110年 - 110 國立清華大學碩士班考試入學試題_數學系:高等微積分#105486

1. (10%) Calculate

2. (10%) Calculate the line integral
where C is the positive oriented curve in R²-{(0,0)} parametrized by

 3. (10%) Determine if converges uniformly on [-π,π]. 

4. (10%) Show that the function f : R² → IR² defned by has a fixed point.

5.(10%) Show that the set is compact.

6. (10%) Let X be a topological space andbe compact subsets. Is A U B always a compact set? Prove or give a counterexample.

7. (10%) Let f :R³ →R be a function defined by
for(x，y，z) . Can f be extended to a differentiable function on R³?

(a) (10%) Is the family equicotinuous on [0,π ]?

(b) (10%) Does the sequence have a convergent subsequence?

(c) (10%) Is the familycompact?