Oriented curves advanced calculus homework solution in 2021
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Can you use green's theorem on the curve C?
Now, since this region has a hole in it we will apparently not be able to use Green’s Theorem on any line integral with the curve C =C1∪C2 C = C 1 ∪ C 2. However, if we cut the disk in half and rename all the various portions of the curves we get the following sketch.
Is the curve C2 C 2 oriented positively?
Notice that both of the curves are oriented positively since the region D D is on the left side as we traverse the curve in the indicated direction. Note as well that the curve C2 C 2 seems to violate the original definition of positive orientation.
When does a curve have a positive orientation?
We originally said that a curve had a positive orientation if it was traversed in a counter-clockwise direction. However, this was only for regions that do not have holes. For the boundary of the hole this definition won’t work and we need to resort to the second definition that we gave above.
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Typical Problems. The Calculus of Variations is concerned with solving Extremal Problems for a Func- tional. That is to say Maximum and Minimum problems for functions whose domain con- tains functions, Y(x) (or Y(x1;¢¢¢x2), or n-tuples of functions).
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