please help me to do the part (c), thanks~ Consider the vector field F(x,y) Writ
ID: 2847016 • Letter: P
Question
please help me to do the part (c), thanks~
Consider the vector field F(x,y) Write F(x, y) = (P, Q). Show that Evaluate and where C1 and C2 are. respectively, the upper half and the lower half of the circle x2 + y2 = 1 from (1, 0) to (-1, 0). Is F independent of path? A simply-connected region does not contain any hole and it cannot consist of two separate pieces. Explain why your result in part does NOT contradict the following theorem: (Theorem) Let F = (P(x, y), Q(x, y)) be a vector field on an open simply connected-region D. If P and Q have continuous first-order derivatives and = throughout the region D. then F is conservative.Explanation / Answer
If the theorem worked, we would have that F is conservative.
If F is conservative, any curves that have starting and ending point would have the same result for the curve integral. However, in your case I suppose in part b)
you got two different results. Why?
It's because the partial derivatives are not continuous at point (0,0)
dQ/dx= dP/dy=(y^2-x^2)/(x^2+y^2)^2
In fact dQ/dx and dP/dy are not even defined at (0,0)
We cannot eliminate (0,0) or make a hole around it because the region wouldn't be simply connected anymore
Also, since the region includes the upper and lower semicircle, it includes the circle, so the region is enclosing (0,0)
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